PyMC Modeling

Modern Bayesian modeling with PyMC v5+. Key defaults: nutpie sampler (2-5x faster), non-centered parameterization for hierarchical models, HSGP over exact GPs, coords/dims for readable InferenceData, and save-early workflow to prevent data loss from late crashes.

Modeling strategy: Build models iteratively — start simple, check prior predictions, fit and diagnose, check posterior predictions, expand one piece at a time. See references/workflow.md for the full workflow.

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(nuts_sampler="nutpie", random_seed=42)

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 preferred)

with model:
    idata = pm.sample(
        draws=1000, tune=1000, chains=4,
        nuts_sampler="nutpie",
        random_seed=42,
    )
idata.to_netcdf("results.nc")  # Save immediately after sampling

Important: nutpie does not store log_likelihood automatically (it silently ignores idata_kwargs={"log_likelihood": True}). If you need LOO-CV or model comparison, compute it after sampling:

pm.compute_log_likelihood(idata, model=model)

When to Use PyMC's Default NUTS Instead

nutpie cannot handle discrete parameters or certain transforms (e.g., ordered transform with OrderedLogistic/OrderedProbit). For these models, omit nuts_sampler="nutpie":

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

Never change the model specification to work around sampler limitations.

If nutpie is not installed, install it (pip install nutpie) or fall back to nuts_sampler="numpyro".

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

Minimum workflow checklist — every model script should include:

Prior predictive check (pm.sample_prior_predictive)
Save results immediately after sampling (idata.to_netcdf(...))
Divergence count + r_hat + ESS check
Posterior predictive check (pm.sample_posterior_predictive)

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)

az.plot_ppc(prior_pred, group="prior", kind="cumulative")
prior_y = prior_pred.prior_predictive["y"].values.flatten()
print(f"Prior predictive range: [{prior_y.min():.1f}, {prior_y.max():.1f}]")

Rule: Run prior predictive checks before pm.sample() on any new model. If the range is implausible (negative counts, probabilities > 1), adjust priors before proceeding.

Posterior Predictive (After Fitting)

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

az.plot_ppc(idata, kind="cumulative")
az.plot_loo_pit(idata, y="y")

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

Model Debugging

Before sampling, validate the model with model.debug() and model.point_logps(). Use print(model) for structure and pm.model_to_graphviz(model) for a DAG visualization.

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

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

For debugging divergences, use az.plot_pair(idata, divergences=True) to locate clusters. See references/diagnostics.md § Divergence Troubleshooting.

For profiling slow models, see references/troubleshooting.md § Performance Issues.

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

# If using nutpie, compute log-likelihood first (nutpie doesn't store it automatically)
pm.compute_log_likelihood(idata_a, model=model_a)
pm.compute_log_likelihood(idata_b, model=model_b)

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

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

Decision rule: If two models have similar stacking weights, they are effectively equivalent.

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

Iterative Model Building

Build complexity incrementally: fit the simplest plausible model first, diagnose it, check posterior predictions, then add ONE piece of complexity at a time. Compare each expansion via LOO. If stacking weights are similar, the models are effectively equivalent. See references/workflow.md for the full iterative 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")

For compressed storage of large InferenceData objects, see references/workflow.md.

Critical: Save IMMEDIATELY after sampling — late crashes destroy valid results:

with model:
    idata = pm.sample(nuts_sampler="nutpie")
idata.to_netcdf("results.nc")  # Save before any post-processing!

with model:
    pm.sample_posterior_predictive(idata, extend_inferencedata=True)
idata.to_netcdf("results.nc")  # Update with posterior predictive

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)
    beta = pm.Normal("beta", 0, 2.5, dims="features")
    p = pm.math.sigmoid(alpha + pm.math.dot(X, beta))
    y = pm.Bernoulli("y", p=p, observed=y_obs)

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

Gaussian Processes

Always prefer HSGP for GP problems with 1-3D inputs. It's O(nm) instead of O(n³), and even at n=200 exact GP (pm.gp.Marginal) is prohibitively slow for MCMC:

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. Only use pm.gp.Marginal or pm.gp.Latent for very small datasets (n < ~50) where exact inference is specifically needed.

See references/gp.md for HSGP parameter selection (m, c), HSGPPeriodic, covariance functions, and common patterns.

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 AR/ARMA, random walks, structural time series, state space models, and 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/classification, variable importance, and configuration.

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,
                   initval=np.linspace(y_obs.min(), y_obs.max(), K))
    sigma = pm.HalfNormal("sigma", sigma=2, dims="component")

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

Important: Mixture models often need target_accept=0.9 or higher to avoid divergences from the multimodal geometry. Always provide initval on ordered means — without it, components can start overlapping and the sampler struggles to separate them.

See references/mixtures.md for label switching solutions, marginalized mixtures, and mixture diagnostics.

Sparse Regression / Horseshoe

Use the regularized (Finnish) horseshoe prior for high-dimensional regression with expected sparsity. Horseshoe priors create double-funnel geometry — use target_accept=0.95 or higher.

See references/priors.md for full regularized horseshoe code, Laplace, R2D2, and spike-and-slab alternatives.

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)

Note: Don't use the same name for a variable and a dimension. For example, if you have a dimension called "cutpoints", don't also name a variable "cutpoints" — this causes shape errors.

See references/specialized_likelihoods.md for zero-inflated, hurdle, censored/truncated, ordinal, and robust regression models.

Common Pitfalls

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

Shape and dimension errors, initialization failures, mass matrix issues
Divergences and geometry problems (centered vs non-centered)
PyMC API issues (variable naming, deprecated parameters)
Performance issues (GPs, large Deterministics, recompilation)
Identifiability, multicollinearity, prior-data conflict
Discrete variable challenges, data containers, prediction

Causal Inference Operations

PyMC supports do-calculus for causal queries:

# pm.do — intervene (breaks incoming edges)
with pm.do(causal_model, {"x": 2}) as intervention_model:
    idata = pm.sample_prior_predictive()  # P(y, z | do(x=2))

# pm.observe — condition (preserves causal structure)
with pm.observe(causal_model, {"y": 1}) as conditioned_model:
    idata = pm.sample(nuts_sampler="nutpie")  # P(x, z | y=1)

# Combine: P(y | do(x=2), z=0)
with pm.do(causal_model, {"x": 2}) as m1:
    with pm.observe(m1, {"z": 0}) as m2:
        idata = pm.sample(nuts_sampler="nutpie")

See references/causal.md for detailed causal inference patterns.

pymc-extras

Key extensions via import pymc_extras as pmx:

pmx.marginalize(model, ["discrete_var"]) — marginalize discrete parameters for NUTS
pmx.R2D2M2CP(...) — R2D2 prior for regression (see references/priors.md)
pmx.fit_laplace(model) — Laplace approximation for fast inference

Custom Distributions and Model Components

# Soft constraints via Potential
import pytensor.tensor as pt
pm.Potential("sum_to_zero", -100 * pt.sqr(alpha.sum()))

See references/custom_models.md for pm.DensityDist, pm.Potential, pm.Simulator, and pm.CustomDist.

pymc-modeling