PyMC Workflows and Common Patterns

This reference provides standard workflows and patterns for building, validating, and analyzing Bayesian models in PyMC.

Standard Bayesian Workflow

Complete Workflow Template

python

import pymc as pm
import arviz as az
import numpy as np
import matplotlib.pyplot as plt

# 1. PREPARE DATA
# ===============
X = ...  # Predictor variables
y = ...  # Observed outcomes

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

# 2. BUILD MODEL
# ==============
with pm.Model() as model:
    # Define coordinates for named dimensions
    coords = {
        'predictors': ['var1', 'var2', 'var3'],
        'obs_id': np.arange(len(y))
    }

    # 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')

# 3. PRIOR PREDICTIVE CHECK
# ==========================
with model:
    prior_pred = pm.sample_prior_predictive(samples=1000, random_seed=42)

# Visualize prior predictions
az.plot_ppc(prior_pred, group='prior', num_pp_samples=100)
plt.title('Prior Predictive Check')
plt.show()

# 4. FIT MODEL
# ============
with model:
    # Quick VI exploration (optional)
    approx = pm.fit(n=20000, random_seed=42)

    # 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
    )

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

# R-hat and ESS
summary = az.summary(idata)
if (summary['r_hat'] > 1.01).any():
    print("WARNING: Some R-hat values > 1.01, chains may not have converged")

if (summary['ess_bulk'] < 400).any():
    print("WARNING: Some ESS values < 400, consider more samples")

# Check divergences
divergences = idata.sample_stats.diverging.sum().item()
print(f"Number of divergences: {divergences}")

# Trace plots
az.plot_trace(idata, var_names=['alpha', 'beta', 'sigma'])
plt.tight_layout()
plt.show()

# 6. POSTERIOR PREDICTIVE CHECK
# ==============================
with model:
    pm.sample_posterior_predictive(idata, extend_inferencedata=True, random_seed=42)

# Visualize fit
az.plot_ppc(idata, num_pp_samples=100)
plt.title('Posterior Predictive Check')
plt.show()

# 7. ANALYZE RESULTS
# ==================
# Posterior distributions
az.plot_posterior(idata, var_names=['alpha', 'beta', 'sigma'])
plt.tight_layout()
plt.show()

# Forest plot for coefficients
az.plot_forest(idata, var_names=['beta'], combined=True)
plt.title('Coefficient Estimates')
plt.show()

# 8. PREDICTIONS FOR NEW DATA
# ============================
X_new = ...  # New predictor values
X_new_scaled = (X_new - X.mean(axis=0)) / X.std(axis=0)

with model:
    # Update data
    pm.set_data({'X': X_new_scaled})

    # Sample predictions
    post_pred = pm.sample_posterior_predictive(
        idata.posterior,
        var_names=['y_obs'],
        random_seed=42
    )

# 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'])

# 9. SAVE RESULTS
# ===============
idata.to_netcdf('model_results.nc')  # Save for later

Model Building Patterns

Linear Regression

python

with pm.Model() as linear_model:
    # Priors
    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)

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

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

Logistic Regression

python

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

    # Linear predictor
    logit_p = alpha + pm.math.dot(X, beta)

    # Likelihood
    y = pm.Bernoulli('y', logit_p=logit_p, observed=y_obs)

Hierarchical/Multilevel Model

python

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

    mu_beta = pm.Normal('mu_beta', mu=0, sigma=10)
    sigma_beta = pm.HalfNormal('sigma_beta', sigma=1)

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

    beta_offset = pm.Normal('beta_offset', mu=0, sigma=1, dims='group')
    beta = pm.Deterministic('beta', mu_beta + sigma_beta * beta_offset, dims='group')

    # Observation-level model
    mu = alpha[group_idx] + beta[group_idx] * X

    sigma = pm.HalfNormal('sigma', sigma=1)
    y = pm.Normal('y', mu=mu, sigma=sigma, observed=y_obs, dims='obs')

Poisson Regression (Count Data)

python

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

    # Linear predictor on log scale
    log_lambda = alpha + pm.math.dot(X, beta)

    # Likelihood
    y = pm.Poisson('y', mu=pm.math.exp(log_lambda), observed=y_obs)

Time Series (Autoregressive)

python

with pm.Model() as ar_model:
    # Innovation standard deviation
    sigma = pm.HalfNormal('sigma', sigma=1)

    # AR coefficients
    rho = pm.Normal('rho', mu=0, sigma=0.5, shape=ar_order)

    # Initial distribution
    init_dist = pm.Normal.dist(mu=0, sigma=sigma)

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

Mixture Model

python

with pm.Model() as mixture_model:
    # Component weights
    w = pm.Dirichlet('w', a=np.ones(n_components))

    # Component parameters
    mu = pm.Normal('mu', mu=0, sigma=10, shape=n_components)
    sigma = pm.HalfNormal('sigma', sigma=1, shape=n_components)

    # Mixture
    components = [pm.Normal.dist(mu=mu[i], sigma=sigma[i]) for i in range(n_components)]
    y = pm.Mixture('y', w=w, comp_dists=components, observed=y_obs)

Data Preparation Best Practices

Standardization

Standardize continuous predictors for better sampling:

python

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

# Model with scaled data
with pm.Model() as model:
    beta_scaled = pm.Normal('beta_scaled', 0, 1)
    # ... rest of model ...

# Transform back to original scale
beta_original = beta_scaled / X_std
alpha_original = alpha - (beta_scaled * X_mean / X_std).sum()

Handling Missing Data

Treat missing values as parameters:

python

# Identify missing values
missing_idx = np.isnan(X)
X_observed = np.where(missing_idx, 0, X)  # Placeholder

with pm.Model() as model:
    # Prior for missing values
    X_missing = pm.Normal('X_missing', mu=0, sigma=1, shape=missing_idx.sum())

    # Combine observed and imputed
    X_complete = pm.math.switch(missing_idx.flatten(), X_missing, X_observed.flatten())

    # ... rest of model using X_complete ...

Centering and Scaling

For regression models, center predictors and outcome:

python

# Center
X_centered = X - X.mean(axis=0)
y_centered = y - y.mean()

with pm.Model() as model:
    # Simpler prior on intercept
    alpha = pm.Normal('alpha', mu=0, sigma=1)  # Intercept near 0 when centered
    beta = pm.Normal('beta', mu=0, sigma=1, shape=n_predictors)

    mu = alpha + pm.math.dot(X_centered, beta)
    sigma = pm.HalfNormal('sigma', sigma=1)

    y_obs = pm.Normal('y_obs', mu=mu, sigma=sigma, observed=y_centered)

Prior Selection Guidelines

Weakly Informative Priors

Use when you have limited prior knowledge:

python

# For standardized predictors
beta = pm.Normal('beta', mu=0, sigma=1)

# For scale parameters
sigma = pm.HalfNormal('sigma', sigma=1)

# For probabilities
p = pm.Beta('p', alpha=2, beta=2)  # Slight preference for middle values

Informative Priors

Use domain knowledge:

python

# Effect size from literature: Cohen's d ≈ 0.3
beta = pm.Normal('beta', mu=0.3, sigma=0.1)

# Physical constraint: probability between 0.7-0.9
p = pm.Beta('p', alpha=8, beta=2)  # Check with prior predictive!

Prior Predictive Checks

Always validate priors:

python

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

# Check if predictions are reasonable
print(f"Prior predictive range: {prior_pred.prior_predictive['y'].min():.2f} to {prior_pred.prior_predictive['y'].max():.2f}")
print(f"Observed range: {y_obs.min():.2f} to {y_obs.max():.2f}")

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

Model Comparison Workflow

Comparing Multiple Models

python

import arviz as az

# Fit multiple models
models = {}
idatas = {}

# Model 1: Simple linear
with pm.Model() as models['linear']:
    # ... define model ...
    idatas['linear'] = pm.sample(idata_kwargs={'log_likelihood': True})

# Model 2: With interaction
with pm.Model() as models['interaction']:
    # ... define model ...
    idatas['interaction'] = pm.sample(idata_kwargs={'log_likelihood': True})

# Model 3: Hierarchical
with pm.Model() as models['hierarchical']:
    # ... define model ...
    idatas['hierarchical'] = pm.sample(idata_kwargs={'log_likelihood': True})

# Compare using LOO
comparison = az.compare(idatas, ic='loo')
print(comparison)

# Visualize comparison
az.plot_compare(comparison)
plt.show()

# Check LOO reliability
for name, idata in idatas.items():
    loo = az.loo(idata, pointwise=True)
    high_pareto_k = (loo.pareto_k > 0.7).sum().item()
    if high_pareto_k > 0:
        print(f"Warning: {name} has {high_pareto_k} observations with high Pareto-k")

Model Weights

python

# Get model weights (pseudo-BMA)
weights = comparison['weight'].values

print("Model probabilities:")
for name, weight in zip(comparison.index, weights):
    print(f"  {name}: {weight:.2%}")

# Model averaging (weighted predictions)
def weighted_predictions(idatas, weights):
    preds = []
    for (name, idata), weight in zip(idatas.items(), weights):
        pred = idata.posterior_predictive['y_obs'].mean(dim=['chain', 'draw'])
        preds.append(weight * pred)
    return sum(preds)

averaged_pred = weighted_predictions(idatas, weights)

Diagnostics and Troubleshooting

Diagnosing Sampling Problems

python

def diagnose_sampling(idata, var_names=None):
    """Comprehensive sampling diagnostics"""

    # Check convergence
    summary = az.summary(idata, var_names=var_names)

    print("=== Convergence Diagnostics ===")
    bad_rhat = summary[summary['r_hat'] > 1.01]
    if len(bad_rhat) > 0:
        print(f"⚠️  {len(bad_rhat)} variables with R-hat > 1.01")
        print(bad_rhat[['r_hat']])
    else:
        print("✓ All R-hat values < 1.01")

    # Check effective sample size
    print("\n=== Effective Sample Size ===")
    low_ess = summary[summary['ess_bulk'] < 400]
    if len(low_ess) > 0:
        print(f"⚠️  {len(low_ess)} variables with ESS < 400")
        print(low_ess[['ess_bulk', 'ess_tail']])
    else:
        print("✓ All ESS values > 400")

    # Check divergences
    print("\n=== Divergences ===")
    divergences = idata.sample_stats.diverging.sum().item()
    if divergences > 0:
        print(f"⚠️  {divergences} divergent transitions")
        print("   Consider: increase target_accept, reparameterize, or stronger priors")
    else:
        print("✓ No divergences")

    # Check tree depth
    print("\n=== NUTS Statistics ===")
    max_treedepth = idata.sample_stats.tree_depth.max().item()
    hits_max = (idata.sample_stats.tree_depth == max_treedepth).sum().item()
    if hits_max > 0:
        print(f"⚠️  Hit max treedepth {hits_max} times")
        print("   Consider: reparameterize or increase max_treedepth")
    else:
        print(f"✓ No max treedepth issues (max: {max_treedepth})")

    return summary

# Usage
diagnose_sampling(idata, var_names=['alpha', 'beta', 'sigma'])

Common Fixes

Problem	Solution
Divergences	Increase `target_accept=0.95`, use non-centered parameterization
Low ESS	Sample more draws, reparameterize to reduce correlation
High R-hat	Run longer chains, check for multimodality, improve initialization
Slow sampling	Use ADVI initialization, reparameterize, reduce model complexity
Biased posterior	Check prior predictive, ensure likelihood is correct

Using Named Dimensions (dims)

Benefits of dims

More readable code
Easier subsetting and analysis
Better xarray integration

python

# Define coordinates
coords = {
    'predictors': ['age', 'income', 'education'],
    'groups': ['A', 'B', 'C'],
    'time': pd.date_range('2020-01-01', periods=100, freq='D')
}

with pm.Model(coords=coords) as model:
    # Use dims instead of shape
    beta = pm.Normal('beta', mu=0, sigma=1, dims='predictors')
    alpha = pm.Normal('alpha', mu=0, sigma=1, dims='groups')
    y = pm.Normal('y', mu=0, sigma=1, dims=['groups', 'time'], observed=data)

# After sampling, dimensions are preserved
idata = pm.sample()

# Easy subsetting
beta_age = idata.posterior['beta'].sel(predictors='age')
group_A = idata.posterior['alpha'].sel(groups='A')

Saving and Loading Results

python

# Save InferenceData
idata.to_netcdf('results.nc')

# Load InferenceData
loaded_idata = az.from_netcdf('results.nc')

# Save model for later predictions
import pickle

with open('model.pkl', 'wb') as f:
    pickle.dump({'model': model, 'idata': idata}, f)

# Load model
with open('model.pkl', 'rb') as f:
    saved = pickle.load(f)
    model = saved['model']
    idata = saved['idata']