PyMC Modeling

Bayesian modeling workflow for PyMC v5+ with modern API patterns.

Notebook preference: Use marimo for interactive modeling unless the project already uses Jupyter.

Model Specification

Basic Structure

import pymc as pm
import arviz as az

with pm.Model(coords=coords) as model:
    # Data containers (for out-of-sample prediction)
    x = pm.Data("x", x_obs, dims="obs")

    # Priors
    beta = pm.Normal("beta", mu=0, sigma=1, dims="features")
    sigma = pm.HalfNormal("sigma", sigma=1)

    # Likelihood
    mu = pm.math.dot(x, beta)
    y = pm.Normal("y", mu=mu, sigma=sigma, observed=y_obs, dims="obs")

    # Inference
    idata = pm.sample()

Coords and Dims

Use coords/dims for interpretable InferenceData when model has meaningful structure:

coords = {
    "obs": np.arange(n_obs),
    "features": ["intercept", "age", "income"],
    "group": group_labels,
}

Skip for simple models where overhead exceeds benefit.

Parameterization

Prefer non-centered parameterization for hierarchical models with weak data:

# Non-centered (better for divergences)
offset = pm.Normal("offset", 0, 1, dims="group")
alpha = mu_alpha + sigma_alpha * offset

# Centered (better with strong data)
alpha = pm.Normal("alpha", mu_alpha, sigma_alpha, dims="group")

Inference

Default Sampling (nutpie)

Use nutpie as the default sampler—it's Rust-based and typically 2-5x faster:

with model:
    idata = pm.sample(
        draws=1000, tune=1000, chains=4,
        nuts_sampler="nutpie",
        random_seed=42,
    )

PyMC Native Sampling

Fall back to PyMC's NUTS when nutpie unavailable:

with model:
    idata = pm.sample(draws=1000, tune=1000, chains=4, random_seed=42)

Alternative MCMC Backends

See references/inference.md for:

• NumPyro/JAX: GPU acceleration, vectorized chains

Approximate Inference

For fast (but inexact) posterior approximations:

• ADVI/DADVI: Variational inference with Gaussian approximation
• Pathfinder: Quasi-Newton optimization for initialization or screening

Diagnostics and ArviZ Workflow

Follow this systematic workflow after every sampling run:

Phase 1: Immediate Checks (Required)

# 1. Check for divergences (must be 0 or near 0)
n_div = idata.sample_stats["diverging"].sum().item()
print(f"Divergences: {n_div}")

# 2. Summary with convergence diagnostics
summary = az.summary(idata, var_names=["~offset"])  # exclude auxiliary
print(summary[["mean", "sd", "hdi_3%", "hdi_97%", "ess_bulk", "ess_tail", "r_hat"]])

# 3. Visual convergence check
az.plot_trace(idata, compact=True)
az.plot_rank(idata, var_names=["beta", "sigma"])

Pass criteria (all must pass before proceeding):

• Zero divergences (or < 0.1% and randomly scattered)
• r_hat < 1.01 for all parameters
• ess_bulk > 400 and ess_tail > 400
• Trace plots show good mixing (overlapping densities, fuzzy caterpillar)

Phase 2: Deep Convergence (If Phase 1 marginal)

# ESS evolution (should grow linearly)
az.plot_ess(idata, kind="evolution")

# Energy diagnostic (HMC health)
az.plot_energy(idata)

# Autocorrelation (should decay rapidly)
az.plot_autocorr(idata, var_names=["beta"])

Phase 3: Model Criticism (Required)

# Generate posterior predictive
with model:
    pm.sample_posterior_predictive(idata, extend_inferencedata=True)

# Does the model capture the data?
az.plot_ppc(idata, kind="cumulative")

# Calibration check
az.plot_loo_pit(idata, y="y")

Critical rule: Never interpret parameters until Phases 1-3 pass.

Phase 4: Parameter Interpretation

# Posterior summaries
az.plot_posterior(idata, var_names=["beta"], ref_val=0)

# Forest plots for hierarchical parameters
az.plot_forest(idata, var_names=["alpha"], combined=True)

# Parameter correlations (identify non-identifiability)
az.plot_pair(idata, var_names=["alpha", "beta", "sigma"])

See references/arviz.md for comprehensive ArviZ usage. See references/diagnostics.md for troubleshooting.

Prior and Posterior Predictive Checks

Prior Predictive (Before Fitting)

Always check prior implications before fitting:

with model:
    prior_pred = pm.sample_prior_predictive(draws=500)

# Do prior predictions span reasonable outcome range?
az.plot_ppc(prior_pred, group="prior", kind="cumulative")

# Numerical sanity check
prior_y = prior_pred.prior_predictive["y"].values.flatten()
print(f"Prior predictive range: [{prior_y.min():.1f}, {prior_y.max():.1f}]")

Warning signs: Prior predictive covers implausible values (negative counts, probabilities > 1) or is extremely wide/narrow.

Posterior Predictive (After Fitting)

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

# Density comparison
az.plot_ppc(idata, kind="kde")

# Cumulative (better for systematic deviations)
az.plot_ppc(idata, kind="cumulative")

# Calibration diagnostic
az.plot_loo_pit(idata, y="y")

Interpretation: Observed data (dark line) should fall within posterior predictive distribution (light lines). See references/arviz.md for detailed interpretation.

Model Debugging

Inspecting Model Structure

# Print model summary (variables, shapes, distributions)
print(model)

# Visualize model as directed graph
pm.model_to_graphviz(model)

Checking for Specification Errors

Before sampling, validate the model:

# Debug model: checks for common issues
model.debug()

# Check initial point log-probabilities
# Identifies which variables have invalid starting values
model.point_logps()

Common Issues

Symptom	Likely Cause	Fix
ValueError: Shape mismatch	Parameter vs observation dimensions	Use index vectors: alpha[group_idx]
Initial evaluation failed	Data outside distribution support	Check bounds; use init="adapt_diag"
Mass matrix contains zeros	Unscaled predictors or flat priors	Standardize features; use weakly informative priors
High divergence count	Funnel geometry	Non-centered parameterization
NaN in log-probability	Invalid parameter combinations	Check parameter constraints, add bounds
-inf log-probability	Observations outside likelihood support	Verify data matches distribution domain
Slow discrete sampling	NUTS incompatible with discrete	Marginalize discrete variables
Poor out-of-sample prediction	Static data arrays	Use pm.Data containers with mutable=True

See references/troubleshooting.md for comprehensive problem-solution guide.

Debugging Divergences

# Identify where divergences occur in parameter space
az.plot_pair(idata, var_names=["alpha", "beta", "sigma"], divergences=True)

# Check if divergences cluster in specific regions
# Clustering suggests parameterization or prior issues

Profiling Slow Models

# Time individual operations in the log-probability computation
profile = model.profile(model.logp())
profile.summary()

# Identify bottlenecks in gradient computation
import pytensor
grad_profile = model.profile(pytensor.grad(model.logp(), model.continuous_value_vars))
grad_profile.summary()

See references/gotchas.md for additional troubleshooting.

Model Comparison

LOO-CV (Preferred)

# Compute LOO with pointwise diagnostics
loo = az.loo(idata, pointwise=True)
print(f"ELPD: {loo.elpd_loo:.1f} ± {loo.se:.1f}")

# Check Pareto k values (must be < 0.7 for reliable LOO)
print(f"Bad k (>0.7): {(loo.pareto_k > 0.7).sum().item()}")
az.plot_khat(idata)

Comparing Models

comparison = az.compare({
    "model_a": idata_a,
    "model_b": idata_b,
}, ic="loo")

print(comparison[["rank", "elpd_loo", "d_loo", "weight", "dse"]])
az.plot_compare(comparison)

Decision rule: If d_loo < 2*dse, models are effectively equivalent.

See references/arviz.md for detailed model comparison workflow.

Saving and Loading Results

InferenceData Persistence

Save sampling results for later analysis or sharing:

# Save to NetCDF (recommended format)
idata.to_netcdf("results/model_v1.nc")

# Load
idata = az.from_netcdf("results/model_v1.nc")

Compressed Storage

For large InferenceData objects (many draws, large posterior predictive):

# Compress with zlib (reduces file size 50-80%)
idata.to_netcdf(
    "results/model_v1.nc",
    engine="h5netcdf",
    encoding={var: {"zlib": True, "complevel": 4}
              for group in ["posterior", "posterior_predictive"]
              if hasattr(idata, group)
              for var in getattr(idata, group).data_vars}
)

What Gets Saved

InferenceData preserves the full Bayesian workflow:

• posterior: Parameter samples from MCMC
• prior, prior_predictive: Prior samples (if generated)
• posterior_predictive: Predictions (if generated)
• observed_data, constant_data: Data used in fitting
• sample_stats: Diagnostics (divergences, tree depth, energy)
• log_likelihood: Pointwise log-likelihood (for LOO-CV)
• All coordinates and dimensions

Workflow Pattern

# Save after each major step
with model:
    idata = pm.sample(nuts_sampler="nutpie")
idata.to_netcdf("results/step1_posterior.nc")

with model:
    pm.sample_posterior_predictive(idata, extend_inferencedata=True)
idata.to_netcdf("results/step2_with_ppc.nc")

# Resume later
idata = az.from_netcdf("results/step2_with_ppc.nc")
az.plot_ppc(idata)  # Continue analysis

Prior Selection

See references/priors.md for:

• Weakly informative defaults by distribution type
• Prior predictive checking workflow
• Domain-specific recommendations

Common Patterns

Hierarchical/Multilevel

with pm.Model(coords={"group": groups, "obs": obs_idx}) as hierarchical:
    # Hyperpriors
    mu_alpha = pm.Normal("mu_alpha", 0, 1)
    sigma_alpha = pm.HalfNormal("sigma_alpha", 1)

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

    # Likelihood
    y = pm.Normal("y", alpha[group_idx], sigma, observed=y_obs, dims="obs")

GLMs

# Logistic regression
with pm.Model() as logistic:
    alpha = pm.Normal("alpha", 0, 2.5)  # intercept
    beta = pm.Normal("beta", 0, 2.5, dims="features")
    
    # Logit link
    logit_p = alpha + pm.math.dot(X, beta)
    p = pm.math.sigmoid(logit_p)
    
    y = pm.Bernoulli("y", p=p, observed=y_obs)

# Poisson regression
with pm.Model() as poisson:
    beta = pm.Normal("beta", 0, 1, dims="features")
    mu = pm.math.exp(pm.math.dot(X, beta))
    y = pm.Poisson("y", mu=mu, observed=y_obs)

Gaussian Processes

Default to HSGP for most GP problems (n > 500, 1-3D inputs). It's O(nm) instead of O(n³):

with pm.Model() as gp_model:
    # Hyperparameters
    ell = pm.InverseGamma("ell", alpha=5, beta=5)
    eta = pm.HalfNormal("eta", sigma=2)
    sigma = pm.HalfNormal("sigma", sigma=0.5)

    # Covariance function (Matern52 recommended)
    cov = eta**2 * pm.gp.cov.Matern52(1, ls=ell)

    # HSGP approximation
    gp = pm.gp.HSGP(m=[20], c=1.5, cov_func=cov)
    f = gp.prior("f", X=X[:, None])  # X must be 2D

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

For periodic patterns, use pm.gp.HSGPPeriodic. For small datasets (n < 500), use pm.gp.Marginal or pm.gp.Latent.

See references/gp.md for:

• HSGP parameter selection (choosing m and c, automatic heuristics)
• HSGPPeriodic for seasonal/cyclic patterns
• Approximation quality diagnostics
• Covariance functions and priors
• Common patterns (trend + seasonality, classification, heteroscedastic)

Time Series

with pm.Model(coords={"time": range(T)}) as ar_model:
    rho = pm.Uniform("rho", -1, 1)
    sigma = pm.HalfNormal("sigma", sigma=1)

    y = pm.AR("y", rho=[rho], sigma=sigma, constant=True,
              observed=y_obs, dims="time")

See references/timeseries.md for:

• Autoregressive models (AR, ARMA)
• Random walk and local level models
• Structural time series (trend + seasonality)
• State space models
• GPs for time series
• Handling multiple seasonalities
• Forecasting patterns

BART (Bayesian Additive Regression Trees)

import pymc_bart as pmb

with pm.Model() as bart_model:
    mu = pmb.BART("mu", X=X, Y=y, m=50)
    sigma = pm.HalfNormal("sigma", 1)
    y_obs = pm.Normal("y_obs", mu=mu, sigma=sigma, observed=y)

See references/bart.md for:

• Regression and classification
• Variable importance and partial dependence
• Combining BART with parametric components
• Configuration (number of trees, depth priors)

Mixture Models

import numpy as np

coords = {"component": range(K)}

with pm.Model(coords=coords) as gmm:
    # Mixture weights
    w = pm.Dirichlet("w", a=np.ones(K), dims="component")

    # Component parameters (with ordering to avoid label switching)
    mu = pm.Normal("mu", mu=0, sigma=10, dims="component",
                   transform=pm.distributions.transforms.ordered)
    sigma = pm.HalfNormal("sigma", sigma=2, dims="component")

    # Mixture likelihood
    y = pm.NormalMixture("y", w=w, mu=mu, sigma=sigma, observed=y_obs)

See references/mixtures.md for:

• Finite mixture models and mixture of regressions
• Label switching problem and solutions (ordering constraints, relabeling)
• Marginalized mixtures (pymc-extras)
• Diagnostics for mixture models

Specialized Likelihoods

# Zero-Inflated Poisson (excess zeros)
with pm.Model() as zip_model:
    psi = pm.Beta("psi", alpha=2, beta=2)  # P(structural zero)
    mu = pm.Exponential("mu", lam=1)
    y = pm.ZeroInflatedPoisson("y", psi=psi, mu=mu, observed=y_obs)

# Censored data (e.g., right-censored survival)
with pm.Model() as censored_model:
    mu = pm.Normal("mu", mu=0, sigma=10)
    sigma = pm.HalfNormal("sigma", sigma=5)
    y = pm.Censored("y", dist=pm.Normal.dist(mu=mu, sigma=sigma),
                    lower=None, upper=censoring_time, observed=y_obs)

# Ordinal regression
with pm.Model() as ordinal:
    beta = pm.Normal("beta", mu=0, sigma=2, dims="features")
    cutpoints = pm.Normal("cutpoints", mu=0, sigma=2,
                          transform=pm.distributions.transforms.ordered,
                          shape=n_categories - 1)
    y = pm.OrderedLogistic("y", eta=pm.math.dot(X, beta),
                           cutpoints=cutpoints, observed=y_obs)

See references/specialized_likelihoods.md for:

• Zero-inflated models (Poisson, Negative Binomial, Binomial)
• Hurdle models for count data
• Censored and truncated data
• Ordinal regression
• Robust regression with Student-t likelihood

Common Pitfalls

See references/gotchas.md for:

• Centered vs non-centered parameterization
• Priors on scale parameters
• Label switching in mixtures
• Performance issues (GPs, large Deterministics)
• Python conditionals and hard clipping
• Redundant intercepts in hierarchical models

See references/troubleshooting.md for comprehensive problem-solution guide covering:

• Shape and dimension errors
• Initialization failures
• Mass matrix and numerical issues
• Discrete variable challenges
• Data container and prediction issues

Causal Inference Operations

pm.do (Interventions)

Apply do-calculus interventions to set variables to fixed values:

with pm.Model() as causal_model:
    x = pm.Normal("x", 0, 1)
    y = pm.Normal("y", x, 1)
    z = pm.Normal("z", y, 1)

# Intervene: set x = 2 (breaks incoming edges to x)
with pm.do(causal_model, {"x": 2}) as intervention_model:
    idata = pm.sample_prior_predictive()
    # Samples from P(y, z | do(x=2))

pm.observe (Conditioning)

Condition on observed values without intervention:

# Condition: observe y = 1 (doesn't break causal structure)
with pm.observe(causal_model, {"y": 1}) as conditioned_model:
    idata = pm.sample()
    # Samples from P(x, z | y=1)

Combining do and observe

# Intervention + observation for causal queries
with pm.do(causal_model, {"x": 2}) as m1:
    with pm.observe(m1, {"z": 0}) as m2:
        idata = pm.sample()
        # P(y | do(x=2), z=0)

pymc-extras

For specialized models and inference:

import pymc_extras as pmx

# Marginalize discrete parameters from a model
model = pmx.marginalize(model, ["discrete_var"])

# R2D2 prior for regression (requires output_sigma and input_sigma)
residual_sigma, beta = pmx.R2D2M2CP(
    "r2d2",
    output_sigma=y.std(),
    input_sigma=X.std(axis=0),
    dims="features",
    r2=0.5,
)

# Laplace approximation for fast inference
idata = pmx.fit_laplace(model)

Custom Distributions and Model Components

For extending PyMC beyond built-in distributions:

import pymc as pm
import pytensor.tensor as pt

# Custom likelihood via DensityDist
def custom_logp(value, mu, sigma):
    return pm.logp(pm.Normal.dist(mu=mu, sigma=sigma), value)

with pm.Model() as model:
    mu = pm.Normal("mu", 0, 1)
    y = pm.DensityDist("y", mu, 1.0, logp=custom_logp, observed=y_obs)

# Soft constraints via Potential
with pm.Model() as model:
    alpha = pm.Normal("alpha", 0, 1, dims="group")
    pm.Potential("sum_to_zero", -100 * pt.sqr(alpha.sum()))

See references/custom_models.md for:

• pm.DensityDist for custom likelihoods
• pm.Potential for soft constraints and Jacobian adjustments
• pm.Simulator for simulation-based inference (ABC)
• pm.CustomDist for custom prior distributions