Calculating Posterior Distributions and Modal Estimates in Markov Mixture Models

Summary

Develops a full Bayesian Gibbs sampling approach for Markov mixture models (hidden Markov models) — finite mixture distributions where the component populations are selected by an unobserved Markov chain. The central innovation is drawing the entire latent state sequence $S_n=(s_1,\ldots,s_n)$ from its joint distribution in a single Gibbs block via a forward filter–backward sampling algorithm, rather than from $n$ individual full conditionals. This reduces state-related Gibbs blocks from $n$ to 1 and dramatically improves convergence. The paper also derives stochastic expectation-maximization (EM) variants (SEM and MCEM) using the same joint-sampling routine for computing posterior modes.

Key Claims

1. Joint state sampling (main contribution, eq. 5–8): The joint $p(S_n|Y_n,\theta)$ factorizes as $p(s_n|Y_n,\theta)\prod_{t=1}^{n-1}p(s_t|Y_n,S^{t+1},\theta)$; each factor equals (row $t$ of filter $F$) $\times$ (column $s_{t+1}$ of $P$), normalized — requiring only forward-filter output and one transition probability per step.
2. Forward filter: $n\times m$ matrix $F$ where row $t\propto (F_{t-1}P)\odot d_t$ ($d_t$ = row vector of component observation densities $f(y_t|Y_{t-1},\theta_{s_t})$ across all $m$ states); initialized from the chain's stationary distribution; cost $O(nm^2)$.
3. Backward sampler: Draw $s_n$ from final row of $F$; for $t=n-1,\ldots,1$ draw $s_t\propto [F_t]_{s_t}\cdot P_{s_t,s_{t+1}}$ (eq. 7); cost $O(nm)$. Total: one block, replacing $n$ blocks.
4. Convergence improvement: Albert-Chib (1993) and McCulloch-Tsay (1994) used single-move state sampling ($n$ blocks, high autocorrelation). Joint sampling requires truly one additional block for all states.
5. Dirichlet conjugacy for $P$: With prior $p_i\sim\mathcal{D}(\alpha_{i1},\ldots,\alpha_{im})$, the posterior conditional is $p_i|S_n\sim\mathcal{D}(\alpha_{i1}+n_{i1},\ldots,\alpha_{im}+n_{im})$ (eq. 10), where $n_{ij}$ counts $i\to j$ transitions in the sampled $S_n$.
6. SEM (Celeux-Diebolt 1985): one draw of $S_n$ per iteration; iterates form an aperiodic irreducible Markov chain converging to the posterior mode — a stochastic alternative to standard EM.
7. MCEM (Monte Carlo EM; Wei-Tanner 1990): $N$ draws of $S_n$ per iteration to approximate the Q-function; averages over $S_{n,j}$, $j=1,\ldots,N$; deterministic convergence to mode as $N\to\infty$. Recommended strategy: SEM burn-in, MCEM for final iterations.
8. Three applications: (i) Poisson fetal movement ($n=240$, 2/3 states: $\hat\lambda_1\approx0.26$, $\hat\lambda_2\approx2.29$, $\hat p_{11}\approx0.98$ — high persistence); (ii) fourth-order autoregressive AR(4) US gross national product (GNP) 1951:2–1984:4 ($n=134$, 4 states, state-dependent intercepts, common AR and variance: AR(2)+4 states appears parsimonious); (iii) bivariate Gaussian mixture ($n=300$, 3 components: Bayes more accurate than SEM; label-switching observed between states 2 and 3).

Concepts Introduced or Extended

• Markov-Switching VAR — joint state sampling supplants single-move sampling; becomes standard for Bayesian Markov-switching (MS) estimation
• Gibbs Sampler — forward filter–backward sampler for discrete latent states; discrete-state analogue of Carter-Kohn (1994) simulation smoother

Entities Mentioned

• Siddhartha Chib
• James D. Hamilton
• James H. Albert
• Bradley P. Carlin
• Robert E. McCulloch

Quotes

"This new result is extremely significant. Instead of $n$ additional blocks in the Gibbs sampler (the number required if each state is sampled from its full conditional distribution), truly one additional block is required. This dramatically improves the convergence of the Markov chain induced by the Gibbs sampling algorithm."

My Take

Chib (1996) is the discrete-state counterpart of the Carter-Kohn (1994) simulation smoother for Gaussian state-space models: both replace $n$ single-move draws with a joint draw via a forward-backward pass. The algorithm became standard for Bayesian Markov-switching estimation and the forward-filter / backward-sampler structure reappears in all subsequent Markov-switching vector-autoregression (MS-VAR) implementations (Kim-Nelson 1999; Frühwirth-Schnatter 2006; Sims-Zha 2006). The SEM/MCEM discussion is valuable historically but now largely superseded — full Bayes with the Chib (1995) marginal likelihood provides cleaner model selection. The label-switching problem (noted in the bivariate Gaussian example) remains an open challenge for mixture posteriors.