PyMC Bayesian Modeling

Overview

PyMC is a Python library for Bayesian modeling and probabilistic programming. Build, fit, validate, and compare Bayesian models using PyMC's modern API (version 5.x+), including hierarchical models, MCMC sampling (NUTS), variational inference, and model comparison (LOO, WAIC).

When to Use This Skill

This skill should be used when:

• Building Bayesian models (linear/logistic regression, hierarchical models, time series, etc.)
• Performing MCMC sampling or variational inference
• Conducting prior/posterior predictive checks
• Diagnosing sampling issues (divergences, convergence, ESS)
• Comparing multiple models using information criteria (LOO, WAIC)
• Implementing uncertainty quantification through Bayesian methods
• Working with hierarchical/multilevel data structures
• Handling missing data or measurement error in a principled way

Standard Bayesian Workflow

Follow this workflow for building and validating Bayesian models:

1. Data Preparation

import pymc as pm
import arviz as az
import numpy as np

# Load and prepare data
X = ...  # Predictors
y = ...  # Outcomes

# Standardize predictors for better sampling
X_mean = X.mean(axis=0)
X_std = X.std(axis=0)
X_scaled = (X - X_mean) / X_std

Key practices:

• Standardize continuous predictors (improves sampling efficiency)
• Center outcomes when possible
• Handle missing data explicitly (treat as parameters)
• Use named dimensions with coords for clarity

2. Model Building

coords = {
    'predictors': ['var1', 'var2', 'var3'],
    'obs_id': np.arange(len(y))
}

with pm.Model(coords=coords) as model:
    # Priors
    alpha = pm.Normal('alpha', mu=0, sigma=1)
    beta = pm.Normal('beta', mu=0, sigma=1, dims='predictors')
    sigma = pm.HalfNormal('sigma', sigma=1)

    # Linear predictor
    mu = alpha + pm.math.dot(X_scaled, beta)

    # Likelihood
    y_obs = pm.Normal('y_obs', mu=mu, sigma=sigma, observed=y, dims='obs_id')

Key practices:

• Use weakly informative priors (not flat priors)
• Use HalfNormal or Exponential for scale parameters
• Use named dimensions (dims) instead of shape when possible
• Use pm.Data() for values that will be updated for predictions

3. Prior Predictive Check

Always validate priors before fitting:

with model:
    prior_pred = pm.sample_prior_predictive(samples=1000, random_seed=42)

# Visualize
az.plot_ppc(prior_pred, group='prior')

Check:

• Do prior predictions span reasonable values?
• Are extreme values plausible given domain knowledge?
• If priors generate implausible data, adjust and re-check

4. Fit Model

with model:
    # Optional: Quick exploration with ADVI
    # approx = pm.fit(n=20000)

    # Full MCMC inference
    idata = pm.sample(
        draws=2000,
        tune=1000,
        chains=4,
        target_accept=0.9,
        random_seed=42,
        idata_kwargs={'log_likelihood': True}  # For model comparison
    )

Key parameters:

• draws=2000: Number of samples per chain
• tune=1000: Warmup samples (discarded)
• chains=4: Run 4 chains for convergence checking
• target_accept=0.9: Higher for difficult posteriors (0.95-0.99)
• Include log_likelihood=True for model comparison

5. Check Diagnostics

Use the diagnostic script:

from scripts.model_diagnostics import check_diagnostics

results = check_diagnostics(idata, var_names=['alpha', 'beta', 'sigma'])

Check:

• R-hat < 1.01: Chains have converged
• ESS > 400: Sufficient effective samples
• No divergences: NUTS sampled successfully
• Trace plots: Chains should mix well (fuzzy caterpillar)

If issues arise:

• Divergences → Increase target_accept=0.95, use non-centered parameterization
• Low ESS → Sample more draws, reparameterize to reduce correlation
• High R-hat → Run longer, check for multimodality

6. Posterior Predictive Check

Validate model fit:

with model:
    pm.sample_posterior_predictive(idata, extend_inferencedata=True, random_seed=42)

# Visualize
az.plot_ppc(idata)

Check:

• Do posterior predictions capture observed data patterns?
• Are systematic deviations evident (model misspecification)?
• Consider alternative models if fit is poor

7. Analyze Results

# Summary statistics
print(az.summary(idata, var_names=['alpha', 'beta', 'sigma']))

# Posterior distributions
az.plot_posterior(idata, var_names=['alpha', 'beta', 'sigma'])

# Coefficient estimates
az.plot_forest(idata, var_names=['beta'], combined=True)

8. Make Predictions

X_new = ...  # New predictor values
X_new_scaled = (X_new - X_mean) / X_std

with model:
    pm.set_data({'X_scaled': X_new_scaled})
    post_pred = pm.sample_posterior_predictive(
        idata.posterior,
        var_names=['y_obs'],
        random_seed=42
    )

# Extract prediction intervals
y_pred_mean = post_pred.posterior_predictive['y_obs'].mean(dim=['chain', 'draw'])
y_pred_hdi = az.hdi(post_pred.posterior_predictive, var_names=['y_obs'])

Common Model Patterns

Linear Regression

For continuous outcomes with linear relationships:

with pm.Model() as linear_model:
    alpha = pm.Normal('alpha', mu=0, sigma=10)
    beta = pm.Normal('beta', mu=0, sigma=10, shape=n_predictors)
    sigma = pm.HalfNormal('sigma', sigma=1)

    mu = alpha + pm.math.dot(X, beta)
    y = pm.Normal('y', mu=mu, sigma=sigma, observed=y_obs)

Use template: assets/linear_regression_template.py

Logistic Regression

For binary outcomes:

with pm.Model() as logistic_model:
    alpha = pm.Normal('alpha', mu=0, sigma=10)
    beta = pm.Normal('beta', mu=0, sigma=10, shape=n_predictors)

    logit_p = alpha + pm.math.dot(X, beta)
    y = pm.Bernoulli('y', logit_p=logit_p, observed=y_obs)

Hierarchical Models

For grouped data (use non-centered parameterization):

with pm.Model(coords={'groups': group_names}) as hierarchical_model:
    # Hyperpriors
    mu_alpha = pm.Normal('mu_alpha', mu=0, sigma=10)
    sigma_alpha = pm.HalfNormal('sigma_alpha', sigma=1)

    # Group-level (non-centered)
    alpha_offset = pm.Normal('alpha_offset', mu=0, sigma=1, dims='groups')
    alpha = pm.Deterministic('alpha', mu_alpha + sigma_alpha * alpha_offset, dims='groups')

    # Observation-level
    mu = alpha[group_idx]
    sigma = pm.HalfNormal('sigma', sigma=1)
    y = pm.Normal('y', mu=mu, sigma=sigma, observed=y_obs)

Use template: assets/hierarchical_model_template.py Critical: Always use non-centered parameterization for hierarchical models to avoid divergences.

Poisson Regression

For count data:

with pm.Model() as poisson_model:
    alpha = pm.Normal('alpha', mu=0, sigma=10)
    beta = pm.Normal('beta', mu=0, sigma=10, shape=n_predictors)

    log_lambda = alpha + pm.math.dot(X, beta)
    y = pm.Poisson('y', mu=pm.math.exp(log_lambda), observed=y_obs)

For overdispersed counts, use NegativeBinomial instead.

Time Series

For autoregressive processes:

with pm.Model() as ar_model:
    sigma = pm.HalfNormal('sigma', sigma=1)
    rho = pm.Normal('rho', mu=0, sigma=0.5, shape=ar_order)
    init_dist = pm.Normal.dist(mu=0, sigma=sigma)

    y = pm.AR('y', rho=rho, sigma=sigma, init_dist=init_dist, observed=y_obs)

Model Comparison

Comparing Models

Use LOO or WAIC for model comparison:

from scripts.model_comparison import compare_models, check_loo_reliability

# Fit models with log_likelihood
models = {
    'Model1': idata1,
    'Model2': idata2,
    'Model3': idata3
}

# Compare using LOO
comparison = compare_models(models, ic='loo')

# Check reliability
check_loo_reliability(models)

Interpretation:

• Δloo < 2: Models are similar, choose simpler model
• 2 < Δloo < 4: Weak evidence for better model
• 4 < Δloo < 10: Moderate evidence
• Δloo > 10: Strong evidence for better model

Check Pareto-k values:

• k < 0.7: LOO reliable
• k > 0.7: Consider WAIC or k-fold CV

Model Averaging

When models are similar, average predictions:

from scripts.model_comparison import model_averaging

averaged_pred, weights = model_averaging(models, var_name='y_obs')

Distribution Selection Guide

For Priors

Scale parameters (σ, τ):

• pm.HalfNormal('sigma', sigma=1) - Default choice
• pm.Exponential('sigma', lam=1) - Alternative
• pm.Gamma('sigma', alpha=2, beta=1) - More informative

Unbounded parameters:

• pm.Normal('theta', mu=0, sigma=1) - For standardized data
• pm.StudentT('theta', nu=3, mu=0, sigma=1) - Robust to outliers

Positive parameters:

• pm.LogNormal('theta', mu=0, sigma=1)
• pm.Gamma('theta', alpha=2, beta=1)

Probabilities:

• pm.Beta('p', alpha=2, beta=2) - Weakly informative
• pm.Uniform('p', lower=0, upper=1) - Non-informative (use sparingly)

Correlation matrices:

• pm.LKJCorr('corr', n=n_vars, eta=2) - eta=1 uniform, eta>1 prefers identity

For Likelihoods

Continuous outcomes:

• pm.Normal('y', mu=mu, sigma=sigma) - Default for continuous data
• pm.StudentT('y', nu=nu, mu=mu, sigma=sigma) - Robust to outliers

Count data:

• pm.Poisson('y', mu=lambda) - Equidispersed counts
• pm.NegativeBinomial('y', mu=mu, alpha=alpha) - Overdispersed counts
• pm.ZeroInflatedPoisson('y', psi=psi, mu=mu) - Excess zeros

Binary outcomes:

• pm.Bernoulli('y', p=p) or pm.Bernoulli('y', logit_p=logit_p)

Categorical outcomes:

• pm.Categorical('y', p=probs)

See: references/distributions.md for comprehensive distribution reference

Sampling and Inference

MCMC with NUTS

Default and recommended for most models:

idata = pm.sample(
    draws=2000,
    tune=1000,
    chains=4,
    target_accept=0.9,
    random_seed=42
)

Adjust when needed:

• Divergences → target_accept=0.95 or higher
• Slow sampling → Use ADVI for initialization
• Discrete parameters → Use pm.Metropolis() for discrete vars

Variational Inference

Fast approximation for exploration or initialization:

with model:
    approx = pm.fit(n=20000, method='advi')

    # Use for initialization
    start = approx.sample(return_inferencedata=False)[0]
    idata = pm.sample(start=start)

Trade-offs:

• Much faster than MCMC
• Approximate (may underestimate uncertainty)
• Good for large models or quick exploration

See: references/sampling_inference.md for detailed sampling guide

Diagnostic Scripts

Comprehensive Diagnostics

from scripts.model_diagnostics import create_diagnostic_report

create_diagnostic_report(
    idata,
    var_names=['alpha', 'beta', 'sigma'],
    output_dir='diagnostics/'
)

Creates:

• Trace plots
• Rank plots (mixing check)
• Autocorrelation plots
• Energy plots
• ESS evolution
• Summary statistics CSV

Quick Diagnostic Check

from scripts.model_diagnostics import check_diagnostics

results = check_diagnostics(idata)

Checks R-hat, ESS, divergences, and tree depth.

Common Issues and Solutions

Divergences

Symptom: idata.sample_stats.diverging.sum() > 0 Solutions:

• Increase target_accept=0.95 or 0.99
• Use non-centered parameterization (hierarchical models)
• Add stronger priors to constrain parameters
• Check for model misspecification

Low Effective Sample Size

Symptom: ESS < 400 Solutions:

• Sample more draws: draws=5000
• Reparameterize to reduce posterior correlation
• Use QR decomposition for regression with correlated predictors

High R-hat

Symptom: R-hat > 1.01 Solutions:

• Run longer chains: tune=2000, draws=5000
• Check for multimodality
• Improve initialization with ADVI

Slow Sampling

Solutions:

• Use ADVI initialization
• Reduce model complexity
• Increase parallelization: cores=8, chains=8
• Use variational inference if appropriate

Best Practices

Mo