Bayesian Model Choice via Markov Chain Monte Carlo Methods

Summary

Carlin and Chib (1995) address the fundamental difficulty that standard Markov chain Monte Carlo (MCMC) methods for model fitting cannot be naively extended to model comparison: adding a discrete model indicator $M$ to the Gibbs sampler creates absorbing states when parameters are dropped from the model, violating convergence conditions. Their solution — the pseudo-prior (linking density) augmentation — introduces proper but otherwise arbitrary densities for inactive parameters, rendering all full conditionals well-defined and the sampler ergodic. Posterior model probabilities are then estimated directly from visit frequency counts, and Bayes factors follow immediately. The approach is illustrated on non-nested regression and finite mixture models.

Key Claims

• Naive model indicator augmentation fails: at iteration $g$, a parameter forced out of the model creates an absorbing state, violating the irreducibility condition for Gibbs convergence.
• The fix: introduce pseudo-priors (linking densities) $p(\theta_j | M \neq j)$ — proper densities for inactive parameters that are irrelevant to $p(y|M=j)$ (since $y \perp \theta_j | M \neq j$) but keep all states of the chain accessible.
• The full conditional for $M$ is a proper discrete distribution: $P(M=j|\theta, y) \propto f(y|\theta_j, M=j)\,p(\theta_j|M=j)\,\pi_j$, which is computable at every iteration.
• Posterior model probabilities are estimated from visit frequencies: $\hat{P}(M=j|y) = G_j/G$. Bayes factors follow as $B_{12} = \hat{P}(M=1|y)/\hat{P}(M=2|y) \cdot \pi_2/\pi_1$.
• Mixing quality depends critically on pseudo-prior accuracy: pseudo-priors that closely approximate the model-specific posteriors produce well-mixing chains; diffuse pseudo-priors produce high autocorrelation in the $M$-chain and slow convergence.
• Recommended workflow: (1) run $K$ individual model-specific MCMC chains to estimate posteriors; (2) use first-order normal approximations as pseudo-priors; (3) run the combined sampler. Prior model probabilities $\pi_j$ can be tuned to rebalance when one model is overwhelmingly preferred — the Bayes factor estimate adjusts automatically.
• Contrast with Chib (1995): Chib (1995) computes $\log p(y|M_j)$ from a single model's MCMC output via an identity at a point; Carlin-Chib (1995) runs a combined sampler spanning all models and reads off probabilities from visit counts. Both target the same quantity by different routes; Carlin-Chib is more direct for small $K$ but scales poorly.
• Bayes factors are sensitive to the prior when the likelihood does not identify the larger model's parameters — shown explicitly in the mixture model example where replacing Dirichlet(1,1,1,1.5) with Dirichlet(3,8,3,3) changes the Bayes factor from 0.57 to 284.

Concepts Introduced or Extended

• Marginal Data Density — augmented sampler as alternative to Chib (1995) identity for Bayes factor estimation
• Gibbs Sampler — absorbing state problem; pseudo-prior / model augmentation; linking density tuning

Entities Mentioned

• Bradley P. Carlin
• Siddhartha Chib

Quotes

"A pseudoprior is not really a prior but only a conveniently chosen linking density, required to define completely the joint model specification."

My Take

A clean and practically important paper that solves a genuine computational problem elegantly. The pseudo-prior framework is conceptually satisfying: it makes explicit that Bayesian model comparison requires a joint distribution over all model parameters simultaneously, not just the parameters of the active model. The main limitation is practical scalability — running $K$ preliminary chains and designing pseudo-priors becomes onerous as $K$ grows, and the combined sampler's dimension grows as $\sum_j \dim(\theta_j)$. For large $K$, reversible-jump MCMC (Green 1995) or Chib (1995) marginal-data-density (MDD) computation are more scalable alternatives. The mixture model example's sensitivity finding is an important reminder that Bayes factors are prior-dependent in a way that frequentist tests are not, especially when the larger model is weakly identified.