Use this Skill when the user needs to build Bayesian statistical models: prior selection, posterior sampling with MCMC (NUTS), convergence diagnostics, posterior predictive checks, LOO-CV model comparison, hierarchical models, and Bayesian A/B testing. Covers PyMC 5.x and ArviZ.
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name
bayesian-stats
description
Use this Skill when the user needs to build Bayesian statistical models: prior selection, posterior sampling with MCMC (NUTS), convergence diagnostics, posterior predictive checks, LOO-CV model comparison, hierarchical models, and Bayesian A/B testing. Covers PyMC 5.x and ArviZ.
TL;DR — Full Bayesian workflow with PyMC 5.x and ArviZ: prior selection, NUTS
sampling, convergence diagnostics (R-hat, ESS), posterior predictive checks,
LOO-CV model comparison, hierarchical models, and Bayesian A/B testing.
1. Overview
What Problem Does This Skill Solve?
Bayesian inference provides a principled framework for combining prior knowledge with
observed data to produce full posterior distributions over parameters — rather than
single point estimates. This Skill covers the complete modern Bayesian workflow:
import pymc as pm
import arviz as az
with pm.Model():
mu = pm.Normal("mu", mu=, sigma=)
idata = pm.sample(, tune=, chains=, progressbar=)
(az.summary(idata))
0
1
100
100
2
False
print
3. Core Implementation
3.1 Prior Selection Helper
import numpy as np
import pymc as pm
import arviz as az
import matplotlib.pyplot as plt
from scipy import stats
from typing importOptionaldefprior_predictive_check(
model: pm.Model,
n_samples: int = 500,
observed_data: Optional[np.ndarray] = None,
output_path: Optional[str] = None,
) -> az.InferenceData:
"""
Draw from the prior predictive distribution and compare with observed data.
Use this before fitting to verify priors produce plausible outcome ranges.
If the prior predictive generates values far outside the observed range,
consider tighter / more informative priors.
Args:
model: A PyMC model object (not yet sampled).
n_samples: Number of prior predictive draws.
observed_data: Optional observed data array for overlay comparison.
output_path: If given, save the figure to this path.
Returns:
ArviZ InferenceData with prior_predictive group populated.
"""with model:
idata = pm.sample_prior_predictive(samples=n_samples, random_seed=42)
obs_var = list(idata.prior_predictive.data_vars)[-1]
ppc_draws = idata.prior_predictive[obs_var].values.flatten()
fig, ax = plt.subplots(figsize=(8, 4))
ax.hist(ppc_draws, bins=50, density=True, alpha=0.6, label="Prior predictive", color="#4C72B0")
if observed_data isnotNone:
ax.hist(observed_data, bins=30, density=True, alpha=0.5, label="Observed data", color="#DD8452")
ax.set_xlabel("Outcome")
ax.set_title("Prior Predictive Check")
ax.legend()
fig.tight_layout()
if output_path:
fig.savefig(output_path, dpi=150)
print(f"Saved prior predictive check to {output_path}")
return idata
3.2 Linear Regression with Weakly Informative Priors
defbuild_linear_model(
X: np.ndarray,
y: np.ndarray,
feature_names: Optional[list] = None,
) -> pm.Model:
"""
Build a Bayesian linear regression model with weakly informative priors.
Model:
alpha ~ Normal(y_mean, y_sd * 2)
beta_j ~ Normal(0, 1) for each feature (standardized inputs recommended)
sigma ~ HalfNormal(y_sd)
y_i ~ Normal(alpha + X_i @ beta, sigma)
Args:
X: Feature matrix, shape (n, p). Standardize before passing.
y: Target vector, shape (n,).
feature_names: Names for the beta coefficients (for readable summaries).
Returns:
An unsampled PyMC Model.
"""
n, p = X.shape
feature_names = feature_names or [f"x{j}"for j inrange(p)]
y_mean = float(np.mean(y))
y_sd = float(np.std(y))
with pm.Model() as model:
alpha = pm.Normal("alpha", mu=y_mean, sigma=y_sd * 2)
beta = pm.Normal("beta", mu=0, sigma=1, shape=p)
sigma = pm.HalfNormal("sigma", sigma=y_sd)
mu = alpha + pm.math.dot(X, beta)
obs = pm.Normal("y_obs", mu=mu, sigma=sigma, observed=y)
return model
3.3 NUTS Sampling and Diagnostics
defsample_and_diagnose(
model: pm.Model,
draws: int = 2000,
tune: int = 1000,
chains: int = 4,
target_accept: float = 0.9,
random_seed: int = 42,
) -> az.InferenceData:
"""
Sample posterior with NUTS and run convergence diagnostics.
Convergence criteria:
- R-hat (Gelman-Rubin): all parameters should have R-hat <= 1.01
- Effective Sample Size (ESS): all parameters should have ESS >= 400
- No divergences (divergences indicate posterior geometry problems)
Args:
model: PyMC model.
draws: Number of posterior draws per chain.
tune: Number of tuning (warm-up) steps.
chains: Number of independent chains (>= 4 recommended).
target_accept: NUTS target acceptance rate (raise to 0.95 for difficult posteriors).
random_seed: For reproducibility.
Returns:
ArviZ InferenceData with posterior, sample_stats, and posterior_predictive groups.
"""with model:
idata = pm.sample(
draws=draws,
tune=tune,
chains=chains,
target_accept=target_accept,
random_seed=random_seed,
return_inferencedata=True,
idata_kwargs={"log_likelihood": True},
)
pm.sample_posterior_predictive(idata, extend_inferencedata=True)
# Convergence diagnostics
summary = az.summary(idata, round_to=3)
print("=== Convergence Diagnostics ===")
rhat_max = summary["r_hat"].max()
ess_min = summary[["ess_bulk", "ess_tail"]].min().min()
n_divergences = int(idata.sample_stats["diverging"].sum())
print(f" Max R-hat: {rhat_max:.4f} (target: <= 1.01)")
print(f" Min ESS (bulk): {int(summary['ess_bulk'].min())} (target: >= 400)")
print(f" Min ESS (tail): {int(summary['ess_tail'].min())} (target: >= 400)")
print(f" Divergences: {n_divergences} (target: 0)")
if rhat_max > 1.01:
print(" WARNING: R-hat > 1.01 — chains may not have converged. Increase tune/draws.")
if ess_min < 400:
print(" WARNING: Low ESS — increase draws or check for funnel geometry.")
if n_divergences > 0:
print(f" WARNING: {n_divergences} divergences — consider reparameterization or higher target_accept.")
return idata
defplot_diagnostics(
idata: az.InferenceData,
var_names: Optional[list] = None,
output_dir: str = ".",
) -> None:
"""
Generate standard ArviZ diagnostic plots.
Produces: trace plot, posterior plot, rank plot (better than trace for convergence).
Args:
idata: ArviZ InferenceData from sample_and_diagnose().
var_names: Variables to plot; None = all scalar parameters.
output_dir: Directory to save plots (PNG).
"""import os
# Trace plot
ax = az.plot_trace(idata, var_names=var_names, compact=True)
plt.suptitle("Trace Plot", y=1.02)
plt.tight_layout()
plt.savefig(os.path.join(output_dir, "trace_plot.png"), dpi=150, bbox_inches="tight")
plt.close()
# Posterior plot
az.plot_posterior(idata, var_names=var_names, round_to=3)
plt.savefig(os.path.join(output_dir, "posterior_plot.png"), dpi=150, bbox_inches="tight")
plt.close()
# Rank plot (recommended over trace for multi-chain convergence)
az.plot_rank(idata, var_names=var_names)
plt.savefig(os.path.join(output_dir, "rank_plot.png"), dpi=150, bbox_inches="tight")
plt.close()
print(f"Saved diagnostic plots to {output_dir}/")
3.4 LOO-CV Model Comparison
defcompare_models(
idatas: dict,
output_path: Optional[str] = None,
) -> pd.DataFrame:
"""
Compare Bayesian models using Leave-One-Out Cross-Validation (LOO-CV).
LOO-CV estimates expected log predictive density (ELPD). Higher ELPD is better.
Models within 4 ELPD_diff units of each other are considered comparable.
Args:
idatas: Dictionary mapping model names to ArviZ InferenceData objects.
Each must have log_likelihood computed (idata_kwargs={"log_likelihood": True}).
output_path: If given, save the comparison plot to this path.
Returns:
DataFrame of LOO comparison results sorted by ELPD descending.
"""import pandas as pd
loo_results = {}
for name, idata in idatas.items():
loo_results[name] = az.loo(idata, pointwise=True)
comparison = az.compare(loo_results, ic="loo", method="stacking")
print("=== LOO-CV Model Comparison ===")
print(comparison[["elpd_loo", "se", "p_loo", "weight"]].to_string())
if output_path:
az.plot_compare(comparison)
plt.savefig(output_path, dpi=150, bbox_inches="tight")
plt.close()
print(f"Saved model comparison plot to {output_path}")
return comparison
3.5 Hierarchical Model (Partial Pooling)
defbuild_hierarchical_model(
group_ids: np.ndarray,
y: np.ndarray,
X: Optional[np.ndarray] = None,
) -> pm.Model:
"""
Build a hierarchical (multilevel) model with non-centered parameterization.
Non-centered parameterization (NCP) avoids the funnel geometry problem in
hierarchical models and greatly improves NUTS sampling efficiency.
Model (NCP):
mu_alpha ~ Normal(0, 10) # global intercept mean
sigma_alpha ~ HalfNormal(1) # between-group SD
z_alpha_g ~ Normal(0, 1) # standardized group offsets
alpha_g = mu_alpha + sigma_alpha * z_alpha_g # actual group intercepts
sigma_obs ~ HalfNormal(1)
y_i ~ Normal(alpha_{group[i]}, sigma_obs)
Args:
group_ids: Integer array of group indices (0-indexed), shape (n,).
y: Observed outcomes, shape (n,).
X: Optional covariate matrix for group-level predictors, shape (n_groups, q).
Returns:
An unsampled PyMC Model.
"""
n_groups = int(np.max(group_ids)) + 1with pm.Model() as hierarchical_model:
# Hyperpriors
mu_alpha = pm.Normal("mu_alpha", mu=float(np.mean(y)), sigma=float(np.std(y)) * 2)
sigma_alpha = pm.HalfNormal("sigma_alpha", sigma=1.0)
# Non-centered parameterization: z_alpha ~ N(0,1), alpha = mu + sigma * z
z_alpha = pm.Normal("z_alpha", mu=0, sigma=1, shape=n_groups)
alpha = pm.Deterministic("alpha", mu_alpha + sigma_alpha * z_alpha)
sigma_obs = pm.HalfNormal("sigma_obs", sigma=float(np.std(y)))
mu_y = alpha[group_ids]
obs = pm.Normal("y_obs", mu=mu_y, sigma=sigma_obs, observed=y)
return hierarchical_model
3.6 Bayesian A/B Testing
defbayesian_ab_test(
n_control: int,
n_conversions_control: int,
n_treatment: int,
n_conversions_treatment: int,
draws: int = 10000,
random_seed: int = 42,
) -> dict:
"""
Bayesian A/B test for conversion rates using Beta-Binomial conjugacy.
Directly computes P(treatment > control) without p-values or stopping rules.
Uses an uninformative Beta(1,1) prior (uniform over [0,1]).
Args:
n_control: Number of users in control group.
n_conversions_control: Number of conversions in control group.
n_treatment: Number of users in treatment group.
n_conversions_treatment: Number of conversions in treatment group.
draws: Number of posterior samples per group.
random_seed: For reproducibility.
Returns:
Dictionary with: p_control_mean, p_treatment_mean, prob_treatment_better,
relative_uplift_mean, credible_interval_95.
"""
rng = np.random.default_rng(random_seed)
# Posterior: Beta(alpha + conversions, beta + non-conversions)
alpha_prior, beta_prior = 1.0, 1.0
p_control = rng.beta(
alpha_prior + n_conversions_control,
beta_prior + (n_control - n_conversions_control),
size=draws,
)
p_treatment = rng.beta(
alpha_prior + n_conversions_treatment,
beta_prior + (n_treatment - n_conversions_treatment),
size=draws,
)
prob_better = float(np.mean(p_treatment > p_control))
relative_uplift = (p_treatment - p_control) / p_control
ci_low, ci_high = np.percentile(relative_uplift, [2.5, 97.5])
result = {
"p_control_mean": float(np.mean(p_control)),
"p_treatment_mean": float(np.mean(p_treatment)),
"prob_treatment_better": prob_better,
"relative_uplift_mean": float(np.mean(relative_uplift)),
"credible_interval_95": (float(ci_low), float(ci_high)),
"posterior_samples_control": p_control,
"posterior_samples_treatment": p_treatment,
}
print(f"Control conversion rate: {result['p_control_mean']:.4f}")
print(f"Treatment conversion rate: {result['p_treatment_mean']:.4f}")
print(f"P(treatment > control): {prob_better:.4f}")
print(f"Expected relative uplift: {result['relative_uplift_mean']*100:.1f}%")
print(f"95% credible interval: [{ci_low*100:.1f}%, {ci_high*100:.1f}%]")
return result
Increase tune (e.g. 2000); check for multimodal posterior
Many divergences
Posterior funnel geometry
Use non-centered parameterization; increase target_accept to 0.95
ESS < 400
High autocorrelation
Increase draws; reparameterize; check for near-improper priors
ValueError: observed RV
observed= data has wrong shape
Ensure y is a 1D numpy array
Sampling extremely slow
High-dimensional model or bad priors
Standardize inputs; use more informative priors; check pm.model_to_graphviz()
KeyError: log_likelihood
LOO called without computing log likelihood
Add idata_kwargs={"log_likelihood": True} to pm.sample()
Prior predictive all near zero
Priors too tight / misspecified scale
Check data scale; use pm.Normal("sigma", mu=y_sd, sigma=y_sd) pattern
6. Prior Selection Decision Tree
Is there strong domain knowledge?
YES --> Use informative prior (e.g. Normal(literature_mean, literature_sd))
NO --> Use weakly informative prior:
Continuous, unbounded --> Normal(0, 1) on standardized scale
Positive only --> HalfNormal(1) or Exponential(1)
Probability [0,1] --> Beta(2, 2) or Beta(1, 1) uninformative
Count data --> Poisson or NegativeBinomial
Always run prior_predictive_check() to verify priors produce
plausible outcome values before fitting.
Gelman, A. et al. (2013). Bayesian Data Analysis, 3rd ed. — standard reference text
McElreath, R. (2020). Statistical Rethinking, 2nd ed. — accessible introduction
Vehtari, A. et al. (2017). "Practical Bayesian model evaluation using LOO-CV and WAIC."
Statistics and Computing, 27(5), 1413–1432. https://doi.org/10.1007/s11222-016-9696-4
