Siddhartha Chib

Overview

Siddhartha Chib is an econometrician and statistician at Washington University in St. Louis (Olin Business School). He is best known for two foundational contributions to Bayesian computation: the Chib (1995) marginal likelihood identity — which extracts the marginal data density from standard Gibbs output without additional simulation — and the Carlin-Chib (1995) pseudo-prior algorithm for Bayesian model choice via an augmented MCMC sampler. Both contributions became standard tools for Bayesian model comparison in econometrics.

Key Contributions

• Albert and Chib (1997): With James Albert; MCMC-based model diagnostics for CIHMs; discrete mixture perturbations (scale-inflated outlier, partial exchangeability, link family) each add one Bernoulli block to the base Gibbs cycle; MCMC-grid Bayes factor for fixed vs. random effects; cancer mortality and math placement applications. See Bayesian Hierarchical Model.
• Albert and Chib (1993): Data augmentation Gibbs sampler for Markov-switching AR models; treats latent state sequence as missing data; all conditionals conjugate; foundational reference for Bayesian MS-VAR estimation. See Markov-Switching VAR and Gibbs Sampler.
• Albert and Chib (1993b): Data augmentation for binary and polychotomous probit models; latent $Z_i \sim \mathcal{N}(x_i'\beta, 1)$ converts truncated-likelihood probit into a two-block Normal/truncated-Normal Gibbs sampler; extensions to t-link scale mixtures, hierarchical priors, ordered multinomial, and unordered multinomial probit. See Binary Probit.
• Albert and Chib (1995): Bayesian residual analysis for binary regression; defines parametric residual $r_i = y_i - p_i(\beta)$ and latent data residual $\varepsilon_i = Z_i - x_i^T\beta$ (a priori $\mathcal{N}(0,1)$); both free from the Gibbs run; Rao-Blackwellised density estimates; extensions to logistic link and longitudinal random effects probit. See Bayesian Residual Analysis.
• Chib-Greenberg (1998) multivariate probit — with Edward Greenberg; unified simulation-based framework for correlated binary responses $Y_i=(Y_{i1},\ldots,Y_{iJ})'$; three-block Gibbs (latent MVN truncation via Geweke 1991, conjugate Normal for β, tailored independence/reflection M-H for σ); extends Chib (1995) identity to settings where $\pi(\sigma|Z,\beta)$ has unknown normalising constant via kernel density (eq. 10) — precursor to Chib-Jeliazkov (2001); GHK likelihood evaluation (eq. 11); three applications (voter, Six Cities wheezing, PSID 7-year labour supply). See Multinomial Probit and Marginal Data Density.
• Chib-Carlin (1999) blocked Gibbs for longitudinal models: With Bradley Carlin; proposes partially and fully blocked MCMC algorithms for the Gaussian linear mixed model and binary longitudinal probit; Algorithm 2 marginalizes $\beta$ over random effects for essentially iid fixed-effects draws ($\kappa$: 20→1); Algorithm 3 uses the Chib (1995) identity to evaluate $f(y|\sigma^2,D)$ as a ratio of three Normal densities, enabling a single-block M-H update for all variance parameters ($\kappa(D_{21})$: 60→11); Algorithm 7 for binary data uses the GKH method to fully marginalize over random effects and latent data. Universal rule: fixed effects must be sampled marginalized over random effects. See Bayesian Hierarchical Model and Gibbs Sampler.
• Chib (1996) joint state sampling: Shows the latent state sequence $S_n=(s_1,\ldots,s_n)$ of a Markov mixture model can be drawn jointly in one Gibbs block via a forward filter–backward sampler (eq. 5–8); reduces state-related blocks from $n$ to 1; covers SEM and MCEM for modal estimation; applied to Poisson, Gaussian AR(4), and bivariate normal mixtures. Discrete-state analogue of the Carter-Kohn smoother. See Markov-Switching VAR and Chib (1996).
• Chib (1995) marginal likelihood identity — exploits Bayes' theorem in the form $\log p(Y) = \log L(\phi^*) + \log p(\phi^*) - \log p(\phi^*|Y)$ evaluated at a high-density point $\phi^*$; the posterior ordinate $p(\phi^*|Y)$ is estimated from the Gibbs chain via Rao-Blackwellization of the full conditionals. Standard tool for MDD computation in identified VARs (used by Zha 2005, Warne 2006, Amisano-Federico 2004). See Marginal Data Density.
• Carlin-Chib (1995) model-augmented Gibbs sampler — with Bradley Carlin; solves the absorbing state problem in multi-model MCMC by introducing pseudo-priors for inactive parameters; estimates Bayes factors as ratios of model visit frequencies. See Gibbs Sampler.
• Chib-Greenberg (1995): With Edward Greenberg; tutorial derivation of the MH algorithm from the reversibility (detailed balance) condition; unification of five candidate-generating families; Product of Kernels principle (Gibbs sampler = α=1 special case); M-H A-R algorithm (§6.1) embedding rejection sampling inside MH; optimal acceptance rates 0.45 (1D), 0.25 (d≥6), 0.23 (∞-d). See Chib-Greenberg (1995).
• Chib-Jeliazkov (2001): With Ivan Jeliazkov; extends Chib (1995) to M-H samplers via the local reversibility identity $\pi(\theta^*|y) = E_1\{\alpha(\theta,\theta^*|y)q(\theta,\theta^*|y)\}/E_2\{\alpha(\theta^*,\theta|y)\}$ — $E_1$ from main MCMC run, $E_2$ from cheap reduced run; general $B$-block formula (eq. 16–18); NSE via Newey-West Delta method; applied to binary logit, AIDS longitudinal, epileptic Poisson, and PSID multivariate probit (21 correlation parameters). See Marginal Data Density.
• Chib-Greenberg (1995b): With Edward Greenberg; hierarchical SUR via Gibbs (three-level conjugate prior structure); Metropolis-within-Gibbs for VMA(1) errors (Taylor-linearisation candidate, ~50% acceptance); joint FFBS backward simulation for TVP-SUR (early derivation of the Carter-Kohn smoother); partial Bayes factors; OECD GNP application with 40 parameters. See Chib-Greenberg (1995b) and Seemingly Unrelated Regression.
• Albert and Chib (1997b): With James Albert; unified Bayesian MCMC framework for fitting and comparing three non-nested ordinal models — cumulative probit (McCullagh 1980), sequential probit (Tutz 1990–91), and two-step compound (Tutz 1989); log-spacing reparameterization $\alpha_j=\log(\gamma_j-\gamma_{j-1})$ lifts ordered cut-points to an unrestricted real vector, enabling a tuned multivariate-$t$ MH step with rapid mixing; Bayes factors computed from Chib (1995) marginal likelihoods; three prior elicitation methods (training sample, imaginary prior, Dirichlet-on-multinomial); NLSY educational attainment and Canada GSS physician visits applications. See Ordinal Regression.
• Albert and Chib (1998): With James Albert; working paper of the 2001 paper; adds hierarchical sequential model shrinking cutpoints toward a quadratic polynomial via $\tau^2$ hyperparameter; 6-model table (BF₂₁ = 6.4×10¹⁹ sequential vs. cumulative; M₄ quadratic baseline wins, ln m = −2117.4 reduced); confirms log-spacing reparameterization from 1997b is used for the cumulative model's Algorithm 3.
• Albert and Chib (2001): With James Albert; Bayesian framework for the sequential ordinal model; $J-1$ independent binary probit decisions (discrete-time hazard); data augmentation with truncated normals; unordered cutpoints; right censoring handled natively; marginal likelihood via Chib (1995) identity; hospital length-of-stay application ($N=1000$, $J=12$) shows sequential model beats cumulative probit (BF$\approx10^{22}$), Weibull, and log-logistic. See Sequential Ordinal Model.
• Chib (2001) handbook chapter: "Markov Chain Monte Carlo Methods: Computation and Inference," Handbook of Econometrics, Vol. 5, Ch. 57; 82-page synthesis covering M-H and Gibbs theory, 17 explicit algorithms for canonical Bayesian models (probit, SUR, AR, HMM, state space, SV, MVP), and the clearest exposition of the Chib (1995) marginal likelihood identity and Chib-Jeliazkov (2001) M-H extension. See Chib (2001).
• Kim-Shephard-Chib (1998): With Sangjoon Kim and Neil Shephard; foundational multi-move MCMC sampler for SV models — log-squaring transformation + 7-component Gaussian mixture approximation for $\log\chi^2(1)$ errors + Carter-Kohn (1994) simulation smoother draws the full $\{h_t\}$ path in one Gibbs block; importance reweighting corrects the approximation to exact inference; inefficiency factors 1–3 vs. 20–200 for single-move; Bayes factors (Chib 1995) decisively favor SV over GARCH and $t$-GARCH for daily S&P 500. See Stochastic Volatility and Kim-Shephard-Chib (1998).
• Chib-Nardari-Shephard (2002): With Federico Nardari and Neil Shephard; SVt (Student-t SV) and SVJt (Student-t + jumps) models; 4- and 6-block Gibbs samplers using 7-component Gaussian mixture + simulation smoother; marginal likelihood via Chib-Jeliazkov (2001) for M-H blocks; S&P 500 application shows decisive evidence for SVt over both SV0 and SVJ. See Stochastic Volatility.
• Chib-Nardari-Shephard (2006): With Federico Nardari and Neil Shephard; extends the univariate SVt/SVJt framework to high-dimensional multivariate factor-SV ($p=50$ series, $k=8$ factors, 688 parameters); reduced blocking scheme marginalizes $b$ over latent factors $f$ via Newton-Raphson-tuned multivariate-$t$ M-H (inefficiency factors drop from >1,000 to 1–30); auxiliary particle filter + Chib (1995)/Chib-Jeliazkov (2001) for marginal likelihood; Bayes factors compare MSV/MSVt/MSVJ/MSVJt; 10-index application shows MSV outperforms all MGARCH alternatives on MAD and VaR. See Stochastic Volatility.
• Omori-Chib-Shephard-Nakajima (2007): With Yasuhiro Omori, Neil Shephard, and Jouchi Nakajima; extends KSC (1998) to the ASV model with correlated innovations ($\rho \neq 0$) via a 10-component bivariate Gaussian mixture; analytically derived coefficients $a_j = e^{v_j^2/8}$, $b_j = \tfrac{1}{2}e^{v_j^2/8}$ approximate $\exp(\varepsilon_t^*/2)$ within each component; importance reweighting restores exact inference; TOPIX application: $\hat\rho = -0.362$, log Bayes factor $\approx 2.24$ decisively favoring leverage. See Stochastic Volatility and Omori-Chib-Shephard-Nakajima (2007).

Related

• Marginal Data Density
• Gibbs Sampler
• Markov-Switching VAR
• Binary Probit
• Sequential Ordinal Model
• Ordinal Regression
• Bayesian Residual Analysis
• James H. Albert
• Bradley P. Carlin
• Tao Zha
• Anders Warne
• Neil Shephard
• Federico Nardari
• Edward Greenberg
• Ivan Jeliazkov
• Multinomial Probit
• Chib-Greenberg (1998)