Assignment 3

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import pymc as pm

import numpy as np
import pandas as pd
from sklearn.datasets import fetch_california_housing
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression #for standard linear regression
from sklearn.metrics import mean_squared_error
from sklearn.preprocessing import StandardScaler

California Housing Dataset

# Load California housing data
housing = fetch_california_housing()
X, y = housing.data, housing.target

housing

{'data': array([[   8.3252    ,   41.        ,    6.98412698, ...,    2.55555556,
           37.88      , -122.23      ],
        [   8.3014    ,   21.        ,    6.23813708, ...,    2.10984183,
           37.86      , -122.22      ],
        [   7.2574    ,   52.        ,    8.28813559, ...,    2.80225989,
           37.85      , -122.24      ],
        ...,
        [   1.7       ,   17.        ,    5.20554273, ...,    2.3256351 ,
           39.43      , -121.22      ],
        [   1.8672    ,   18.        ,    5.32951289, ...,    2.12320917,
           39.43      , -121.32      ],
        [   2.3886    ,   16.        ,    5.25471698, ...,    2.61698113,
           39.37      , -121.24      ]]),
 'target': array([4.526, 3.585, 3.521, ..., 0.923, 0.847, 0.894]),
 'frame': None,
 'target_names': ['MedHouseVal'],
 'feature_names': ['MedInc',
  'HouseAge',
  'AveRooms',
  'AveBedrms',
  'Population',
  'AveOccup',
  'Latitude',
  'Longitude'],
 'DESCR': '.. _california_housing_dataset:\n\nCalifornia Housing dataset\n--------------------------\n\n**Data Set Characteristics:**\n\n    :Number of Instances: 20640\n\n    :Number of Attributes: 8 numeric, predictive attributes and the target\n\n    :Attribute Information:\n        - MedInc        median income in block group\n        - HouseAge      median house age in block group\n        - AveRooms      average number of rooms per household\n        - AveBedrms     average number of bedrooms per household\n        - Population    block group population\n        - AveOccup      average number of household members\n        - Latitude      block group latitude\n        - Longitude     block group longitude\n\n    :Missing Attribute Values: None\n\nThis dataset was obtained from the StatLib repository.\nhttps://www.dcc.fc.up.pt/~ltorgo/Regression/cal_housing.html\n\nThe target variable is the median house value for California districts,\nexpressed in hundreds of thousands of dollars ($100,000).\n\nThis dataset was derived from the 1990 U.S. census, using one row per census\nblock group. A block group is the smallest geographical unit for which the U.S.\nCensus Bureau publishes sample data (a block group typically has a population\nof 600 to 3,000 people).\n\nA household is a group of people residing within a home. Since the average\nnumber of rooms and bedrooms in this dataset are provided per household, these\ncolumns may take surprisingly large values for block groups with few households\nand many empty houses, such as vacation resorts.\n\nIt can be downloaded/loaded using the\n:func:`sklearn.datasets.fetch_california_housing` function.\n\n.. topic:: References\n\n    - Pace, R. Kelley and Ronald Barry, Sparse Spatial Autoregressions,\n      Statistics and Probability Letters, 33 (1997) 291-297\n'}

# Split the data into training and test sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# Standardize the features
scaler = StandardScaler()
X_train_scaled = scaler.fit_transform(X_train)
X_test_scaled = scaler.transform(X_test)

#X_train_scaled = X_train
#X_test_scaled = X_test

print(np.shape(X_test))

(4128, 8)

import matplotlib.pyplot as plt

# Assuming X_test is defined and has the shape (4128, 8)

# Set up the matplotlib figure
fig, axs = plt.subplots(nrows=2, ncols=4, figsize=(20, 10))  # Adjust size as needed
fig.subplots_adjust(hspace=0.5, wspace=0.3)  # Adjust spacing between plots as needed

# Iterate over all features (columns) in X_test
for i in range(8):
    # Determine the position of the subplot
    row = i // 4
    col = i % 4
    ax = axs[row, col]

    # Plot the histogram for the i-th feature
    ax.hist(X_test_scaled[:, i], bins=30, alpha=0.7, log=True)
    ax.set_title(f'Feature {i+1}')

# Display the plot
plt.show()

# Plotting the histogram for y_test
plt.figure(figsize=(6, 4))  # Adjust figure size as needed
plt.hist(y_test.ravel(), bins=30, alpha=0.7)  # Use .ravel() to flatten y_test to 1D
plt.title('Histogram of y_test')
plt.xlabel('Value')
plt.ylabel('Frequency')
plt.show()

Linear Regression

# Initialize and train a linear regression model on the scaled data
lin_reg = LinearRegression()
lin_reg.fit(X_train_scaled, y_train)

# Predict on the scaled training set and the scaled test set
y_train_pred = lin_reg.predict(X_train_scaled)
y_test_pred = lin_reg.predict(X_test_scaled)

# Evaluate the model
train_mse = mean_squared_error(y_train, y_train_pred)
test_mse = mean_squared_error(y_test, y_test_pred)

print(f'Training Mean Squared Error: {train_mse:.4f}')
print(f'Test Mean Squared Error: {test_mse:.4f}')

# Coefficients and Intercept
print('Intercept:', lin_reg.intercept_)
print('Coefficients:', lin_reg.coef_)

Training Mean Squared Error: 0.5179
Test Mean Squared Error: 0.5559
Intercept: 2.071946937378619
Coefficients: [ 0.85438303  0.12254624 -0.29441013  0.33925949 -0.00230772 -0.0408291
 -0.89692888 -0.86984178]

# without scaling

#Training Mean Squared Error: 0.5179
#Test Mean Squared Error: 0.5559
#Intercept: -37.02327770606391
#Coefficients: [ 4.48674910e-01  9.72425752e-03 -1.23323343e-01  7.83144907e-01  -2.02962058e-06 -3.52631849e-03 -4.19792487e-01 -4.33708065e-01]

Bayesian Linear Regression with PyMC

# Define the model
with pm.Model() as model_lin:
    # Priors for unknown model parameters
    alpha = pm.Normal('alpha', mu=0, sigma=10)
    beta = pm.Normal('beta', mu=0, sigma=10, shape=X_train.shape[1])
    sigma = pm.HalfNormal('sigma', sigma=1)

    # Expected value of outcome
    mu = alpha + pm.math.dot(X_train_scaled, beta)

    # Likelihood (sampling distribution) of observations
    Y_obs = pm.Normal('Y_obs', mu=mu, sigma=sigma, observed=y_train)

    # Sample from the posterior
    idata_lin = pm.sample(500, chains=1) # lengthy: use burn-in of 1000 and 1000 draws for 1 chain

100.00% [1500/1500 00:14<00:00 Sampling chain 0, 0 divergences]

import arviz as az
az.plot_trace(idata_lin)

array([[<Axes: title={'center': 'alpha'}>,
        <Axes: title={'center': 'alpha'}>],
       [<Axes: title={'center': 'beta'}>,
        <Axes: title={'center': 'beta'}>],
       [<Axes: title={'center': 'sigma'}>,
        <Axes: title={'center': 'sigma'}>]], dtype=object)

pos_alpha = idata_lin.posterior['alpha'].mean(axis=0).values
pos_betas = idata_lin.posterior['beta'].mean(axis=0).values

#print(np.shape(pos_alpha))
print(np.shape(pos_betas))


#pos_alpha = pos_alpha.reshape((np.shape(pos_alpha)[1],))

#pos_betas = pos_betas.reshape((np.shape(pos_betas)[1],np.shape(pos_betas)[2]))

(500, 8)
(500,) (500, 8) (8, 4128)
<class 'numpy.ndarray'> <class 'numpy.ndarray'> <class 'numpy.ndarray'>

pos_y_test = pos_alpha[:, np.newaxis] + np.dot(pos_betas, X_test_scaled.T)
pos_y_train = pos_alpha[:, np.newaxis] + np.dot(pos_betas, X_train_scaled.T)

print(np.shape(pos_y_test), type(pos_y_test))

pos_y_test_mean = np.mean(pos_y_test, axis=0)
pos_y_train_mean = np.mean(pos_y_train, axis=0)

print(np.shape(pos_y_test_mean))

pos_y_test_lower = np.percentile(pos_y_test,3, axis=0)
pos_y_test_upper = np.percentile(pos_y_test,97, axis=0)
pos_y_train_lower = np.percentile(pos_y_train,3, axis=0)
pos_y_train_upper = np.percentile(pos_y_train,97, axis=0)
print(np.shape(pos_y_test_lower))

for i in range(len(y_test)):
  if(i%50==0):
    print(f"true: {y_test[i]:1.4f}, y_test_pred: {y_test_pred[i]:1.4f}, Bayes mean: {pos_y_test_mean[i]:1.4f}, range: ({pos_y_test_lower[i]:1.4f}, {pos_y_test_upper[i]:1.4f})")

(500, 4128) <class 'numpy.ndarray'>
(4128,)
(4128,)
true: 0.7100, y_test_pred: 0.7191, Bayes mean: 0.8282, range: (0.7996, 0.8556)
true: 1.8800, y_test_pred: 2.4037, Bayes mean: 2.5199, range: (2.5014, 2.5369)
true: 2.2940, y_test_pred: 1.5178, Bayes mean: 2.5583, range: (2.5374, 2.5781)
true: 1.5630, y_test_pred: 1.5097, Bayes mean: 2.1514, range: (2.1311, 2.1708)
true: 1.9640, y_test_pred: 2.3937, Bayes mean: 2.2789, range: (2.2600, 2.2955)
true: 0.7020, y_test_pred: 1.0502, Bayes mean: 0.6899, range: (0.6695, 0.7124)
true: 1.8310, y_test_pred: 0.6687, Bayes mean: 2.2953, range: (2.2688, 2.3216)
true: 1.1610, y_test_pred: 1.4168, Bayes mean: 1.5943, range: (1.5747, 1.6132)
true: 1.7710, y_test_pred: 2.5336, Bayes mean: 2.3466, range: (2.3202, 2.3735)
true: 4.3930, y_test_pred: 2.8634, Bayes mean: 2.7801, range: (2.7474, 2.8080)
true: 2.6050, y_test_pred: 2.2113, Bayes mean: 2.4015, range: (2.3789, 2.4223)
true: 2.9520, y_test_pred: 0.9937, Bayes mean: 2.1671, range: (2.1510, 2.1836)
true: 1.6770, y_test_pred: 1.4220, Bayes mean: 2.0871, range: (2.0717, 2.1032)
true: 2.3740, y_test_pred: 1.1735, Bayes mean: 3.4368, range: (3.4020, 3.4742)
true: 1.9840, y_test_pred: 1.9858, Bayes mean: 2.3951, range: (2.3671, 2.4208)
true: 1.6660, y_test_pred: 0.7142, Bayes mean: 1.6425, range: (1.6200, 1.6645)
true: 1.1110, y_test_pred: 0.9356, Bayes mean: 0.8779, range: (0.8339, 0.9273)
true: 5.0000, y_test_pred: 1.5730, Bayes mean: 2.9698, range: (2.9380, 2.9988)
true: 0.2750, y_test_pred: 2.6333, Bayes mean: -0.3888, range: (-0.4442, -0.3366)
true: 2.2540, y_test_pred: 3.3560, Bayes mean: 1.1468, range: (1.1229, 1.1718)
true: 1.8750, y_test_pred: 2.0999, Bayes mean: 1.6768, range: (1.6614, 1.6945)
true: 3.6400, y_test_pred: 1.8346, Bayes mean: 2.4500, range: (2.4240, 2.4795)
true: 5.0000, y_test_pred: 1.1454, Bayes mean: 4.0948, range: (4.0632, 4.1243)
true: 4.4290, y_test_pred: 4.8032, Bayes mean: 2.3340, range: (2.3027, 2.3668)
true: 2.0700, y_test_pred: 1.5665, Bayes mean: 1.8228, range: (1.8042, 1.8443)
true: 0.8760, y_test_pred: 4.2362, Bayes mean: 1.7924, range: (1.7595, 1.8254)
true: 0.9800, y_test_pred: 2.0385, Bayes mean: 1.8739, range: (1.8551, 1.8937)
true: 3.9300, y_test_pred: 1.5520, Bayes mean: 3.5645, range: (3.5330, 3.5971)
true: 0.7970, y_test_pred: 1.2078, Bayes mean: 1.0687, range: (1.0462, 1.0900)
true: 1.6900, y_test_pred: 1.6035, Bayes mean: 2.0719, range: (2.0553, 2.0884)
true: 2.2020, y_test_pred: 1.9321, Bayes mean: 2.8197, range: (2.7981, 2.8393)
true: 2.4560, y_test_pred: 1.6540, Bayes mean: 2.2592, range: (2.1960, 2.3228)
true: 1.4000, y_test_pred: 2.5838, Bayes mean: 0.6081, range: (0.5811, 0.6368)
true: 1.9100, y_test_pred: 2.4022, Bayes mean: 2.1911, range: (2.1763, 2.2058)
true: 2.1740, y_test_pred: 2.6450, Bayes mean: 2.6469, range: (2.6290, 2.6651)
true: 1.6050, y_test_pred: 2.9621, Bayes mean: 2.3035, range: (2.2867, 2.3214)
true: 0.4550, y_test_pred: 2.6647, Bayes mean: 0.4170, range: (0.3904, 0.4454)
true: 1.0160, y_test_pred: 2.1881, Bayes mean: 1.3898, range: (1.3684, 1.4115)
true: 1.9220, y_test_pred: 2.2026, Bayes mean: 2.5246, range: (2.4956, 2.5541)
true: 1.8580, y_test_pred: 2.7611, Bayes mean: 2.0160, range: (1.9812, 2.0522)
true: 2.5220, y_test_pred: 2.0046, Bayes mean: 2.0188, range: (1.9890, 2.0467)
true: 5.0000, y_test_pred: 3.1471, Bayes mean: 2.5501, range: (2.5269, 2.5721)
true: 2.7860, y_test_pred: 0.9452, Bayes mean: 2.4931, range: (2.4648, 2.5211)
true: 1.9140, y_test_pred: 1.0731, Bayes mean: 2.3644, range: (2.3502, 2.3780)
true: 2.3310, y_test_pred: 2.0241, Bayes mean: 2.1435, range: (2.1238, 2.1625)
true: 1.0360, y_test_pred: 2.4118, Bayes mean: 1.7932, range: (1.7759, 1.8109)
true: 0.9920, y_test_pred: 1.6802, Bayes mean: 1.1980, range: (1.1633, 1.2338)
true: 1.8840, y_test_pred: 3.1414, Bayes mean: 1.9298, range: (1.9040, 1.9579)
true: 1.7100, y_test_pred: 1.4824, Bayes mean: 2.1509, range: (2.1310, 2.1700)
true: 2.5080, y_test_pred: 2.2120, Bayes mean: 3.3363, range: (3.3029, 3.3652)
true: 1.5740, y_test_pred: 1.2696, Bayes mean: 2.3000, range: (2.2859, 2.3141)
true: 5.0000, y_test_pred: 2.3738, Bayes mean: 4.3889, range: (4.3570, 4.4205)
true: 0.8750, y_test_pred: 2.7551, Bayes mean: 1.2609, range: (1.2242, 1.2969)
true: 0.4660, y_test_pred: 1.7705, Bayes mean: 0.8677, range: (0.8392, 0.8951)
true: 1.9830, y_test_pred: 3.1743, Bayes mean: 1.4168, range: (1.3683, 1.4679)
true: 0.4390, y_test_pred: 1.7385, Bayes mean: 0.8814, range: (0.8499, 0.9138)
true: 1.9090, y_test_pred: 2.1866, Bayes mean: 1.9746, range: (1.9516, 1.9974)
true: 0.6750, y_test_pred: 1.6092, Bayes mean: 0.9138, range: (0.8939, 0.9363)
true: 1.3130, y_test_pred: 2.7028, Bayes mean: 2.0743, range: (2.0456, 2.1044)
true: 3.9000, y_test_pred: 2.3851, Bayes mean: 3.3703, range: (3.3418, 3.3979)
true: 1.5280, y_test_pred: 2.0705, Bayes mean: 1.7734, range: (1.7571, 1.7923)
true: 2.3280, y_test_pred: 1.7752, Bayes mean: 2.3420, range: (2.3203, 2.3645)
true: 2.4480, y_test_pred: 2.2463, Bayes mean: 2.3642, range: (2.3416, 2.3861)
true: 2.3330, y_test_pred: 1.4475, Bayes mean: 2.3559, range: (2.3381, 2.3740)
true: 3.7000, y_test_pred: 2.4799, Bayes mean: 2.5330, range: (2.5048, 2.5611)
true: 1.6230, y_test_pred: 1.9872, Bayes mean: 2.3838, range: (2.3631, 2.4051)
true: 1.3490, y_test_pred: 2.5150, Bayes mean: 1.7489, range: (1.7024, 1.7955)
true: 0.5750, y_test_pred: 2.8794, Bayes mean: -0.2778, range: (-0.3344, -0.2209)
true: 3.3240, y_test_pred: 1.6917, Bayes mean: 3.2900, range: (3.2610, 3.3192)
true: 2.2500, y_test_pred: 1.5525, Bayes mean: 2.4180, range: (2.3937, 2.4432)
true: 1.7800, y_test_pred: 4.6456, Bayes mean: 1.8437, range: (1.8193, 1.8682)
true: 1.6650, y_test_pred: 2.3504, Bayes mean: 2.0658, range: (2.0288, 2.1008)
true: 1.5820, y_test_pred: 1.7014, Bayes mean: 2.5318, range: (2.5130, 2.5504)
true: 4.1610, y_test_pred: 1.9083, Bayes mean: 2.3244, range: (2.3057, 2.3425)
true: 5.0000, y_test_pred: 1.9192, Bayes mean: 3.4275, range: (3.4025, 3.4504)
true: 1.4750, y_test_pred: 1.6407, Bayes mean: 2.0675, range: (2.0492, 2.0857)
true: 1.0830, y_test_pred: 1.7796, Bayes mean: 2.2980, range: (2.2721, 2.3258)
true: 4.5000, y_test_pred: 3.0174, Bayes mean: 1.3462, range: (1.3035, 1.3865)
true: 3.9770, y_test_pred: 1.6099, Bayes mean: 3.8453, range: (3.8079, 3.8849)
true: 1.5040, y_test_pred: 3.2436, Bayes mean: 2.1676, range: (2.1168, 2.2205)
true: 2.3800, y_test_pred: 2.6613, Bayes mean: 3.0320, range: (3.0101, 3.0539)
true: 1.7810, y_test_pred: 1.6718, Bayes mean: 2.1605, range: (2.1338, 2.1861)
true: 2.0880, y_test_pred: 1.1509, Bayes mean: 1.5704, range: (1.5461, 1.5969)

#--- comparing Linear Regression with Bayesian Linear Regression

print("LINEAR REGRESSION")
print(f'Training Mean Squared Error: {train_mse:.4f}')
print(f'Test Mean Squared Error: {test_mse:.4f}')
print("\n")

# Evaluate the model
train_mse_bayes = mean_squared_error(y_train, pos_y_train_mean)
test_mse_bayes  = mean_squared_error(y_test, pos_y_test_mean)

print("BAYESIAN LINEAR REGRESSION")
print(f'Training Mean Squared Error: {train_mse_bayes:.4f}')
print(f'Test Mean Squared Error: {test_mse_bayes:.4f}')

LINEAR REGRESSION
Training Mean Squared Error: 0.5179
Test Mean Squared Error: 0.5559


BAYESIAN LINEAR REGRESSION
Training Mean Squared Error: 0.5212
Test Mean Squared Error: 0.5418

az.plot_trace(idata_lin)

array([[<Axes: title={'center': 'alpha'}>,
        <Axes: title={'center': 'alpha'}>],
       [<Axes: title={'center': 'beta'}>,
        <Axes: title={'center': 'beta'}>],
       [<Axes: title={'center': 'sigma'}>,
        <Axes: title={'center': 'sigma'}>]], dtype=object)

az.summary(idata_pol)

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
alpha	1.930	0.005	1.922	1.940	0.000	0.000	1249.0	839.0	1.00
beta[0]	0.886	0.009	0.870	0.902	0.000	0.000	659.0	739.0	1.00
beta[1]	0.113	0.005	0.102	0.122	0.000	0.000	1011.0	610.0	1.00
beta[2]	-0.337	0.017	-0.368	-0.305	0.001	0.000	616.0	697.0	1.01
beta[3]	0.400	0.017	0.367	0.433	0.001	0.000	732.0	417.0	1.01
beta[4]	0.036	0.004	0.028	0.044	0.000	0.000	1289.0	741.0	1.01
beta[5]	-2.814	0.077	-2.944	-2.649	0.002	0.001	1319.0	572.0	1.01
beta[6]	-0.773	0.014	-0.796	-0.745	0.001	0.000	598.0	754.0	1.00
beta[7]	-0.730	0.013	-0.757	-0.706	0.001	0.000	620.0	736.0	1.00
sigma	0.448	0.005	0.439	0.458	0.000	0.000	720.0	647.0	1.01
ν_	2.977	0.079	2.827	3.116	0.003	0.002	735.0	681.0	1.01
ν	2.977	0.079	2.827	3.116	0.003	0.002	735.0	681.0	1.01

Polynomial Linear regression

# Define the model
with pm.Model() as model_pol:
    # Priors for unknown model parameters
    alpha = pm.Normal('alpha', mu=0, sigma=10)
    beta = pm.Normal('beta', mu=0, sigma=10, shape=X_train.shape[1])
    gamma = pm.Normal('gamma', mu=0, sigma=10, shape=X_train.shape[1])
    sigma = pm.HalfNormal('sigma', sigma=1)

    ν_ = pm.Exponential('ν_', 1/29)
    ν = pm.Deterministic('ν', ν_ + 1)

    # Expected value of outcome
    mu = alpha + pm.math.dot(X_train_scaled, beta) + pm.math.dot(X_train_scaled**2, gamma)

    # Likelihood (sampling distribution) of observations
    #Y_obs = pm.Normal('Y_obs', mu=mu, sigma=sigma, observed=y_train)

    Y_obs = pm.StudentT('Y_obs', mu=mu, sigma=sigma, nu=ν, observed=y_train) #ϵ



    # Sample from the posterior
    idata_pol = pm.sample(500, chains=2) # lengthy: use burn-in of 1000 and 1000 draws for 1 chain

100.00% [1500/1500 04:58<00:00 Sampling chain 0, 0 divergences]

100.00% [1500/1500 05:40<00:00 Sampling chain 1, 0 divergences]

az.summary(idata_pol)

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
alpha	1.943	0.011	1.922	1.963	0.004	0.003	8.0	479.0	1.20
beta[0]	0.868	0.013	0.844	0.891	0.005	0.004	6.0	189.0	1.30
beta[1]	0.110	0.005	0.100	0.120	0.000	0.000	193.0	633.0	1.03
beta[2]	-0.330	0.024	-0.373	-0.285	0.013	0.010	4.0	271.0	1.54
beta[3]	0.480	0.030	0.429	0.532	0.017	0.013	4.0	91.0	1.61
beta[4]	0.052	0.008	0.038	0.067	0.003	0.002	6.0	202.0	1.26
beta[5]	-1.768	0.167	-1.993	-1.540	0.114	0.095	3.0	221.0	1.83
beta[6]	-0.870	0.017	-0.903	-0.840	0.001	0.001	352.0	508.0	1.01
beta[7]	-0.824	0.016	-0.854	-0.796	0.001	0.001	364.0	529.0	1.01
gamma[0]	0.007	0.004	0.000	0.015	0.001	0.001	25.0	444.0	1.06
gamma[1]	0.002	0.005	-0.007	0.011	0.001	0.001	11.0	194.0	1.14
gamma[2]	0.029	0.004	0.023	0.036	0.001	0.001	8.0	325.0	1.19
gamma[3]	-0.035	0.004	-0.042	-0.026	0.002	0.001	8.0	318.0	1.20
gamma[4]	-0.003	0.001	-0.005	-0.001	0.000	0.000	18.0	249.0	1.08
gamma[5]	0.639	0.628	0.016	1.399	0.437	0.368	3.0	125.0	1.83
gamma[6]	0.066	0.007	0.054	0.078	0.000	0.000	492.0	604.0	1.02
gamma[7]	-0.080	0.008	-0.095	-0.066	0.001	0.000	118.0	586.0	1.04
sigma	0.437	0.006	0.426	0.447	0.002	0.002	7.0	198.0	1.22
ν_	1.916	0.086	1.759	2.079	0.021	0.015	17.0	435.0	1.09
ν	2.916	0.086	2.759	3.079	0.021	0.015	17.0	435.0	1.09

az.plot_trace(idata_pol, compact=True)

array([[<Axes: title={'center': 'alpha'}>,
        <Axes: title={'center': 'alpha'}>],
       [<Axes: title={'center': 'beta'}>,
        <Axes: title={'center': 'beta'}>],
       [<Axes: title={'center': 'gamma'}>,
        <Axes: title={'center': 'gamma'}>],
       [<Axes: title={'center': 'sigma'}>,
        <Axes: title={'center': 'sigma'}>],
       [<Axes: title={'center': 'ν_'}>, <Axes: title={'center': 'ν_'}>],
       [<Axes: title={'center': 'ν'}>, <Axes: title={'center': 'ν'}>]],
      dtype=object)

pos_alpha_pol = idata_pol.posterior['alpha'].mean(axis=0).values
pos_betas_pol = idata_pol.posterior['beta'].mean(axis=0).values
pos_gammas_pol = idata_pol.posterior['gamma'].mean(axis=0).values

pos_y_test_pol = pos_alpha_pol[:, np.newaxis] + np.dot(pos_betas_pol, X_test_scaled.T) + np.dot(pos_gammas_pol, (X_test_scaled**2).T)
pos_y_test_mean_pol = np.mean(pos_y_test_pol, axis=0)

pos_y_train_pol = pos_alpha_pol[:, np.newaxis] + np.dot(pos_betas_pol, X_train_scaled.T) + np.dot(pos_gammas_pol, (X_train_scaled**2).T)
pos_y_train_mean_pol = np.mean(pos_y_train_pol, axis=0)

pos_y_test_lower_pol = np.percentile(pos_y_test_pol,3, axis=0)
pos_y_test_upper_pol = np.percentile(pos_y_test_pol,97, axis=0)

pos_y_train_lower_pol = np.percentile(pos_y_train_pol,3, axis=0)
pos_y_train_upper_pol = np.percentile(pos_y_train_pol,97, axis=0)


for i in range(len(y_test)):
  if(i%50==0):
    print(f"true: {y_test[i]:1.4f}, y_test_pred: {y_test_pred[i]:1.4f}, Bayes mean: {pos_y_test_mean_pol[i]:1.4f}, range: ({pos_y_test_lower_pol[i]:1.4f}, {pos_y_test_upper_pol[i]:1.4f})")

true: 0.7100, y_test_pred: 0.7191, Bayes mean: 0.8540, range: (0.8388, 0.8722)
true: 1.8800, y_test_pred: 2.4037, Bayes mean: 2.0451, range: (2.0291, 2.0622)
true: 2.2940, y_test_pred: 1.5178, Bayes mean: 2.3091, range: (2.2954, 2.3226)
true: 1.5630, y_test_pred: 1.5097, Bayes mean: 2.0410, range: (2.0267, 2.0553)
true: 1.9640, y_test_pred: 2.3937, Bayes mean: 2.3225, range: (2.3085, 2.3389)
true: 0.7020, y_test_pred: 1.0502, Bayes mean: 0.7358, range: (0.7131, 0.7576)
true: 1.8310, y_test_pred: 0.6687, Bayes mean: 1.9189, range: (1.9022, 1.9385)
true: 1.1610, y_test_pred: 1.4168, Bayes mean: 1.5029, range: (1.4898, 1.5159)
true: 1.7710, y_test_pred: 2.5336, Bayes mean: 2.2480, range: (2.2301, 2.2658)
true: 4.3930, y_test_pred: 2.8634, Bayes mean: 3.0842, range: (3.0581, 3.1112)
true: 2.6050, y_test_pred: 2.2113, Bayes mean: 2.2908, range: (2.2762, 2.3061)
true: 2.9520, y_test_pred: 0.9937, Bayes mean: 2.3018, range: (2.2846, 2.3199)
true: 1.6770, y_test_pred: 1.4220, Bayes mean: 1.8248, range: (1.8134, 1.8361)
true: 2.3740, y_test_pred: 1.1735, Bayes mean: 3.4531, range: (3.4253, 3.4830)
true: 1.9840, y_test_pred: 1.9858, Bayes mean: 2.6404, range: (2.6144, 2.6696)
true: 1.6660, y_test_pred: 0.7142, Bayes mean: 1.3694, range: (1.3554, 1.3853)
true: 1.1110, y_test_pred: 0.9356, Bayes mean: 1.1600, range: (1.1228, 1.1951)
true: 5.0000, y_test_pred: 1.5730, Bayes mean: 3.0222, range: (2.9955, 3.0486)
true: 0.2750, y_test_pred: 2.6333, Bayes mean: -0.5147, range: (-0.5928, -0.4481)
true: 2.2540, y_test_pred: 3.3560, Bayes mean: 1.3710, range: (1.3464, 1.3942)
true: 1.8750, y_test_pred: 2.0999, Bayes mean: 1.3067, range: (1.2926, 1.3196)
true: 3.6400, y_test_pred: 1.8346, Bayes mean: 2.5451, range: (2.5224, 2.5701)
true: 5.0000, y_test_pred: 1.1454, Bayes mean: 4.0484, range: (4.0255, 4.0739)
true: 4.4290, y_test_pred: 4.8032, Bayes mean: 2.3408, range: (2.3165, 2.3672)
true: 2.0700, y_test_pred: 1.5665, Bayes mean: 1.5708, range: (1.5576, 1.5822)
true: 0.8760, y_test_pred: 4.2362, Bayes mean: 1.8450, range: (1.8214, 1.8696)
true: 0.9800, y_test_pred: 2.0385, Bayes mean: 1.5912, range: (1.5802, 1.6022)
true: 3.9300, y_test_pred: 1.5520, Bayes mean: 3.5345, range: (3.5136, 3.5538)
true: 0.7970, y_test_pred: 1.2078, Bayes mean: 1.0205, range: (1.0037, 1.0372)
true: 1.6900, y_test_pred: 1.6035, Bayes mean: 1.6370, range: (1.6257, 1.6486)
true: 2.2020, y_test_pred: 1.9321, Bayes mean: 2.5729, range: (2.5561, 2.5893)
true: 2.4560, y_test_pred: 1.6540, Bayes mean: 2.3260, range: (2.2929, 2.3588)
true: 1.4000, y_test_pred: 2.5838, Bayes mean: 0.7366, range: (0.7090, 0.7600)
true: 1.9100, y_test_pred: 2.4022, Bayes mean: 2.2168, range: (2.2048, 2.2301)
true: 2.1740, y_test_pred: 2.6450, Bayes mean: 2.1959, range: (2.1798, 2.2119)
true: 1.6050, y_test_pred: 2.9621, Bayes mean: 2.3587, range: (2.3457, 2.3722)
true: 0.4550, y_test_pred: 2.6647, Bayes mean: 0.4610, range: (0.4433, 0.4789)
true: 1.0160, y_test_pred: 2.1881, Bayes mean: 1.4007, range: (1.3885, 1.4143)
true: 1.9220, y_test_pred: 2.2026, Bayes mean: 2.6259, range: (2.6045, 2.6484)
true: 1.8580, y_test_pred: 2.7611, Bayes mean: 1.8910, range: (1.8713, 1.9131)
true: 2.5220, y_test_pred: 2.0046, Bayes mean: 2.1495, range: (2.1294, 2.1696)
true: 5.0000, y_test_pred: 3.1471, Bayes mean: 2.6115, range: (2.5935, 2.6301)
true: 2.7860, y_test_pred: 0.9452, Bayes mean: 2.6628, range: (2.6410, 2.6866)
true: 1.9140, y_test_pred: 1.0731, Bayes mean: 2.0632, range: (2.0520, 2.0745)
true: 2.3310, y_test_pred: 2.0241, Bayes mean: 2.1976, range: (2.1825, 2.2152)
true: 1.0360, y_test_pred: 2.4118, Bayes mean: 1.3813, range: (1.3706, 1.3934)
true: 0.9920, y_test_pred: 1.6802, Bayes mean: 1.1199, range: (1.0816, 1.1583)
true: 1.8840, y_test_pred: 3.1414, Bayes mean: 2.0106, range: (1.9924, 2.0292)
true: 1.7100, y_test_pred: 1.4824, Bayes mean: 2.1493, range: (2.1320, 2.1675)
true: 2.5080, y_test_pred: 2.2120, Bayes mean: 3.0535, range: (3.0279, 3.0807)
true: 1.5740, y_test_pred: 1.2696, Bayes mean: 2.0070, range: (1.9961, 2.0185)
true: 5.0000, y_test_pred: 2.3738, Bayes mean: 4.3064, range: (4.2808, 4.3301)
true: 0.8750, y_test_pred: 2.7551, Bayes mean: 1.6257, range: (1.5975, 1.6536)
true: 0.4660, y_test_pred: 1.7705, Bayes mean: 0.4762, range: (0.4573, 0.4954)
true: 1.9830, y_test_pred: 3.1743, Bayes mean: 1.3224, range: (1.2957, 1.3498)
true: 0.4390, y_test_pred: 1.7385, Bayes mean: 0.9923, range: (0.9595, 1.0260)
true: 1.9090, y_test_pred: 2.1866, Bayes mean: 2.0339, range: (2.0166, 2.0522)
true: 0.6750, y_test_pred: 1.6092, Bayes mean: 1.0142, range: (0.9944, 1.0345)
true: 1.3130, y_test_pred: 2.7028, Bayes mean: 1.7823, range: (1.7528, 1.8130)
true: 3.9000, y_test_pred: 2.3851, Bayes mean: 3.0825, range: (3.0640, 3.1012)
true: 1.5280, y_test_pred: 2.0705, Bayes mean: 1.0998, range: (1.0784, 1.1206)
true: 2.3280, y_test_pred: 1.7752, Bayes mean: 2.1859, range: (2.1713, 2.2014)
true: 2.4480, y_test_pred: 2.2463, Bayes mean: 2.4089, range: (2.3949, 2.4240)
true: 2.3330, y_test_pred: 1.4475, Bayes mean: 2.1611, range: (2.1481, 2.1735)
true: 3.7000, y_test_pred: 2.4799, Bayes mean: 2.5793, range: (2.5536, 2.6039)
true: 1.6230, y_test_pred: 1.9872, Bayes mean: 2.4077, range: (2.3896, 2.4262)
true: 1.3490, y_test_pred: 2.5150, Bayes mean: 1.5193, range: (1.4931, 1.5476)
true: 0.5750, y_test_pred: 2.8794, Bayes mean: 0.1773, range: (0.1321, 0.2225)
true: 3.3240, y_test_pred: 1.6917, Bayes mean: 3.0592, range: (3.0410, 3.0769)
true: 2.2500, y_test_pred: 1.5525, Bayes mean: 2.4350, range: (2.4156, 2.4551)
true: 1.7800, y_test_pred: 4.6456, Bayes mean: 1.3513, range: (1.3320, 1.3701)
true: 1.6650, y_test_pred: 2.3504, Bayes mean: 2.1178, range: (2.0840, 2.1518)
true: 1.5820, y_test_pred: 1.7014, Bayes mean: 2.1128, range: (2.0980, 2.1287)
true: 4.1610, y_test_pred: 1.9083, Bayes mean: 2.1967, range: (2.1834, 2.2114)
true: 5.0000, y_test_pred: 1.9192, Bayes mean: 3.4709, range: (3.4533, 3.4893)
true: 1.4750, y_test_pred: 1.6407, Bayes mean: 1.6578, range: (1.6418, 1.6741)
true: 1.0830, y_test_pred: 1.7796, Bayes mean: 2.0512, range: (2.0325, 2.0719)
true: 4.5000, y_test_pred: 3.0174, Bayes mean: 1.7865, range: (1.7490, 1.8230)
true: 3.9770, y_test_pred: 1.6099, Bayes mean: 3.8891, range: (3.8598, 3.9218)
true: 1.5040, y_test_pred: 3.2436, Bayes mean: 2.1547, range: (2.1249, 2.1854)
true: 2.3800, y_test_pred: 2.6613, Bayes mean: 2.6496, range: (2.6309, 2.6693)
true: 1.7810, y_test_pred: 1.6718, Bayes mean: 1.9610, range: (1.9419, 1.9828)
true: 2.0880, y_test_pred: 1.1509, Bayes mean: 1.4923, range: (1.4754, 1.5084)

#--- comparing Linear Regression with Polynomial Bayesian Linear Regression

print("LINEAR REGRESSION")
print(f'Training Mean Squared Error: {train_mse:.4f}')
print(f'Test Mean Squared Error: {test_mse:.4f}')
print("\n")

# Evaluate the model
train_mse_bayes_rob = mean_squared_error(y_train, pos_y_train_mean_rob)
test_mse_bayes_rob  = mean_squared_error(y_test, pos_y_test_mean_rob)

print("BAYESIAN LINEAR REGRESSION")
print(f'Training Mean Squared Error: {train_mse_bayes:.4f}')
print(f'Test Mean Squared Error: {test_mse_bayes:.4f}')
print("\n")

print("BAYESIAN POLYNOMIAL REGRESSION")
print(f'Training Mean Squared Error: {train_mse_bayes_rob:.4f}')
print(f'Test Mean Squared Error: {test_mse_bayes_rob:.4f}')

LINEAR REGRESSION
Training Mean Squared Error: 0.5179
Test Mean Squared Error: 0.5559


BAYESIAN LINEAR REGRESSION
Training Mean Squared Error: 0.5212
Test Mean Squared Error: 0.5418


BAYESIAN POLYNOMIAL REGRESSION
Training Mean Squared Error: 0.5231
Test Mean Squared Error: 0.5892

Model Comparison

#---------- Posterior Predictive Checks

y_l = pm.sample_posterior_predictive(idata_lin,model=model_lin)

100.00% [500/500 00:00<00:00]

y_p = pm.sample_posterior_predictive(idata_pol,model=model_pol)

100.00% [1000/1000 00:02<00:00]

y_l

arviz.InferenceData

• posterior_predictive

  <xarray.Dataset>
  Dimensions:      (chain: 1, draw: 500, Y_obs_dim_2: 16512)
  Coordinates:
    * chain        (chain) int64 0
    * draw         (draw) int64 0 1 2 3 4 5 6 7 ... 493 494 495 496 497 498 499
    * Y_obs_dim_2  (Y_obs_dim_2) int64 0 1 2 3 4 ... 16507 16508 16509 16510 16511
  Data variables:
      Y_obs        (chain, draw, Y_obs_dim_2) float64 2.126 3.056 ... 3.535 2.128
  Attributes:
      created_at:                 2024-03-10T00:10:15.997634
      arviz_version:              0.15.1
      inference_library:          pymc
      inference_library_version:  5.10.4

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 500
    • Y_obs_dim_2: 16512
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 495 496 497 498 499

      array([  0,   1,   2, ..., 497, 498, 499])

    • Y_obs_dim_2
      (Y_obs_dim_2)
      int64
      0 1 2 3 ... 16508 16509 16510 16511

      array([    0,     1,     2, ..., 16509, 16510, 16511])

  • Data variables: (1)
    • Y_obs
      (chain, draw, Y_obs_dim_2)
      float64
      2.126 3.056 2.645 ... 3.535 2.128

      array([[[2.12630733, 3.05561347, 2.64505204, ..., 2.06987289,
               1.44676801, 2.34132864],
              [2.37858337, 2.59042201, 1.80878   , ..., 1.45089206,
               2.73707928, 2.1967672 ],
              [1.25287714, 2.0184837 , 3.32303402, ..., 2.19433913,
               2.7359095 , 1.6066616 ],
              ...,
              [1.7527359 , 2.29811353, 4.05035153, ..., 2.26074127,
               2.07547475, 2.77371468],
              [2.77195012, 3.30396036, 3.43008295, ..., 2.06142076,
               3.84274256, 2.73249698],
              [1.754206  , 1.52501967, 3.82949001, ..., 1.58057289,
               3.53525552, 2.12760942]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Int64Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Int64Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
                  ...
                  490, 491, 492, 493, 494, 495, 496, 497, 498, 499],
                 dtype='int64', name='draw', length=500))

    • Y_obs_dim_2
      PandasIndex

      PandasIndex(Int64Index([    0,     1,     2,     3,     4,     5,     6,     7,     8,
                      9,
                  ...
                  16502, 16503, 16504, 16505, 16506, 16507, 16508, 16509, 16510,
                  16511],
                 dtype='int64', name='Y_obs_dim_2', length=16512))

  • Attributes: (4)

    created_at :

    2024-03-10T00:10:15.997634

    arviz_version :

    0.15.1

    inference_library :

    pymc

    inference_library_version :

    5.10.4

  
  
• observed_data

  <xarray.Dataset>
  Dimensions:      (Y_obs_dim_0: 16512)
  Coordinates:
    * Y_obs_dim_0  (Y_obs_dim_0) int64 0 1 2 3 4 ... 16507 16508 16509 16510 16511
  Data variables:
      Y_obs        (Y_obs_dim_0) float64 1.03 3.821 1.726 ... 2.221 2.835 3.25
  Attributes:
      created_at:                 2024-03-10T00:10:15.999430
      arviz_version:              0.15.1
      inference_library:          pymc
      inference_library_version:  5.10.4

  xarray.Dataset

  • Dimensions:
    • Y_obs_dim_0: 16512
  • Coordinates: (1)
    • Y_obs_dim_0
      (Y_obs_dim_0)
      int64
      0 1 2 3 ... 16508 16509 16510 16511

      array([    0,     1,     2, ..., 16509, 16510, 16511])

  • Data variables: (1)
    • Y_obs
      (Y_obs_dim_0)
      float64
      1.03 3.821 1.726 ... 2.835 3.25

      array([1.03 , 3.821, 1.726, ..., 2.221, 2.835, 3.25 ])

  • Indexes: (1)
    • Y_obs_dim_0
      PandasIndex

      PandasIndex(Int64Index([    0,     1,     2,     3,     4,     5,     6,     7,     8,
                      9,
                  ...
                  16502, 16503, 16504, 16505, 16506, 16507, 16508, 16509, 16510,
                  16511],
                 dtype='int64', name='Y_obs_dim_0', length=16512))

  • Attributes: (4)

    created_at :

    2024-03-10T00:10:15.999430

    arviz_version :

    0.15.1

    inference_library :

    pymc

    inference_library_version :

    5.10.4

  
  

plt.figure(figsize=(8,3))

y_o   = y_train
y_l  = y_l.posterior_predictive['Y_obs'].mean(axis=0).mean(axis=0).values
y_p  = y_p.posterior_predictive['y_pred'].mean(axis=0).mean(axis=0).values


data = [y_o, y_l, y_p]
labels = ['data', 'linear model', 'pol. order 2']

for i, d in enumerate(data):
  mean = d.mean()
  err = np.percentile(d, [25,75])
  plt.errorbar(mean,-i,xerr=[[err[0]],[err[1]]], fmt='o')
  plt.text(mean,-i+0.2, labels[i], ha='center', fontsize=14)

plt.ylim([-i-0.5,0.5])
plt.yticks([])

([], [])

#---------- Information Criteria

with model_lin:
    pm.compute_log_likelihood(idata_lin)


with model_pol:
    pm.compute_log_likelihood(idata_pol)

100.00% [1000/1000 00:01<00:00]

idata_pol

arviz.InferenceData

• posterior

  <xarray.Dataset>
  Dimensions:      (chain: 2, draw: 500, beta_dim_0: 8, gamma_dim_0: 8)
  Coordinates:
    * chain        (chain) int64 0 1
    * draw         (draw) int64 0 1 2 3 4 5 6 7 ... 493 494 495 496 497 498 499
    * beta_dim_0   (beta_dim_0) int64 0 1 2 3 4 5 6 7
    * gamma_dim_0  (gamma_dim_0) int64 0 1 2 3 4 5 6 7
  Data variables:
      alpha        (chain, draw) float64 1.942 1.948 1.942 ... 1.937 1.942 1.95
      beta         (chain, draw, beta_dim_0) float64 0.8741 0.1163 ... -0.8466
      gamma        (chain, draw, gamma_dim_0) float64 0.005186 ... -0.08653
      sigma        (chain, draw) float64 0.4335 0.4367 0.4426 ... 0.4418 0.4355
      ν_           (chain, draw) float64 1.823 2.003 2.07 ... 1.966 1.942 1.847
      ν            (chain, draw) float64 2.823 3.003 3.07 ... 2.966 2.942 2.847
  Attributes:
      created_at:                 2024-03-10T00:08:27.922198
      arviz_version:              0.15.1
      inference_library:          pymc
      inference_library_version:  5.10.4
      sampling_time:              638.5446758270264
      tuning_steps:               1000

  xarray.Dataset

  • Dimensions:
    • chain: 2
    • draw: 500
    • beta_dim_0: 8
    • gamma_dim_0: 8
  • Coordinates: (4)
    • chain
      (chain)
      int64
      0 1

      array([0, 1])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 495 496 497 498 499

      array([  0,   1,   2, ..., 497, 498, 499])

    • beta_dim_0
      (beta_dim_0)
      int64
      0 1 2 3 4 5 6 7

      array([0, 1, 2, 3, 4, 5, 6, 7])

    • gamma_dim_0
      (gamma_dim_0)
      int64
      0 1 2 3 4 5 6 7

      array([0, 1, 2, 3, 4, 5, 6, 7])

  • Data variables: (6)
    • alpha
      (chain, draw)
      float64
      1.942 1.948 1.942 ... 1.942 1.95

      array([[1.9423071 , 1.94758879, 1.94220918, 1.94024201, 1.94386833,
              1.93501266, 1.94963023, 1.94101071, 1.9381218 , 1.93519718,
              1.92932453, 1.9272789 , 1.94501854, 1.94815634, 1.928738  ,
              1.95131349, 1.92781214, 1.92172101, 1.91712935, 1.95626775,
              1.95030508, 1.94721483, 1.95016785, 1.94662497, 1.93288419,
              1.93022519, 1.92880209, 1.93878469, 1.94062149, 1.93985951,
              1.92830209, 1.95106154, 1.92733764, 1.94725058, 1.9241799 ,
              1.94622069, 1.93754619, 1.93660123, 1.93779346, 1.93341188,
              1.93422514, 1.93281179, 1.93228819, 1.93218797, 1.92708783,
              1.94635246, 1.93736961, 1.93743215, 1.95135743, 1.93121431,
              1.92762492, 1.94433752, 1.9371623 , 1.94989354, 1.93953227,
              1.94033521, 1.93027403, 1.93214428, 1.93046198, 1.93513372,
              1.9261226 , 1.92332221, 1.92185959, 1.93335187, 1.9226825 ,
              1.92494281, 1.94104102, 1.94253732, 1.92843085, 1.92770001,
              1.93148579, 1.93008432, 1.9555811 , 1.96528573, 1.96926528,
              1.95815065, 1.9487418 , 1.93801048, 1.94984459, 1.94900616,
              1.93836255, 1.92814474, 1.92373829, 1.92779911, 1.93457903,
              1.94809908, 1.93456844, 1.93876423, 1.94241817, 1.93675975,
              1.93671368, 1.93205444, 1.93212292, 1.92697962, 1.9526169 ,
              1.9345231 , 1.93628497, 1.92763752, 1.93519695, 1.93511642,
      ...
              1.93961182, 1.9626116 , 1.95076638, 1.95840731, 1.94480618,
              1.96780384, 1.93329142, 1.96250519, 1.93651273, 1.93981442,
              1.95460558, 1.95215213, 1.95471158, 1.95090022, 1.95056952,
              1.93031344, 1.95669333, 1.91779305, 1.93091217, 1.95098187,
              1.93878586, 1.93478606, 1.95792575, 1.96273358, 1.94767621,
              1.95007735, 1.96127366, 1.94042591, 1.94255224, 1.95436317,
              1.95071333, 1.94653186, 1.94601464, 1.94220227, 1.94911067,
              1.9394562 , 1.94724608, 1.94047473, 1.95328293, 1.93778875,
              1.95764979, 1.95584736, 1.96302143, 1.94482295, 1.94639235,
              1.94008271, 1.94555183, 1.95794784, 1.93570486, 1.9329362 ,
              1.9371681 , 1.94451629, 1.93998887, 1.96091636, 1.95692044,
              1.95649686, 1.95245252, 1.96145657, 1.9350308 , 1.94575064,
              1.93944088, 1.95005048, 1.95189995, 1.95355662, 1.9470346 ,
              1.94196019, 1.95724529, 1.96365588, 1.94645071, 1.95042145,
              1.94760977, 1.94292966, 1.94845321, 1.95505694, 1.93768923,
              1.93505391, 1.95923921, 1.9422276 , 1.95088131, 1.94356392,
              1.95803704, 1.95239057, 1.9649617 , 1.9402656 , 1.96037837,
              1.9418358 , 1.93344455, 1.93244095, 1.93955593, 1.94363965,
              1.95471767, 1.94651225, 1.96667881, 1.96601857, 1.95875089,
              1.95222408, 1.94087546, 1.93665088, 1.94176168, 1.95031181]])

    • beta
      (chain, draw, beta_dim_0)
      float64
      0.8741 0.1163 ... -0.8853 -0.8466

      array([[[ 0.87413949,  0.11631376, -0.3243267 , ..., -1.89515826,
               -0.88848722, -0.8444707 ],
              [ 0.86598335,  0.10082556, -0.31420801, ..., -2.03681535,
               -0.86501135, -0.80849164],
              [ 0.86052339,  0.11445497, -0.31423486, ..., -1.97070335,
               -0.84792706, -0.80133986],
              ...,
              [ 0.86610775,  0.10846038, -0.32620061, ..., -1.90804637,
               -0.87154812, -0.81885036],
              [ 0.85911989,  0.11016433, -0.32017586, ..., -1.9286874 ,
               -0.8811929 , -0.82894981],
              [ 0.85560434,  0.110208  , -0.30809449, ..., -1.90884879,
               -0.87820845, -0.82811281]],
      
             [[ 0.88243869,  0.10561501, -0.37320832, ..., -1.64310071,
               -0.86174381, -0.8200379 ],
              [ 0.88011079,  0.12079701, -0.32772341, ..., -1.5936536 ,
               -0.84922015, -0.80672663],
              [ 0.88687172,  0.11253209, -0.35271495, ..., -1.56495478,
               -0.85808599, -0.81985018],
              ...,
              [ 0.87416937,  0.10379781, -0.33754102, ..., -1.60699287,
               -0.88086082, -0.83169085],
              [ 0.86444544,  0.10375916, -0.3468429 , ..., -1.60036623,
               -0.88339036, -0.84553553],
              [ 0.87701578,  0.10931882, -0.34097414, ..., -1.59647332,
               -0.88528426, -0.84656604]]])

    • gamma
      (chain, draw, gamma_dim_0)
      float64
      0.005186 0.001412 ... -0.08653

      array([[[ 5.18647746e-03,  1.41218422e-03,  2.56740703e-02, ...,
                1.24540027e+00,  7.87688573e-02, -9.28061325e-02],
              [ 9.53620935e-03, -6.15434442e-03,  3.36905844e-02, ...,
                1.39498825e+00,  5.91017641e-02, -8.03617721e-02],
              [ 6.92623956e-03, -3.95713600e-03,  3.45810549e-02, ...,
                1.46325503e+00,  5.65232112e-02, -7.62708017e-02],
              ...,
              [ 8.92667961e-03,  8.63550929e-03,  2.51781482e-02, ...,
                1.33801019e+00,  7.52961477e-02, -7.83879399e-02],
              [ 1.22961550e-02, -2.58323250e-03,  2.90418213e-02, ...,
                1.22655737e+00,  6.46914753e-02, -7.63651285e-02],
              [ 8.77896480e-03, -7.99387339e-04,  2.99105211e-02, ...,
                1.13442868e+00,  6.71648167e-02, -7.77104107e-02]],
      
             [[ 1.35343964e-02,  7.03007137e-03,  3.54702246e-02, ...,
                1.77130299e-02,  6.77443962e-02, -8.16207311e-02],
              [ 3.78679932e-03, -8.16447120e-04,  2.95646616e-02, ...,
                1.70310358e-02,  6.43722177e-02, -6.66628897e-02],
              [ 7.10935333e-03,  5.50093128e-03,  2.99377212e-02, ...,
                1.67520395e-02,  6.29765161e-02, -7.29071707e-02],
              ...,
              [ 9.40072348e-03,  8.25369758e-03,  3.02347180e-02, ...,
                1.73258399e-02,  7.56321331e-02, -8.44975752e-02],
              [ 8.42533449e-03,  8.46183624e-03,  3.29579906e-02, ...,
                1.72255055e-02,  7.37752192e-02, -8.41675597e-02],
              [ 4.68576042e-03,  3.22218400e-03,  2.78699598e-02, ...,
                1.71877980e-02,  7.62086452e-02, -8.65264012e-02]]])

    • sigma
      (chain, draw)
      float64
      0.4335 0.4367 ... 0.4418 0.4355

      array([[0.4334857 , 0.43665399, 0.44258265, 0.41596015, 0.4305951 ,
              0.43707811, 0.43564553, 0.43588759, 0.43616776, 0.43945102,
              0.42448731, 0.43184099, 0.43660072, 0.43632477, 0.43501365,
              0.43003093, 0.43232602, 0.43154498, 0.42668137, 0.4338882 ,
              0.439098  , 0.43669488, 0.44152029, 0.44190127, 0.43336127,
              0.44184875, 0.4425734 , 0.43209339, 0.43292372, 0.433416  ,
              0.42653854, 0.43108386, 0.43272865, 0.43488155, 0.43687128,
              0.4399331 , 0.43320643, 0.43591158, 0.43919795, 0.42826546,
              0.43603714, 0.43373049, 0.43404284, 0.43237147, 0.42702296,
              0.44661338, 0.438514  , 0.4360879 , 0.43714621, 0.43784371,
              0.4386591 , 0.43676829, 0.43375508, 0.43863667, 0.43929613,
              0.43349541, 0.42986632, 0.42730822, 0.43038251, 0.43518053,
              0.43338704, 0.436584  , 0.43399605, 0.43054495, 0.43215223,
              0.43379208, 0.43248941, 0.4398293 , 0.43249547, 0.43269195,
              0.42957127, 0.43013889, 0.43864577, 0.43350751, 0.43155381,
              0.4331101 , 0.42485792, 0.43606983, 0.43310311, 0.43341455,
              0.43210911, 0.43163266, 0.43776718, 0.43893686, 0.42883108,
              0.43613834, 0.43690278, 0.43512747, 0.43683772, 0.42905146,
              0.43247395, 0.43066067, 0.4339311 , 0.43496356, 0.43281025,
              0.42815011, 0.43227133, 0.42821463, 0.43638839, 0.44659634,
      ...
              0.44506248, 0.44246091, 0.43383038, 0.44901667, 0.43975533,
              0.44640703, 0.43279491, 0.45073067, 0.43472082, 0.4334498 ,
              0.44770402, 0.43144515, 0.43582474, 0.43779721, 0.43747067,
              0.43391104, 0.43900911, 0.4311576 , 0.4385956 , 0.43419213,
              0.42438909, 0.43519322, 0.43438732, 0.43661279, 0.44158965,
              0.44350453, 0.44322889, 0.44284433, 0.4464022 , 0.43772976,
              0.43701814, 0.43888471, 0.43898255, 0.43426065, 0.43778845,
              0.4349343 , 0.44228431, 0.43613306, 0.44181079, 0.440121  ,
              0.43810096, 0.4449856 , 0.44498306, 0.4373465 , 0.43606665,
              0.43597959, 0.43603546, 0.43259457, 0.43482758, 0.43511763,
              0.43710461, 0.43386813, 0.44335444, 0.43631255, 0.44754266,
              0.44761097, 0.45086072, 0.4388855 , 0.43982281, 0.44006657,
              0.43709157, 0.43332718, 0.44441296, 0.44454767, 0.43647284,
              0.43757015, 0.43576112, 0.43504573, 0.44591998, 0.43810324,
              0.43152414, 0.43871491, 0.43832891, 0.44526703, 0.44166398,
              0.43240046, 0.43474812, 0.44295521, 0.44143938, 0.44592195,
              0.44100328, 0.44581188, 0.44354028, 0.43216261, 0.44394018,
              0.44063642, 0.43781745, 0.44400596, 0.43886527, 0.43002023,
              0.42993131, 0.43191235, 0.44059498, 0.43987112, 0.43023804,
              0.42951309, 0.44685277, 0.44137672, 0.44175037, 0.43554389]])

    • ν_
      (chain, draw)
      float64
      1.823 2.003 2.07 ... 1.942 1.847

      array([[1.82291899, 2.00257762, 2.07008847, 1.67397051, 1.78592763,
              1.94885428, 1.91545816, 1.85986116, 1.90708034, 2.00409613,
              1.79737478, 1.84421477, 1.93556145, 1.94103157, 1.86719236,
              1.83222568, 1.87040274, 1.7939492 , 1.82587799, 1.92271415,
              1.92300702, 2.01103591, 1.98045447, 2.09815179, 1.91255694,
              1.99255   , 1.99433267, 1.93717622, 1.88636433, 1.88621378,
              1.73544024, 1.88470294, 1.86793012, 1.86383566, 1.97158589,
              1.85719203, 1.90852051, 1.84030038, 1.85417503, 1.90053969,
              1.93708043, 1.87767953, 1.88354099, 1.89607018, 1.75254304,
              1.97170677, 1.85918232, 1.88747162, 1.9014691 , 1.98202266,
              1.99394512, 1.85735827, 1.8769968 , 1.89523985, 1.94709322,
              1.89922067, 1.8233326 , 1.81611114, 1.83062144, 1.90568185,
              1.89358099, 1.87997868, 1.83736532, 1.85782208, 1.87139067,
              1.85608506, 1.79564606, 2.03419597, 1.79683823, 1.79295511,
              1.84699355, 1.84867566, 1.96102161, 1.92267764, 1.91630637,
              1.84160163, 1.79444526, 1.86042973, 1.879361  , 1.89138866,
              1.84874065, 1.83873274, 1.96735568, 1.94111608, 1.84863088,
              1.85934678, 2.02885716, 1.88655773, 1.91598369, 1.78802044,
              1.81349045, 1.85638785, 1.84876066, 1.89024961, 1.85192993,
              1.85917655, 1.83290581, 1.82428982, 1.97715847, 1.98741101,
      ...
              1.95128914, 1.88833609, 1.96679919, 1.97503586, 1.83972911,
              2.0791386 , 1.78577647, 2.20288837, 1.87873258, 1.75903007,
              2.0762794 , 1.89977385, 1.85735323, 1.80960967, 1.79550961,
              1.98290369, 1.89889879, 1.8146199 , 1.85120558, 1.86719732,
              1.83011765, 1.72397731, 1.94178865, 1.90512766, 2.02008292,
              2.03923417, 1.89598805, 2.1010057 , 1.93218842, 1.89533272,
              1.90689147, 1.91192455, 1.97658442, 1.8758781 , 1.82001793,
              1.95442161, 2.04192638, 1.83971834, 1.90626294, 1.90145177,
              2.0014294 , 1.96733317, 2.06670913, 1.93453038, 1.90536342,
              1.94672018, 1.98046351, 1.74764286, 1.9465643 , 1.93248918,
              1.91144878, 1.83501957, 1.93807012, 1.93929936, 2.05049657,
              2.16531132, 2.11547357, 1.94408425, 1.964666  , 1.9163563 ,
              1.86243581, 1.8648438 , 2.04300493, 1.96415943, 1.92986496,
              1.95758876, 1.91179607, 2.00332508, 1.89979078, 1.88504307,
              1.9222446 , 2.04965245, 1.98039535, 1.88301376, 1.85826012,
              1.97491952, 1.97385162, 1.93429   , 2.1738406 , 2.09567937,
              1.97349141, 1.9616953 , 1.92832517, 1.94890702, 2.03663294,
              1.83882505, 2.03000436, 1.97566926, 1.99716747, 1.84454749,
              1.80589173, 1.76589953, 2.00812612, 2.05276346, 1.69187222,
              1.7894263 , 2.06943535, 1.96598147, 1.9419061 , 1.84711736]])

    • ν
      (chain, draw)
      float64
      2.823 3.003 3.07 ... 2.942 2.847

      array([[2.82291899, 3.00257762, 3.07008847, 2.67397051, 2.78592763,
              2.94885428, 2.91545816, 2.85986116, 2.90708034, 3.00409613,
              2.79737478, 2.84421477, 2.93556145, 2.94103157, 2.86719236,
              2.83222568, 2.87040274, 2.7939492 , 2.82587799, 2.92271415,
              2.92300702, 3.01103591, 2.98045447, 3.09815179, 2.91255694,
              2.99255   , 2.99433267, 2.93717622, 2.88636433, 2.88621378,
              2.73544024, 2.88470294, 2.86793012, 2.86383566, 2.97158589,
              2.85719203, 2.90852051, 2.84030038, 2.85417503, 2.90053969,
              2.93708043, 2.87767953, 2.88354099, 2.89607018, 2.75254304,
              2.97170677, 2.85918232, 2.88747162, 2.9014691 , 2.98202266,
              2.99394512, 2.85735827, 2.8769968 , 2.89523985, 2.94709322,
              2.89922067, 2.8233326 , 2.81611114, 2.83062144, 2.90568185,
              2.89358099, 2.87997868, 2.83736532, 2.85782208, 2.87139067,
              2.85608506, 2.79564606, 3.03419597, 2.79683823, 2.79295511,
              2.84699355, 2.84867566, 2.96102161, 2.92267764, 2.91630637,
              2.84160163, 2.79444526, 2.86042973, 2.879361  , 2.89138866,
              2.84874065, 2.83873274, 2.96735568, 2.94111608, 2.84863088,
              2.85934678, 3.02885716, 2.88655773, 2.91598369, 2.78802044,
              2.81349045, 2.85638785, 2.84876066, 2.89024961, 2.85192993,
              2.85917655, 2.83290581, 2.82428982, 2.97715847, 2.98741101,
      ...
              2.95128914, 2.88833609, 2.96679919, 2.97503586, 2.83972911,
              3.0791386 , 2.78577647, 3.20288837, 2.87873258, 2.75903007,
              3.0762794 , 2.89977385, 2.85735323, 2.80960967, 2.79550961,
              2.98290369, 2.89889879, 2.8146199 , 2.85120558, 2.86719732,
              2.83011765, 2.72397731, 2.94178865, 2.90512766, 3.02008292,
              3.03923417, 2.89598805, 3.1010057 , 2.93218842, 2.89533272,
              2.90689147, 2.91192455, 2.97658442, 2.8758781 , 2.82001793,
              2.95442161, 3.04192638, 2.83971834, 2.90626294, 2.90145177,
              3.0014294 , 2.96733317, 3.06670913, 2.93453038, 2.90536342,
              2.94672018, 2.98046351, 2.74764286, 2.9465643 , 2.93248918,
              2.91144878, 2.83501957, 2.93807012, 2.93929936, 3.05049657,
              3.16531132, 3.11547357, 2.94408425, 2.964666  , 2.9163563 ,
              2.86243581, 2.8648438 , 3.04300493, 2.96415943, 2.92986496,
              2.95758876, 2.91179607, 3.00332508, 2.89979078, 2.88504307,
              2.9222446 , 3.04965245, 2.98039535, 2.88301376, 2.85826012,
              2.97491952, 2.97385162, 2.93429   , 3.1738406 , 3.09567937,
              2.97349141, 2.9616953 , 2.92832517, 2.94890702, 3.03663294,
              2.83882505, 3.03000436, 2.97566926, 2.99716747, 2.84454749,
              2.80589173, 2.76589953, 3.00812612, 3.05276346, 2.69187222,
              2.7894263 , 3.06943535, 2.96598147, 2.9419061 , 2.84711736]])

  • Indexes: (4)
    • chain
      PandasIndex

      PandasIndex(Int64Index([0, 1], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Int64Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
                  ...
                  490, 491, 492, 493, 494, 495, 496, 497, 498, 499],
                 dtype='int64', name='draw', length=500))

    • beta_dim_0
      PandasIndex

      PandasIndex(Int64Index([0, 1, 2, 3, 4, 5, 6, 7], dtype='int64', name='beta_dim_0'))

    • gamma_dim_0
      PandasIndex

      PandasIndex(Int64Index([0, 1, 2, 3, 4, 5, 6, 7], dtype='int64', name='gamma_dim_0'))

  • Attributes: (6)

    created_at :

    2024-03-10T00:08:27.922198

    arviz_version :

    0.15.1

    inference_library :

    pymc

    inference_library_version :

    5.10.4

    sampling_time :

    638.5446758270264

    tuning_steps :

    1000

  
  
• sample_stats

  <xarray.Dataset>
  Dimensions:                (chain: 2, draw: 500)
  Coordinates:
    * chain                  (chain) int64 0 1
    * draw                   (draw) int64 0 1 2 3 4 5 ... 494 495 496 497 498 499
  Data variables: (12/17)
      process_time_diff      (chain, draw) float64 0.173 0.1644 ... 0.2309 0.2423
      n_steps                (chain, draw) float64 127.0 127.0 ... 127.0 127.0
      diverging              (chain, draw) bool False False False ... False False
      largest_eigval         (chain, draw) float64 nan nan nan nan ... nan nan nan
      energy                 (chain, draw) float64 1.581e+04 ... 1.59e+04
      max_energy_error       (chain, draw) float64 -0.2077 5.983 ... -0.3784
      ...                     ...
      acceptance_rate        (chain, draw) float64 0.9707 0.6147 ... 0.9992 0.9931
      step_size              (chain, draw) float64 0.0274 0.0274 ... 0.03957
      step_size_bar          (chain, draw) float64 0.03393 0.03393 ... 0.03759
      reached_max_treedepth  (chain, draw) bool False False False ... False False
      smallest_eigval        (chain, draw) float64 nan nan nan nan ... nan nan nan
      perf_counter_start     (chain, draw) float64 1.221e+04 ... 1.264e+04
  Attributes:
      created_at:                 2024-03-10T00:08:27.941327
      arviz_version:              0.15.1
      inference_library:          pymc
      inference_library_version:  5.10.4
      sampling_time:              638.5446758270264
      tuning_steps:               1000

  xarray.Dataset

  • Dimensions:
    • chain: 2
    • draw: 500
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1

      array([0, 1])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 495 496 497 498 499

      array([  0,   1,   2, ..., 497, 498, 499])

  • Data variables: (17)
    • process_time_diff
      (chain, draw)
      float64
      0.173 0.1644 ... 0.2309 0.2423

      array([[0.17302689, 0.16441164, 0.19387699, 0.2476848 , 0.12420204,
              0.24373115, 0.23916003, 0.23381701, 0.22670605, 0.23043602,
              0.23766666, 0.12169144, 0.24868886, 0.17901795, 0.33964718,
              0.16787137, 0.1988652 , 0.17160574, 0.09015809, 0.17233718,
              0.17825652, 0.0899997 , 0.17451827, 0.17847325, 0.18121552,
              0.16618154, 0.09348595, 0.17430661, 0.1812483 , 0.18053168,
              0.18711317, 0.17970967, 0.18089183, 0.20369953, 0.17310428,
              0.17135242, 0.17209571, 0.08795963, 0.1625655 , 0.09388189,
              0.14146899, 0.1754771 , 0.25862047, 0.33687669, 0.10092443,
              0.17033766, 0.19623864, 0.0927941 , 0.17876031, 0.17535835,
              0.1858917 , 0.17118526, 0.19041603, 0.17567598, 0.17538759,
              0.17222555, 0.18215635, 0.09175981, 0.10091226, 0.1698019 ,
              0.09328706, 0.08603258, 0.24710293, 0.17960159, 0.35583638,
              0.09016312, 0.17115025, 0.17360523, 0.17454373, 0.20076417,
              0.19751252, 0.46245715, 0.23433651, 0.23499119, 0.22672413,
              0.12186906, 0.24478193, 0.47615718, 0.21155485, 0.17292561,
              0.1772252 , 0.20133268, 0.18351648, 0.16705555, 0.16824396,
              0.17064236, 0.19530669, 0.1740923 , 0.17886524, 0.16734249,
              0.33639822, 0.18577242, 0.08739605, 0.09087417, 0.17945627,
              0.17852345, 0.18111673, 0.17306125, 0.1850282 , 0.18452689,
      ...
              0.17596806, 0.18470594, 0.18655772, 0.18062214, 0.17485119,
              0.18271411, 0.1793955 , 0.17771801, 0.18386102, 0.18227017,
              0.19076082, 0.16253473, 0.09199522, 0.17353138, 0.17745787,
              0.18801049, 0.20944793, 0.17921117, 0.18298227, 0.18211784,
              0.17634574, 0.09346284, 0.18494341, 0.17429637, 0.17908324,
              0.0870943 , 0.1835697 , 0.18769457, 0.17798806, 0.19219497,
              0.24004022, 0.11764835, 0.24436986, 0.23937493, 0.23516189,
              0.23104239, 0.24543269, 0.24094184, 0.24945389, 0.23294926,
              0.19356248, 0.18246705, 0.17719671, 0.183358  , 0.17911846,
              0.1762957 , 0.19579163, 0.0903457 , 0.17189839, 0.17600899,
              0.17587767, 0.17940108, 0.18975479, 0.18014049, 0.17245157,
              0.1746134 , 0.17777822, 0.18692546, 0.19741697, 0.17522401,
              0.18249081, 0.09307075, 0.36804919, 0.18364611, 0.34982807,
              0.17832616, 0.17666167, 0.19654899, 0.1736547 , 0.17861801,
              0.17369139, 0.17565783, 0.09452411, 0.18712106, 0.1713729 ,
              0.09334714, 0.17733781, 0.16794613, 0.17920293, 0.1842586 ,
              0.19950206, 0.17313683, 0.17122531, 0.17617281, 0.17433891,
              0.19684464, 0.17737062, 0.17428214, 0.17531974, 0.17813017,
              0.20025265, 0.35153095, 0.17513121, 0.10769127, 0.24591486,
              0.12431936, 0.23789711, 0.23596141, 0.23090355, 0.24226584]])

    • n_steps
      (chain, draw)
      float64
      127.0 127.0 127.0 ... 127.0 127.0

      array([[127., 127., 127., 127.,  63., 127., 127., 127., 127., 127., 127.,
               63., 127., 127., 255., 127., 127., 127.,  63., 127., 127.,  63.,
              127., 127., 127., 127.,  63., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127.,  63., 127.,  63.,  95., 127., 191., 255.,
               63., 127., 127.,  63., 127., 127., 127., 127., 127., 127., 127.,
              127., 127.,  63.,  63., 127.,  63.,  63., 191., 127., 255.,  63.,
              127., 127., 127., 127., 127., 255., 127., 127., 127.,  63., 127.,
              255., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 255., 127.,  63.,  63., 127., 127., 127., 127., 127.,
              127.,  63., 127., 127.,  63., 191., 127., 127., 127., 127., 127.,
              255., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 255., 127.,  63., 127., 127., 127., 127.,
              255., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 255., 255., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127.,  63., 127., 255., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 255., 127., 127., 127.,  63., 127.,
              127., 127., 127., 127., 127., 127.,  63.,  63., 255., 127., 127.,
      ...
               63.,  63., 127., 127., 127., 127.,  63., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127.,  63., 127., 127., 127., 127.,  63.,  63., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 191., 191., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 127.,  63., 127., 127.,
              127., 127., 127., 127., 127., 127.,  63., 127., 127.,  63., 255.,
              127., 127., 127., 127., 127., 127., 127., 127.,  63., 127., 127.,
              127., 127.,  63., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127.,  63., 127., 127., 127., 127., 127.,
              127., 127., 127.,  63., 127., 127., 127.,  63., 127., 127., 127.,
              127., 127.,  63., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127.,  63., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 127., 127., 127.,  63.,
              255., 127., 255., 127., 127., 127., 127., 127., 127., 127.,  63.,
              127., 127.,  63., 127., 127., 127., 127., 127., 127., 127., 127.,
              127., 127., 127., 127., 127., 127., 127., 255., 127.,  63., 127.,
               63., 127., 127., 127., 127.]])

    • diverging
      (chain, draw)
      bool
      False False False ... False False

      array([[False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
      ...
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False]])

    • largest_eigval
      (chain, draw)
      float64
      nan nan nan nan ... nan nan nan nan

      array([[nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
      ...
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan]])

    • energy
      (chain, draw)
      float64
      1.581e+04 1.582e+04 ... 1.59e+04

      array([[15807.45167901, 15816.84014644, 15818.5234746 , 15811.21146231,
              15809.65829425, 15803.48757869, 15806.21219932, 15806.80252919,
              15805.67616317, 15810.04558963, 15810.30919184, 15809.35141655,
              15811.75953196, 15807.38170928, 15809.35619588, 15804.88116239,
              15806.4321962 , 15808.88206017, 15809.84905887, 15813.87412903,
              15820.29672949, 15820.01574708, 15817.21329756, 15807.12695222,
              15806.83613855, 15801.20046579, 15808.85311198, 15805.32768334,
              15807.73386417, 15812.34906466, 15808.62955543, 15807.45253453,
              15802.75014816, 15803.0664487 , 15802.22385749, 15809.02358377,
              15818.72388487, 15807.05632521, 15818.03178007, 15812.30671871,
              15803.74352731, 15803.89923877, 15811.53765736, 15822.9120357 ,
              15823.18509371, 15824.42532693, 15815.56363983, 15809.58497673,
              15807.67901216, 15810.92251287, 15809.23931818, 15808.38700688,
              15803.18126615, 15805.14705258, 15814.88366637, 15805.77144013,
              15802.70154913, 15799.83926073, 15804.24547652, 15806.01742749,
              15819.27748299, 15817.56702553, 15817.24632832, 15814.44808023,
              15812.93941718, 15815.68209288, 15815.9695397 , 15804.97082948,
              15805.67305728, 15810.48509767, 15806.2836636 , 15807.66009654,
              15810.19096128, 15807.53667557, 15813.55448238, 15812.65474025,
              15818.53869721, 15820.92973748, 15822.32890599, 15819.08593669,
      ...
              15917.58944939, 15912.2848047 , 15906.99869866, 15906.64853494,
              15913.45689761, 15904.58465994, 15907.71454969, 15909.54536353,
              15910.00497022, 15905.01425271, 15904.09211343, 15900.57311279,
              15895.620128  , 15902.65965258, 15901.05973198, 15902.51881186,
              15901.68281089, 15907.20075331, 15903.17379865, 15899.60542116,
              15902.57829484, 15904.04431954, 15907.52559589, 15904.73465404,
              15898.41698664, 15901.39518452, 15898.11055013, 15903.74808065,
              15907.8106372 , 15903.46679559, 15905.95486091, 15901.81049822,
              15901.63725881, 15906.1029907 , 15900.8761947 , 15903.28877232,
              15902.77974156, 15907.02071151, 15903.25380495, 15906.22084784,
              15901.28533946, 15906.25635996, 15911.50822069, 15905.84502091,
              15904.60570442, 15899.33641038, 15899.82890336, 15908.1640899 ,
              15911.36582523, 15907.23156258, 15904.34253766, 15904.51376691,
              15905.07008514, 15905.83302417, 15907.41151611, 15906.73423848,
              15903.80841996, 15903.84541906, 15907.7449399 , 15913.86736323,
              15909.27742331, 15907.82183576, 15904.12522005, 15910.91266288,
              15909.72961501, 15906.55293647, 15904.52814733, 15908.40012233,
              15903.96226882, 15903.24781274, 15905.93948116, 15904.99277999,
              15915.09894971, 15911.51596805, 15918.94824919, 15916.18560323,
              15902.1036086 , 15902.03329429, 15904.07647521, 15901.28811461]])

    • max_energy_error
      (chain, draw)
      float64
      -0.2077 5.983 ... -0.4971 -0.3784

      array([[-2.07685190e-01,  5.98260691e+00,  3.82188797e+00,
              -5.01637814e-02,  4.35399863e-01, -2.96543051e-01,
               4.76975002e+00,  1.68243473e+00, -1.84831596e-01,
               3.84017747e-01, -2.97272484e-01, -2.70549502e-02,
              -3.18950012e-02,  1.28961178e-01,  5.38271146e+00,
              -2.79059283e-01,  1.07670680e-01,  2.35799113e-01,
               5.69081048e-01, -1.69090533e+00, -6.93237932e-01,
              -8.23689348e-01, -2.27911130e+00,  2.29176541e+00,
              -2.29590736e+00, -1.71515710e-01,  2.13588835e+00,
               3.81004813e-01,  6.04651696e+00, -1.50517911e+00,
               7.27528439e-01,  1.17471259e-01,  1.62948520e-02,
               2.64604225e-01, -5.83439381e-02,  2.68167995e-02,
               1.87284516e+00,  7.55916174e-02,  3.93481316e-01,
              -6.41831435e-01,  2.58805605e-01,  2.78436664e-02,
               7.52466786e+00,  1.53490829e+01, -3.16663803e+00,
               2.30865582e+00, -2.50302069e+00,  2.35049761e+00,
              -1.43086436e+00, -3.06777456e-01,  8.61044363e-02,
               1.45659858e+00,  3.23991022e-01,  6.25656840e-01,
               2.57863583e-01,  9.25030977e-01, -8.79094094e-01,
              -3.92568160e-01,  9.58380479e-01,  2.48467945e+00,
      ...
               1.31616819e-01,  3.32392048e-01,  6.15232788e-02,
               5.30867907e-01,  2.71242917e+00, -8.12045823e-01,
               3.19222485e-01, -1.89271925e-01,  1.17913946e-01,
               1.89240199e+00, -8.90032386e-02,  4.38553679e-02,
               5.26670054e-02, -1.88776332e-02,  1.59350378e-01,
              -3.53297087e-01,  7.33814806e-01,  1.61257103e+00,
               3.74429552e+00, -2.27866568e-01, -1.93073377e-01,
               3.96176825e-01, -2.42317460e-01, -7.40334477e-01,
              -6.13957476e-02,  5.42689535e-02, -7.59589756e-02,
              -9.27064096e-02, -2.13367869e-01, -2.21673655e-01,
              -1.96442557e-01,  6.03657984e-02,  1.00586932e-01,
              -4.96259090e-02,  2.64055546e-02,  5.04182419e-01,
               2.16163882e-01,  1.86702974e+00,  1.92640046e+00,
              -1.16054307e+00,  2.36313756e+00, -2.89307712e-01,
               1.82418067e+00, -4.01588241e-01,  2.82861236e-01,
               9.04285563e-02,  1.78104139e+00,  5.12678416e-01,
              -4.21498034e-01,  2.56606888e+00,  4.55757534e-01,
              -9.28716569e-02,  2.08229860e+00,  2.43065079e-01,
              -1.88561614e-01, -1.62734602e-01,  9.15807614e-01,
              -4.97124786e-01, -3.78407768e-01]])

    • lp
      (chain, draw)
      float64
      -1.58e+04 -1.581e+04 ... -1.589e+04

      array([[-15800.94147706, -15807.71936867, -15801.26070202,
              -15804.20008265, -15795.85508613, -15796.26553741,
              -15797.98453476, -15798.39875436, -15797.75961149,
              -15800.21875942, -15802.12764833, -15801.84696912,
              -15802.42305778, -15798.27671273, -15798.16343744,
              -15799.48805392, -15799.21489441, -15794.80855684,
              -15804.22101952, -15805.02632872, -15810.20565904,
              -15807.9012773 , -15799.8862597 , -15801.14772561,
              -15796.49909497, -15798.28644168, -15797.93327979,
              -15798.84886087, -15801.03829604, -15798.16043344,
              -15799.60126839, -15797.52450254, -15794.70626023,
              -15795.24292254, -15798.54757663, -15801.48901922,
              -15799.77611552, -15802.00456678, -15807.22118323,
              -15797.5115429 , -15795.81557461, -15797.39244898,
              -15802.32172207, -15807.94890546, -15813.7544265 ,
              -15807.98154365, -15802.12580537, -15800.90285402,
              -15800.98702753, -15799.26763346, -15800.81133547,
              -15796.549372  , -15797.06801035, -15798.09916455,
              -15797.43619263, -15798.6220799 , -15794.96568741,
              -15797.97480199, -15796.96893484, -15800.98917737,
      ...
              -15895.82633533, -15898.63939164, -15890.66734728,
              -15890.65601209, -15892.96079892, -15893.68973272,
              -15898.39276301, -15897.01383216, -15897.23756038,
              -15895.62704436, -15891.18236967, -15895.33388967,
              -15894.43122338, -15891.57251864, -15896.46902122,
              -15894.56647786, -15896.96679717, -15898.22840926,
              -15896.16809699, -15893.99737867, -15899.52560986,
              -15896.89391214, -15897.12327494, -15891.3187885 ,
              -15892.36067472, -15895.4518863 , -15898.472656  ,
              -15895.16578498, -15894.99741581, -15896.2261028 ,
              -15896.36774455, -15893.48504497, -15898.8430757 ,
              -15896.41791598, -15893.58309303, -15895.4658275 ,
              -15893.14589029, -15899.14350003, -15902.6576737 ,
              -15898.67853761, -15896.93331999, -15897.35054921,
              -15903.66591765, -15893.72465956, -15895.11082284,
              -15895.45711955, -15896.76613256, -15895.3668858 ,
              -15894.56138063, -15896.78487498, -15896.57127556,
              -15902.74377844, -15901.43724447, -15906.05024317,
              -15898.37513532, -15895.84554381, -15893.96331117,
              -15894.10169586, -15893.90610026]])

    • perf_counter_diff
      (chain, draw)
      float64
      0.1752 0.1644 ... 0.2338 0.2517

      array([[0.17515749, 0.16444984, 0.19542112, 0.2521478 , 0.13030161,
              0.25182888, 0.24548717, 0.2401305 , 0.23205773, 0.24165432,
              0.25237873, 0.1270251 , 0.2556236 , 0.1811194 , 0.34480434,
              0.16922209, 0.19476834, 0.17293576, 0.0909844 , 0.17304966,
              0.17975156, 0.09032817, 0.17443963, 0.17981379, 0.18084216,
              0.16782531, 0.09341933, 0.17647265, 0.18061427, 0.17935657,
              0.18702068, 0.18100036, 0.18131943, 0.20472671, 0.17606442,
              0.17188225, 0.17365653, 0.08810319, 0.16287671, 0.09455447,
              0.14084715, 0.1759533 , 0.25999536, 0.33771619, 0.10157176,
              0.17074634, 0.19382263, 0.09189205, 0.17847043, 0.17633677,
              0.18626445, 0.17099836, 0.18788275, 0.17612928, 0.17549398,
              0.17081577, 0.18342389, 0.09244418, 0.10037402, 0.17017195,
              0.09325517, 0.08654857, 0.24796908, 0.1791855 , 0.35731066,
              0.09005806, 0.1713141 , 0.17426835, 0.17492085, 0.20272712,
              0.19967845, 0.47098169, 0.25416251, 0.23949751, 0.23459102,
              0.12464916, 0.25328568, 0.49370194, 0.21263348, 0.17295877,
              0.17728822, 0.20147865, 0.18445296, 0.16752908, 0.16837686,
              0.17058337, 0.19603667, 0.17399681, 0.17993783, 0.16900557,
              0.33695664, 0.19003809, 0.08807628, 0.09134971, 0.18002979,
              0.17875683, 0.18117952, 0.17498647, 0.1869844 , 0.18495543,
      ...
              0.1749519 , 0.18554368, 0.18725029, 0.18098616, 0.17655755,
              0.18326544, 0.18000039, 0.17736038, 0.18435826, 0.18336373,
              0.18896316, 0.164623  , 0.09173122, 0.17373925, 0.17849327,
              0.18628887, 0.20916346, 0.18049847, 0.18378401, 0.18137263,
              0.18227327, 0.09261363, 0.18500807, 0.17515618, 0.17934395,
              0.08770088, 0.18397117, 0.18491147, 0.1779045 , 0.19545522,
              0.24882567, 0.1185574 , 0.25980623, 0.24325794, 0.24075002,
              0.23261108, 0.25862852, 0.25457043, 0.26194254, 0.23736145,
              0.19304151, 0.18349359, 0.17870262, 0.18282185, 0.17951492,
              0.17728893, 0.20131846, 0.09095638, 0.17153255, 0.17722707,
              0.17614117, 0.17938346, 0.18984399, 0.1795941 , 0.17338722,
              0.17438406, 0.17756106, 0.18622373, 0.19436359, 0.1772724 ,
              0.18371192, 0.09344213, 0.37230382, 0.18599976, 0.35212559,
              0.17930111, 0.17628155, 0.19807164, 0.17405353, 0.18031898,
              0.17411003, 0.17492779, 0.09318977, 0.18947817, 0.17080593,
              0.09280821, 0.1770516 , 0.16834412, 0.17991196, 0.18718314,
              0.19472068, 0.17508316, 0.17120458, 0.17628597, 0.17545222,
              0.19813667, 0.17698902, 0.1744591 , 0.17488159, 0.17800948,
              0.19970839, 0.35278113, 0.17602886, 0.10805785, 0.25846301,
              0.1336206 , 0.24318539, 0.24165348, 0.2337709 , 0.25174701]])

    • index_in_trajectory
      (chain, draw)
      int64
      -74 -52 -29 74 44 ... 74 33 19 65

      array([[ -74,  -52,  -29,   74,   44,   67,   51,   32,  -85,   50,  -72,
                30,   88,  -11,  -86,   57,   68,  -21,  -41,   92,  -46,  -27,
                35,  -26,  -63,  -52,   -3,  -33,  -14,    8,  -71,   54,  -70,
                39,   64,   76,   77,  -26,  -15,   39,   24,  -65,    3,   -3,
                37,  -78,  -93,  -12,   92,   58,   55,  -72,   29,  -67, -102,
                54,  -43,  -24,   13,  -72,  -32,   41,   28,  -78,   20,   -8,
                85,  -61,  -67,    1,  -15,   -7,  -93,   40,  -17,  -38,  -34,
               -33,   59,   37,  -76,  -74,  -25,   10,   42,  -36,  -65,   73,
                96, -105,  -32,  -22,   -9,   39,  -70,  -34,   70,  -80, -103,
               -28,  -18,   65,  -72,  -12,  -61,   35,  -48,   31,  -52,   65,
                42,  -80,   73,   39,   55,   33,   41,   91,   44,  -39,   35,
                66,   62,  -25,  -21,   16,   51,   35,   18,  -95,  -26,   59,
                57,  -31,   61, -101,   36, -100,  -68,  -82,   35,    2,   -7,
               -36,   54,   56,   50,   29,  -46,   72,  -69,  -55,  -47,  -66,
               116,  134,   38,  -68,    3,  -64,   36,   51,   34,  -41,    6,
                28,  -17,   40,   29,  -78,  -14, -122,   14,  -61,   39,  -64,
               -80,  -68,  -37,   99,  -52,   68,  -71,   77,  -56,   92,  -21,
                62,  -39,    7,  -81,  -48,  -51,  -48,  -74,  -15,  -88,  -17,
                77,  -40,    5,   48,   26,  -44,  -65,  -31,  -44,   -3,  -70,
                85,  -50,  -57,   22,   55,  -66,   25,   35,   38,   59,  -53,
      ...
               -44,   33,   58,  -12,  -51,   66,   46,  -26,   49,  105,   32,
               -27,   21,   19,   37,  -28,   43,  -91,   32,   27,  -97,   75,
               -51,   87,   12,  -40,    8,   19,   76,   42,  -38,  -89,  -54,
                69,   41,   90,  -76,   54,  -36,   31,   38,   68,  -43,   15,
                88, -121,   79,   84,   57,  -64,   97,  101,  -70,   70,  -73,
                72,  -47,  -80,   28,   66,  -33,   49,   62,   21,   88,  -75,
                49,   50,  -52,  -23,   74,   22,  -55,   93,   16,  108,   69,
               -26,  -31,  -71,   43,  -57,   21,  -30,  -51,   66,   38,  -41,
               -34,   45,   95,   13,  -32,   41,   49,   35,  -35,   40,  -91,
               -67,   87,  -42,  -53,   73,  -51, -105,   64,  -70,  -68,  -21,
                60,   82,  -94,   -7,  -51,   94,   35,  -48,   82,   59,  104,
                65,  -75,  -17, -123,   62,  -16,   21,   -5,  -40,  -74,   66,
                21,   76,   15,  -44,   64,  -17,   30,  -14,   32,  -64,   27,
               -73,   46,   37,  -52,   13,  -28,   39,  -75,  -27,   82,  -78,
              -107,   29,  -65,  -53,   21,   15,  -68,   40,   46,   10,  -31,
               -64,   70,  127,   59,  -14,  -17,  -62,  -88,   27,  -84,   50,
               -88,  -61,  -65,  -69, -104,  -16,   46,   26,   35,  -89,  -25,
                36,  -81,   31,  103,   62,  -24,  -19,   58,  -23,  -85,   54,
               -72,  -70,  -44,  -30,   31,   88,  -13,   36,  -51,   14,   40,
                16,   74,   33,   19,   65]])

    • tree_depth
      (chain, draw)
      int64
      7 7 7 7 6 7 7 7 ... 7 6 7 6 7 7 7 7

      array([[7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 8, 7, 7, 7, 6, 7, 7, 6,
              7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 8, 8,
              6, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 7, 6, 6, 8, 7, 8, 6,
              7, 7, 7, 7, 7, 8, 7, 7, 7, 6, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 8, 7, 6, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 6, 8, 7, 7, 7, 7, 7,
              8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 6, 7, 7, 7, 7,
              8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 8, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 8, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 6, 6, 8, 7, 7,
              6, 7, 7, 7, 7, 8, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 8, 7, 7, 7,
              6, 7, 7, 6, 8, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6,
              7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 6, 7, 6,
              7, 8, 6, 7, 6, 8, 7, 7, 7, 6, 7, 7, 7, 7, 8, 7, 7, 7, 7, 6, 7, 7,
              6, 7, 7, 7, 7, 6, 7, 8, 7, 7, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 7, 7, 7, 6, 6, 7, 7, 6, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 6, 7, 7, 7, 7, 6, 6, 7, 7,
              7, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 6, 7, 7, 6, 6, 7, 7, 7, 7, 8, 7, 8,
              8, 7, 7, 8, 6, 7, 6, 7, 7, 7, 7, 8, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7,
      ...
              7, 8, 8, 6, 7, 6, 7, 6, 7, 7, 7, 7, 6, 7, 7, 7, 8, 7, 7, 7, 7, 8,
              7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 6, 6, 7, 7, 6,
              7, 7, 7, 7, 7, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 6, 7, 7, 6, 8, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 8, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6,
              6, 6, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 6, 6, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 6, 8,
              7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7,
              7, 7, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6,
              8, 7, 8, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7,
              7, 7, 7, 7, 7, 7, 7, 8, 7, 6, 7, 6, 7, 7, 7, 7]])

    • energy_error
      (chain, draw)
      float64
      0.01872 0.8268 ... -0.01391 0.05923

      array([[ 1.87217532e-02,  8.26824504e-01, -4.63607085e-01,
               4.14150175e-03,  2.50071719e-01,  5.83103592e-02,
               2.66563887e-01, -3.36707330e-01,  2.77984464e-03,
              -1.10144524e-02, -2.54917714e-01,  3.08086343e-04,
              -7.50723412e-03,  1.01285284e-01, -8.35500262e-01,
              -2.71743769e-01,  1.07670680e-01,  8.76235607e-02,
               4.12613308e-01, -9.73296476e-01,  1.52368891e-01,
              -4.92556456e-01, -2.00845612e+00, -8.42220660e-02,
              -1.65332988e+00, -3.67016585e-02,  2.22395379e-01,
              -7.34110260e-02,  1.36554718e-01, -3.65764136e-02,
              -4.09698008e-01, -3.39538484e-02, -3.55768502e-03,
              -1.07454925e-01, -4.76028676e-02,  5.20883756e-03,
               6.73248997e-03, -4.42601303e-02,  2.82940332e-01,
              -3.98561722e-01, -7.78483375e-02,  9.25985692e-03,
               1.23045590e+00, -8.78319678e-02, -9.11307796e-01,
              -1.59871631e+00, -1.49783638e+00, -6.43333545e-01,
              -1.26048174e+00, -2.97638751e-01,  4.98896052e-02,
              -4.52916132e-01,  1.95665318e-01, -3.38996687e-01,
               1.33320529e-01,  3.90560414e-01, -1.78605458e-01,
              -1.02713477e-01, -1.97331174e-01,  3.04045990e-01,
      ...
               2.08720921e-02,  1.10540405e-02,  1.58683617e-02,
               1.58961450e-01, -2.18540638e-01, -7.05090837e-01,
               1.24025015e-01, -1.25510480e-01,  4.74565576e-02,
              -2.92847862e-01, -7.10788229e-02,  1.59833446e-02,
              -1.08777474e-02, -5.31731515e-04,  3.23251606e-02,
               9.49186776e-02,  2.89709577e-01,  1.59795386e+00,
              -1.32669005e+00,  1.68199057e-02,  2.04168653e-02,
               6.47457330e-03,  1.70102934e-01, -6.42960177e-01,
              -2.74985021e-02,  4.04263562e-02,  2.77738552e-03,
               4.67997175e-02,  2.21505580e-02,  9.69753567e-02,
              -1.37239876e-01,  2.58014321e-03,  5.50995727e-02,
              -4.54423583e-02,  6.07651879e-03,  2.84594242e-01,
               7.53808987e-03,  2.43609009e-01,  5.62539331e-01,
              -4.39888940e-01, -1.72080134e-01, -1.00374640e-01,
               8.99166924e-01, -3.00537824e-01, -2.65366699e-02,
               3.39154490e-02,  6.52998044e-01,  1.90335815e-01,
              -3.80390871e-01,  8.37732718e-01,  1.10076527e-01,
              -3.18910541e-02, -2.75812377e-02,  1.65735646e-01,
              -1.48258961e-02, -3.09363876e-02,  2.35409853e-01,
              -1.39097891e-02,  5.92280129e-02]])

    • acceptance_rate
      (chain, draw)
      float64
      0.9707 0.6147 ... 0.9992 0.9931

      array([[0.97068296, 0.61467541, 0.72659499, 0.99767827, 0.84788997,
              0.98385822, 0.61566634, 0.62683179, 0.96613181, 0.91662706,
              0.99892422, 0.99887261, 0.99960175, 0.96657685, 0.42158268,
              0.97864996, 0.95176069, 0.95059541, 0.78237099, 0.99210379,
              0.94561773, 0.93330572, 0.74885332, 0.55603106, 1.        ,
              0.9852411 , 0.35615094, 0.94849329, 0.38695742, 0.90296149,
              0.83012965, 0.96973537, 0.99519169, 0.93931807, 0.99131402,
              0.98871967, 0.50984194, 0.98184322, 0.80915629, 0.99369776,
              0.90205586, 0.98774453, 0.05912586, 0.5205142 , 0.95728137,
              0.79244615, 0.91486539, 0.74110665, 0.91117335, 0.99702253,
              0.96577568, 0.74722592, 0.86227825, 0.87493762, 0.8882772 ,
              0.86378038, 0.93698329, 0.96586884, 0.91131189, 0.54522363,
              0.50554976, 0.77329597, 0.99998828, 0.91501189, 0.75376608,
              0.67243119, 0.61446491, 0.98872047, 0.61261716, 0.52023423,
              0.78862828, 0.22563903, 0.79014948, 0.99104559, 0.42680782,
              0.98626197, 0.51681612, 0.18557425, 0.69902407, 0.99103686,
              0.91105148, 0.93414988, 0.37426519, 0.30896834, 0.63377694,
              0.51202261, 0.54936294, 0.77558269, 0.40662779, 0.94677455,
              0.24241085, 0.50503484, 0.95685623, 0.99778697, 0.65360107,
              0.84348575, 0.65516343, 0.86179687, 0.96478395, 0.5895483 ,
      ...
              0.94448098, 0.99272037, 0.86178282, 0.9713985 , 0.99584906,
              0.9994971 , 0.99602833, 0.75831646, 0.98097317, 0.27269885,
              0.83983706, 0.99739981, 0.99799613, 0.86438327, 0.9852624 ,
              0.98883846, 0.98801427, 0.98070843, 0.9982868 , 0.97446199,
              0.50715056, 0.9753332 , 0.93897421, 0.9840293 , 0.47993539,
              0.72058411, 0.99729776, 0.9273156 , 0.94739903, 0.99941814,
              0.96002297, 0.82707316, 0.98931698, 0.99250036, 0.99718291,
              0.99252517, 0.99660135, 0.75546776, 0.99199394, 0.98009947,
              0.90806618, 0.93599417, 0.82419801, 0.97606636, 0.80064569,
              0.53759074, 0.99478503, 0.85465657, 0.9998393 , 0.95053546,
              0.51378447, 0.99886133, 0.98304279, 0.97866842, 0.99971803,
              0.92907778, 0.91512319, 0.84805715, 0.58285522, 0.50001172,
              0.99555755, 0.98938531, 0.85705324, 0.94725396, 0.9517758 ,
              0.99900443, 0.9808169 , 0.99690317, 0.97822113, 0.98866208,
              0.93904982, 0.9998058 , 0.97589594, 0.94152932, 0.99600365,
              0.98901582, 0.83015874, 0.91387206, 0.45756205, 0.54831006,
              0.94103393, 0.5833964 , 0.99456607, 0.53044656, 0.987455  ,
              0.91150264, 0.96080726, 0.55468377, 0.91854976, 0.99851703,
              0.30861241, 0.87948452, 0.99573683, 0.41376483, 0.86002868,
              0.97953244, 0.99721182, 0.69719863, 0.99915281, 0.99307078]])

    • step_size
      (chain, draw)
      float64
      0.0274 0.0274 ... 0.03957 0.03957

      array([[0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
              0.02739745, 0.02739745, 0.02739745, 0.02739745, 0.02739745,
      ...
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007,
              0.03957007, 0.03957007, 0.03957007, 0.03957007, 0.03957007]])

    • step_size_bar
      (chain, draw)
      float64
      0.03393 0.03393 ... 0.03759 0.03759

      array([[0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
              0.03392959, 0.03392959, 0.03392959, 0.03392959, 0.03392959,
      ...
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964,
              0.03758964, 0.03758964, 0.03758964, 0.03758964, 0.03758964]])

    • reached_max_treedepth
      (chain, draw)
      bool
      False False False ... False False

      array([[False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
      ...
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False]])

    • smallest_eigval
      (chain, draw)
      float64
      nan nan nan nan ... nan nan nan nan

      array([[nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
      ...
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
              nan, nan, nan, nan, nan, nan]])

    • perf_counter_start
      (chain, draw)
      float64
      1.221e+04 1.221e+04 ... 1.264e+04

      array([[12208.22074254, 12208.39809857, 12208.56452055, 12208.76313881,
              12209.01833796, 12209.15159917, 12209.41119466, 12209.66012936,
              12209.9030865 , 12210.13820526, 12210.38297396, 12210.6376415 ,
              12210.76927339, 12211.02832289, 12211.21216375, 12211.55990353,
              12211.73142816, 12211.92916856, 12212.10437014, 12212.19780338,
              12212.37297917, 12212.55495766, 12212.64754581, 12212.82609769,
              12213.00812238, 12213.19108381, 12213.36114985, 12213.45719924,
              12213.63593204, 12213.81878036, 12214.00011622, 12214.18998633,
              12214.37360158, 12214.55733137, 12214.7644338 , 12214.94318371,
              12215.11731262, 12215.29318492, 12215.38531324, 12215.55033162,
              12215.64726051, 12215.79049199, 12215.97037165, 12216.23259802,
              12216.57377636, 12216.67756526, 12216.85031336, 12217.04650853,
              12217.14121535, 12217.32147288, 12217.4999269 , 12217.68847831,
              12217.86166312, 12218.05222994, 12218.23117711, 12218.40898732,
              12218.5821711 , 12218.76781097, 12218.86310827, 12218.96616716,
              12219.13842357, 12219.23396385, 12219.32318365, 12219.5738085 ,
              12219.75526239, 12220.11477885, 12220.20705019, 12220.380527  ,
              12220.55660929, 12220.73493514, 12220.93973841, 12221.14351067,
              12221.61742497, 12221.87454205, 12222.11703173, 12222.35466686,
              12222.48332445, 12222.74246919, 12223.24225742, 12223.45720525,
      ...
              12629.49081582, 12629.6753405 , 12629.77030827, 12629.95783863,
              12630.13539325, 12630.31695531, 12630.40671706, 12630.5930664 ,
              12630.78062381, 12630.96068488, 12631.15927597, 12631.41083424,
              12631.53242673, 12631.79528797, 12632.04157312, 12632.285367  ,
              12632.52093528, 12632.7825802 , 12633.04013878, 12633.30543575,
              12633.54495394, 12633.74031484, 12633.9264658 , 12634.10757053,
              12634.2927135 , 12634.47460138, 12634.65509347, 12634.85887398,
              12634.95367695, 12635.12743522, 12635.30691241, 12635.48556467,
              12635.66741521, 12635.85943899, 12636.04138128, 12636.21706995,
              12636.39374458, 12636.57350951, 12636.76247403, 12636.95881292,
              12637.13862598, 12637.32466282, 12637.42035516, 12637.79506278,
              12637.9833916 , 12638.33757601, 12638.51928637, 12638.6978731 ,
              12638.89804778, 12639.07442597, 12639.25822546, 12639.43464989,
              12639.61184303, 12639.70950561, 12639.90115435, 12640.07436217,
              12640.1696337 , 12640.34877874, 12640.51935254, 12640.70154082,
              12640.8915642 , 12641.08849228, 12641.26573496, 12641.43925708,
              12641.61787229, 12641.79538802, 12641.99578738, 12642.17507941,
              12642.35210809, 12642.52930749, 12642.70984791, 12642.91176665,
              12643.26681201, 12643.44525042, 12643.5565817 , 12643.82186517,
              12643.95896535, 12644.20504086, 12644.449823  , 12644.6892051 ]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Int64Index([0, 1], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Int64Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
                  ...
                  490, 491, 492, 493, 494, 495, 496, 497, 498, 499],
                 dtype='int64', name='draw', length=500))

  • Attributes: (6)

    created_at :

    2024-03-10T00:08:27.941327

    arviz_version :

    0.15.1

    inference_library :

    pymc

    inference_library_version :

    5.10.4

    sampling_time :

    638.5446758270264

    tuning_steps :

    1000

  
  
• observed_data

  <xarray.Dataset>
  Dimensions:       (y_pred_dim_0: 16512)
  Coordinates:
    * y_pred_dim_0  (y_pred_dim_0) int64 0 1 2 3 4 ... 16508 16509 16510 16511
  Data variables:
      y_pred        (y_pred_dim_0) float64 1.33 0.375 0.613 ... 2.641 4.869 2.826
  Attributes:
      created_at:                 2024-03-10T00:08:27.950929
      arviz_version:              0.15.1
      inference_library:          pymc
      inference_library_version:  5.10.4

  xarray.Dataset

  • Dimensions:
    • y_pred_dim_0: 16512
  • Coordinates: (1)
    • y_pred_dim_0
      (y_pred_dim_0)
      int64
      0 1 2 3 ... 16508 16509 16510 16511

      array([    0,     1,     2, ..., 16509, 16510, 16511])

  • Data variables: (1)
    • y_pred
      (y_pred_dim_0)
      float64
      1.33 0.375 0.613 ... 4.869 2.826

      array([1.33 , 0.375, 0.613, ..., 2.641, 4.869, 2.826])

  • Indexes: (1)
    • y_pred_dim_0
      PandasIndex

      PandasIndex(Int64Index([    0,     1,     2,     3,     4,     5,     6,     7,     8,
                      9,
                  ...
                  16502, 16503, 16504, 16505, 16506, 16507, 16508, 16509, 16510,
                  16511],
                 dtype='int64', name='y_pred_dim_0', length=16512))

  • Attributes: (4)

    created_at :

    2024-03-10T00:08:27.950929

    arviz_version :

    0.15.1

    inference_library :

    pymc

    inference_library_version :

    5.10.4

  
  

df_compare = az.compare({"model_l": idata_lin, "model_p2": idata_pol}, ic='waic') #loo is recommended
df_compare

/usr/local/lib/python3.10/dist-packages/arviz/stats/stats.py:1645: UserWarning: For one or more samples the posterior variance of the log predictive densities exceeds 0.4. This could be indication of WAIC starting to fail. 
See http://arxiv.org/abs/1507.04544 for details
  warnings.warn(
/usr/local/lib/python3.10/dist-packages/arviz/stats/stats.py:1645: UserWarning: For one or more samples the posterior variance of the log predictive densities exceeds 0.4. This could be indication of WAIC starting to fail. 
See http://arxiv.org/abs/1507.04544 for details
  warnings.warn(

	rank	elpd_waic	p_waic	elpd_diff	weight	se	dse	warning	scale
model_p2	0	-16378.827016	690.666675	0.000000	0.634616	394.119950	0.000000	True	log
model_l	1	-18016.084000	24.350585	1637.256984	0.365384	144.157057	420.237166	True	log

az.plot_compare(df_compare, insample_dev=False)

<Axes: title={'center': 'Model comparison\nhigher is better'}, xlabel='elpd_waic (log)', ylabel='ranked models'>