---
name: pymc-modeling
description: >
  Bayesian statistical modeling with PyMC v5+. Use when building probabilistic models,
  specifying priors, running MCMC inference, diagnosing convergence, or comparing models.
  Covers PyMC, ArviZ, pymc-bart, pymc-extras, nutpie, and JAX/NumPyro backends. Triggers
  on tasks involving: Bayesian inference, posterior sampling, hierarchical/multilevel models,
  GLMs, time series, Gaussian processes, BART, mixture models, prior/posterior predictive
  checks, MCMC diagnostics, LOO-CV, WAIC, model comparison, or causal inference with do/observe.
---

# 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

```python
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:

```python
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:

```python
# 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:

```python
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:

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

### Alternative MCMC Backends

See [references/inference.md](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)

```python
# 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)

```python
# 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)

```python
# 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

```python
# 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](references/arviz.md) for comprehensive ArviZ usage.
See [references/diagnostics.md](references/diagnostics.md) for troubleshooting.

## Prior and Posterior Predictive Checks

### Prior Predictive (Before Fitting)

Always check prior implications before fitting:

```python
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)

```python
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](references/arviz.md) for detailed interpretation.

## Model Debugging

### Inspecting Model Structure

```python
# 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:

```python
# 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 |
|---------|--------------|-----|
| `NaN` in log-probability | Invalid parameter combinations | Check parameter constraints, add bounds |
| `-inf` log-probability | Parameter outside distribution support | Verify observed data matches likelihood support |
| Very large/small logp | Scaling issues | Standardize data, use appropriate priors |
| Slow compilation | Large model graph | Reduce Deterministics, use vectorized ops |

### Debugging Divergences

```python
# 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

```python
# 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](references/gotchas.md) for additional troubleshooting.

## Model Comparison

### LOO-CV (Preferred)

```python
# 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

```python
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](references/arviz.md) for detailed model comparison workflow.

## Saving and Loading Results

### InferenceData Persistence

Save sampling results for later analysis or sharing:

```python
# 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):

```python
# 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

```python
# 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](references/priors.md) for:
- Weakly informative defaults by distribution type
- Prior predictive checking workflow
- Domain-specific recommendations

## Common Patterns

### Hierarchical/Multilevel

```python
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

```python
# 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):

```python
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](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

```python
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](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)

```python
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](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

```python
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](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

```python
# 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](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](references/gotchas.md) for:
- Centered vs non-centered parameterization
- Priors on scale parameters
- Label switching in mixtures
- Performance issues (GPs, large Deterministics)

## Causal Inference Operations

### pm.do (Interventions)

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

```python
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:

```python
# 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

```python
# 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:

```python
import pymc_extras as pmx

# Marginalizing discrete parameters
with pm.Model() as marginal:
    pmx.MarginalMixture(...)

# R2D2 prior for regression
pmx.R2D2M2CP(...)
```

## Custom Distributions and Model Components

For extending PyMC beyond built-in distributions:

```python
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](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