Markov Chain Monte Carlo Methods: Computation and Inference

Summary

Chib's handbook chapter is the canonical reference for Markov chain Monte Carlo (MCMC) methods in econometrics, covering the full toolkit from Markov chain theory through explicit algorithms for seventeen model classes. It unifies the Metropolis-Hastings (M-H) and Gibbs sampling algorithms under a common detailed-balance framework, catalogs data augmentation strategies across canonical Bayesian models, and surveys marginal likelihood computation methods including the Basic Marginal Likelihood Identity (Chib 1995), the Chib-Jeliazkov (2001) extension to M-H chains, and competing approaches (harmonic mean, Laplace, bridge sampling). Published as Chapter 57 of the Handbook of Econometrics, Volume 5, pp. 3569–3649.

Key Claims

Markov Chain Foundations

A chain has invariant distribution $\pi$ iff it satisfies $\pi P = \pi$ ; sufficient condition is detailed balance $\pi(x)P(x,y) = \pi(y)P(y,x)$ , satisfied by construction in the M-H step (Section 3).
Key theorems (Tierney 1994): (1) law of large numbers (LLN) — irreducibility + invariance $\Rightarrow$ $M^{-1}\sum h(\theta^{(i)}) \to E_\pi[h]$ a.s.; (2) central limit theorem (CLT) — geometric ergodicity $\Rightarrow$ $\sqrt{M}(\bar h - E_\pi[h]) \to \mathcal{N}(0, \sigma^2_h)$ where $\sigma^2_h$ includes all autocorrelation lags.
Inefficiency factor $\hat\tau = \text{Var}(\hat h_M)/(s^2/M)$ : ratio of numerical variance to the i.i.d. variance. Estimated by batch means (batch size $m$ chosen so first-order autocorrelation of batch means $< 0.05$ ).

Metropolis-Hastings Algorithm (Section 4)

Acceptance probability $\alpha(y,y') = \min\{p(y')q(y',y)/[p(y)q(y,y')],\, 1\}$ — requires only unnormalized $p$ .
Candidate families: random walk $q(y,y') = q(y'-y)$ ; independence chain $q(y,y') = q(y')$ ; M-H A-R (accept-reject embedded in M-H); tailored proposals from posterior mode and Hessian.
Multiple-block M-H (Section 4.4): group parameters into $p$ blocks; update each block sequentially conditioning on the rest; each block-kernel satisfies local reversibility; combined kernel leaves $\pi$ invariant via product-of-kernels principle.

Gibbs Sampler (Section 5)

Draw each block from its full conditional $p(\theta_k|\theta_{-k}, y)$ — special case of M-H with $\alpha = 1$ .
Connection to M-H: the Gibbs transition kernel is a specific M-H kernel; invariance follows from local reversibility of each full conditional step.
Collapsed/Rao-Blackwellised density estimation: $\hat p(\theta^*) = M^{-1}\sum K_G(\theta^{(g)}, \theta^*)$ where $K_G$ is the Gibbs kernel (Ritter-Tanner 1992) — requires known normalizing constants for all full conditionals.

Algorithm Catalog (Section 8) — 17 explicit algorithms

#	Model	Blocks	Key feature
5	Gaussian regression	$\beta, \sigma^2$	Conjugate normal-inverse-gamma (NIG) updates
6	Student-t regression	$\beta, \sigma^2, \{λ_i\}$	Scale mixture representation (Carlin-Polson 1991)
7	Gaussian SUR (seemingly unrelated regression)	$\beta, \Omega^{-1}$	Normal-Wishart conjugate
8	Binary probit	$\beta, \{z_i\}$	Albert-Chib (1993a) truncated-Normal augmentation
9	Ordinal probit	$\beta, \{z_i\}, \gamma$	M-H for cut-point log-transform (Albert-Chib 1998)
10	Tobit	$\beta, \sigma^2, \{z_i\}$	Truncated-Normal augmentation for censored obs
11	Change point	$\beta_1,\beta_2,\sigma^2_1,\sigma^2_2,\tau$	Full conditional of $\tau$ computed by enumeration
12	AR( $p$ ) (autoregressive) model	$\beta, \phi, \sigma^2$	M-H for $\phi$ with stationarity constraint via Chib-Greenberg (1994) proposal
13	Hidden Markov model	$\theta, S_n$	Joint state sampling via Chib (1996) forward-filter backward-sampling (FFBS)
14	State space (Carter-Kohn)	$\beta, \Sigma, Q, S_n$	FFBS smoother for Gaussian state space
15	Stochastic volatility	$\mu, \phi, \sigma^2, h$	7-component Gaussian mixture + simulation smoother
16	Gaussian panel	$\beta, D, \{b_i\}$	Block vs. marginalised sampling for random effects
17	MVP (multivariate probit)	$\Sigma, \beta, \{Z_i\}$	Tailored independence M-H for $\Sigma$ with Geweke-Hajivassiliou-Keane (GHK) evaluation

Predictive Density (Section 9)

Sample predictive density by composition: for each posterior draw $\theta^{(j)}$ , simulate $y_f^{(j)} \sim f(y_f|\theta^{(j)}, y)$ . For latent-data models, first draw $z_f^{(j)} \sim p(z_f|\theta^{(j)})$ then $y_f^{(j)} \sim f(y_f|z_f^{(j)}, \theta^{(j)})$ .

Marginal Likelihood (Section 10.2)

Basic Marginal Likelihood Identity (Chib 1995, eq. 36): $m(y|M) = \frac{f(y|\theta^*)p(\theta^*)}{p(\theta^*|y,M)}$ identity in $\theta^*$ ; estimate the posterior ordinate $\hat p(\theta^*|y)$ via a marginal/conditional decomposition requiring $B-1$ reduced Gibbs runs for a $B$ -block sampler.

Chib-Jeliazkov (2001) extension for M-H chains (eq. 41): $\hat p(\theta^*|y) = \frac{M^{-1}\sum_g \alpha(\theta^{(g)},\theta^*)q(\theta^{(g)},\theta^*)}{J^{-1}\sum_j \alpha(\theta^*,\theta^{(j)})}$ numerator from main run, denominator from a short run conditioned at $\theta^*$ .

Competing estimators (Section 10.2):

Harmonic mean (Newton-Raftery 1994): $\hat m_{\text{NR}} = [\bar{1/L}]^{-1}$ — unstable, infinite variance
Gelfand-Dey (1994): truncated importance sampling version — stable but requires good truncation
Laplace: $O(n^{-1})$ error — fast but large-sample only
Bridge sampling (Meng-Wong 1996): ratio of two normalizing constants — general but complex

Model-space algorithms (Section 10.3):

Carlin-Chib (1995) pseudo-prior: augment with pseudo-priors for inactive parameters; estimate Bayes factors as visit frequencies
Green (1995) reversible-jump MCMC (RJMCMC): dimension-matching bijections via Jacobian corrections; handles varying-dimension models

Variable selection (Section 10.4): George-McCulloch (1993) stochastic search variable selection (SSVS) — spike-and-slab mixture prior on regression coefficients; Gibbs cycling over $(γ_i,β_i)$ pairs.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"MCMC methods which deal with the simulation of high dimensional probability distributions, have over the last decade gained enormous prominence, sparked intense research interest, and energized Bayesian statistics." (p. 3571)

"The second fact about marginal likelihoods, highlighted in a paper by Chib (1995), is that the marginal likelihood by virtue of being the normalizing constant of the posterior density can be expressed as $m(y|M_k) = f(y|\theta_k, M_k)p(\theta_k|M_k)/p(\theta_k|y, M_k)$ . This expression is an identity in $\theta_k$ because the left hand side is free of $\theta_k$ ." (p. 3629)

My Take

The best single reference for the practitioner who wants to understand how a specific model's Gibbs/M-H sampler is derived. The algorithm catalog (Section 8) is the chapter's primary value: each model's augmented state space, blocking structure, and full conditionals are derived from scratch in consistent notation. The model-choice section (Section 10) is the clearest written exposition of Chib (1995) and Chib-Jeliazkov (2001) available. As a review chapter it introduces no new theory, but the synthesis is authoritative.