Bayesian Statistics Guide

A skill for applying Bayesian statistical methods to research data analysis. Covers prior specification, Markov chain Monte Carlo (MCMC) sampling, posterior interpretation, model comparison, and reporting standards.

Bayesian Framework Overview

Bayes' Theorem in Practice

Posterior = (Likelihood x Prior) / Evidence

P(theta | data) = P(data | theta) * P(theta) / P(data)

In practice:
  P(theta | data) is proportional to P(data | theta) * P(theta)
  (the denominator is a normalizing constant)

When to Use Bayesian Methods

| Scenario | Bayesian Advantage | |----------|-------------------| | Small sample sizes | Priors regularize estimates | | Complex hierarchical models | Natural framework for multilevel data | | Sequential data collection | Update beliefs as data arrives | | Prior knowledge available | Formally incorporate existing evidence | | Model comparison | Bayes factors and posterior model probabilities | | Prediction | Full posterior predictive distributions |

Prior Specification

Types of Priors

python
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

def visualize_priors(parameter_name: str, prior_type: str = 'weakly_informative'):
    """
    Visualize common prior choices for a parameter.
    """
    x = np.linspace(-10, 10, 1000)

    priors = {
        'flat': {
            'dist': stats.uniform(loc=-100, scale=200),
            'description': 'Flat/Uniform: minimal prior info (often improper)',
            'recommendation': 'Avoid -- can lead to improper posteriors'
        },
        'weakly_informative': {
            'dist': stats.norm(loc=0, scale=2.5),
            'description': 'Weakly informative: Normal(0, 2.5)',
            'recommendation': 'Good default for regression coefficients'
        },
        'informative': {
            'dist': stats.norm(loc=0.5, scale=0.2),
            'description': 'Informative: based on previous studies',
            'recommendation': 'Use when strong prior evidence exists'
        },
        'horseshoe': {
            'dist': stats.cauchy(loc=0, scale=1),
            'description': 'Horseshoe-like (Cauchy): sparsity-inducing',
            'recommendation': 'Good for variable selection problems'
        }
    }

    prior = priors.get(prior_type, priors['weakly_informative'])
    return prior

# Recommended default priors (Gelman et al., 2008):
# Intercept: Normal(0, 10)
# Coefficients: Normal(0, 2.5) on standardized predictors
# Standard deviation: Half-Cauchy(0, 2.5) or Exponential(1)
# Correlation: LKJ(2) for correlation matrices

MCMC with PyMC

Linear Regression Example

python
import pymc as pm
import arviz as az

def bayesian_regression(X, y, feature_names=None):
    """
    Fit a Bayesian linear regression model using PyMC.

    Args:
        X: Feature matrix (n_samples, n_features)
        y: Response variable (n_samples,)
        feature_names: List of feature names
    """
    n_features = X.shape[1]
    if feature_names is None:
        feature_names = [f'x{i}' for i in range(n_features)]

    with pm.Model() as model:
        # Priors
        intercept = pm.Normal('intercept', mu=0, sigma=10)
        betas = pm.Normal('betas', mu=0, sigma=2.5, shape=n_features)
        sigma = pm.HalfCauchy('sigma', beta=2.5)

        # Linear predictor
        mu = intercept + pm.math.dot(X, betas)

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

        # MCMC sampling
        trace = pm.sample(
            draws=2000,
            tune=1000,
            chains=4,
            cores=4,
            target_accept=0.9,
            return_inferencedata=True
        )

    return model, trace

# After fitting, analyze results:
# az.summary(trace, var_names=['intercept', 'betas', 'sigma'])
# az.plot_trace(trace)
# az.plot_forest(trace, var_names=['betas'])

Diagnostics

MCMC Convergence Checks

python
def check_mcmc_diagnostics(trace) -> dict:
    """
    Check MCMC convergence diagnostics.
    """
    summary = az.summary(trace)

    diagnostics = {
        'r_hat': {
            'values': summary['r_hat'].to_dict(),
            'threshold': 1.01,
            'pass': (summary['r_hat'] < 1.01).all(),
            'interpretation': 'R-hat < 1.01 indicates convergence'
        },
        'ess_bulk': {
            'min_value': summary['ess_bulk'].min(),
            'threshold': 400,
            'pass': (summary['ess_bulk'] > 400).all(),
            'interpretation': 'ESS > 400 ensures reliable posterior estimates'
        },
        'ess_tail': {
            'min_value': summary['ess_tail'].min(),
            'threshold': 400,
            'pass': (summary['ess_tail'] > 400).all(),
            'interpretation': 'Tail ESS > 400 ensures reliable credible intervals'
        }
    }

    # Overall assessment
    diagnostics['converged'] = all(
        d['pass'] for d in diagnostics.values() if 'pass' in d
    )

    return diagnostics

Model Comparison

Bayesian Model Selection

python
def compare_models(traces: dict) -> dict:
    """
    Compare Bayesian models using LOO-CV and WAIC.

    Args:
        traces: Dict mapping model names to InferenceData objects
    """
    comparison = az.compare(traces, ic='loo')

    return {
        'ranking': comparison.index.tolist(),
        'loo_values': comparison['loo'].to_dict(),
        'weights': comparison['weight'].to_dict(),
        'interpretation': (
            f"Best model: {comparison.index[0]} "
            f"(weight = {comparison['weight'].iloc[0]:.2f})"
        )
    }

Reporting Bayesian Results

Follow the WAMBS checklist (Depaoli & van de Schoot, 2017):

1. Priors: Report all prior distributions and justify choices
2. Convergence: Report R-hat, ESS, and trace plots (in supplement)
3. Posteriors: Report posterior mean/median, 95% credible interval (HDI preferred)
4. Sensitivity: Show results are robust to reasonable prior changes
5. Model fit: Report LOO-IC, WAIC, or posterior predictive checks

Example results sentence: "The effect of treatment on outcome was estimated at beta = 0.45, 95% HDI 0.21, 0.68], with a posterior probability of 0.99 that the effect is positive."

References

• Gelman, A., et al. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.
• McElreath, R. (2020). Statistical Rethinking (2nd ed.). CRC Press.