Chapter 13

Chapter 13: Bayesian Model Averaging

Instead of picking one "best" model, Bayesian Model Averaging (BMA) accounts for model uncertainty by averaging predictions across multiple plausible models, weighted by their posterior probability.

13.1 Model Comparison with WAIC and LOO

Calculating the true model evidence for Bayes factors is often hard. Instead, we can use information criteria like WAIC (Watanabe-Akaike Information Criterion) or LOO (Leave-One-Out Cross-Validation) to compare models. These metrics estimate a model's out-of-sample predictive accuracy. The model with the lower WAIC/LOO score is generally preferred.

13.2 Code Example: Comparing Two Models

Let's compare our linear model to a simpler, intercept-only model.
Python

# The original linear model is already defined and fitted

# Define an intercept-only model
with pm.Model() as intercept_only_model:
    intercept = pm.Normal('intercept', mu=np.mean(y), sigma=5)
    sigma = pm.HalfNormal('sigma', sigma=5)
    y_obs = pm.Normal('y_obs', mu=intercept, sigma=sigma, observed=y)
    trace_intercept_only = pm.sample(1000, tune=1000)
    # Calculate LOO
    loo_intercept_only = az.loo(trace_intercept_only)

# Calculate LOO for the original linear model
with linear_model:
    loo_linear = az.loo(trace_linear)

# Compare the two models
comparison_df = az.compare({'linear': trace_linear, 'intercept_only': trace_intercept_only})
print(comparison_df)

az.plot_compare(comparison_df)
plt.show()
The az.compare function will rank the models by their LOO score, clearly showing that the linear model has far better predictive accuracy.