Block Gibbs Sampler

• Block Gibbs Sampler is an MCMC algorithm that partitions high-dimensional parameter spaces to jointly update strongly correlated parameters.
• Its block updates in Bayesian models improve computational efficiency and mixing compared to single-site updates.
• Rigorous theoretical guarantees, including geometric ergodicity and rapid convergence, make it robust for various hierarchical and latent variable models.

Block Gibbs Sampler is a Markov Chain Monte Carlo (MCMC) algorithmic strategy for sampling from high-dimensional posterior distributions by partitioning the parameter space into subsets ("blocks") and iteratively updating each block from its full conditional distribution. Blocking is motivated both by computational efficiency and by reducing autocorrelation in the generated Markov chain, especially in models where parameters exhibit strong posterior dependence.

1. Definition and Rationale for Block Gibbs Sampling

In Bayesian hierarchical and latent variable models, the full joint posterior π(θ, ψ, λ, …) is typically intractable due to complex dependencies between parameter groups. The block Gibbs sampler addresses this by grouping highly dependent parameters into blocks and cycling through joint draws from their full conditionals while conditioning on the remaining parameters. This substantially improves mixing over single-site updates when parameters within a block exhibit strong posterior correlations, as the chain more efficiently explores the joint posterior (Wang et al., 2017). Blocking is also essential in models with a natural partition—such as random effects versus fixed effects, latent variables, or hyperparameters—to respect inherent model structure.

2. Mathematical Formulation

Suppose the parameter vector θ = (θ₁,…,θ_K) is partitioned into m blocks θ = (θ^{[1]},… θ^{[m]}). The block Gibbs sampler iteratively generates a Markov chain θ^{(t)} by, at each iteration t, cycling through blocks and drawing each θ^{[k]}^{(t+1)} ~ π(θ^{[k]} | θ^{−k}), where θ^{−k} denotes all parameters except block k, fixed at the current values. In many applications, particularly hierarchical models or models with latent variables, the blocks may correspond to (i) structural parameters, (ii) latent variables, (iii) variance/hyperparameter components (0712.3056, Wang et al., 2017).

Formally, one-step transition kernel:

$$K(\theta, d\theta') = \prod_{k=1}^m \pi(\theta'^{[k]}\,|\,\theta'^{[1]},…,\theta'^{[k-1]},\theta^{[k+1]},…,\theta^{[m]})\,d\theta'^{[k]}$$

Under mild regularity, this yields a Harris-recurrent Markov chain with π invariant.

3. Algorithmic Structure and Key Examples

The canonical block Gibbs procedure consists of:

• Initialization at θ^{(0)}.
• For t = 0,…,T−1:
  • For each block k = 1,…,m, set θ^{k} ~ π(θ^{[k]} | θ^{[−k]}^{(t)}).
  • (Optionally) update blocks in random, systematic, or hybrid order (Backlund et al., 2018).

Notable block Gibbs constructions in specific models include:

• Polya-Gamma block sampler for logistic mixed models: Alternates between joint draws of regression and random effects (η = (β, u)) and blocks of PG auxiliary variables and variance parameters (ω, τ), achieving uniform ergodicity (Wang et al., 2017).
• Hierarchical Gaussian models: Blocks partitioned by fixed/random effects and precision parameters, with blockwise updates enabling geometric ergodicity proofs (0712.3056).
• Latent Dirichlet Allocation (LDA): Blocked collapsed sampler groups all topic indicators for identical (document,word)-pairs, allowing joint sampling of multinomial vectors per block for improved mixing and, via nested/backward algorithms, reduced complexity (Zhang et al., 2016).

4. Convergence Properties and Theoretical Guarantees

Block Gibbs samplers often satisfy geometric (or uniform) ergodicity, ensuring rapid convergence of averages to true posterior expectations. This is established via:

• Drift conditions: Construction of Lyapunov functions V(θ) showing expected contraction in each iteration, typically of the form E[V(θ^{(t+1)})|θ^{(t)}] ≤ γV(θ^{(t)}) + L, γ < 1 (0712.3056, Wang et al., 2017).
• Minorization conditions: Existence of small-sets on which the Markov kernel dominates a fixed probability measure, yielding spectral gap bounds.
• Sandwich parameter expansion: Inserted reversible steps that further contract operator norm and reduce asymptotic variance—see hybrid-scan and PX-DA variants (Backlund et al., 2018, Wang et al., 2017).

In certain formulations, even with improper priors or rank-deficient designs, block Gibbs samplers preserve invariant measures and admit explicit conditions for posterior propriety (Wang et al., 2017). Uniform ergodicity implies valid application of central limit theorems (CLTs) and consistent estimation of Monte Carlo standard errors by batch means or spectral methods.

5. Computational Efficiency and Practical Implementation

Blocked updates are designed to exploit model-specific conditional independence and conjugacy, drastically reducing per-iteration cost and autocorrelation:

• Smart amortization: Only recompute predictions for entries affected by the current block, yielding cost O(n + L_k) for random effect blocks or O(C_{dv}\log K) in LDA via nested-simulation (Johnson et al., 2016, Zhang et al., 2016).
• Scalable architectures: Distributed block-split Gibbs combines blocking with parallel execution, exploiting hypergraph structure to minimize communication and achieve linear scaling in high dimensions (Thouvenin et al., 2022).
• Advanced blocking: Group updates of latent variables (collapsed sampling, split samplers) always improve mixing by increasing chain spectral gap compared to single-site updates (Geirsson et al., 2015, Zhang et al., 2016).

Specialized methods ensure positive definiteness (e.g., graphical LASSO HRS approach via Schur-complement truncation) and robustness in graphical models (Oya et al., 2020). In neural network posteriors, blockwise Gaussian and univariate updates enable efficient Gibbs sampling for millions of parameters (Piccioli et al., 2023).

6. Applications and Model Innovations

Block Gibbs sampling is foundational across statistical domains:

• Bayesian Generalized Linear Mixed Models: Joint prior/hyperparameter and random effect updates for Poisson and Gaussian regression with millions of effect levels (Johnson et al., 2016).
• Nonparametric Bayes and HDP: Blocked Gibbs with auxiliary variable augmentation yields scalable inference for hierarchical Dirichlet processes, outperforming CRF-label samplers and unstable slice methods (Das et al., 2023).
• Graphical Model Estimation: Positive-definiteness-assured block Gibbs for graphical LASSO and MGIG distributions, with proven improvements in estimation and support recovery (Oya et al., 2020, Hamura et al., 2023).
• MCMC for Latent Gaussian Models: Split sampling partitions the latent field into data-rich and data-poor blocks, harnessing Gaussian structures for efficient high-dimensional inference (Geirsson et al., 2015).

7. Limitations, Controversies, and Analysis

Block order and block definition impact both invariance and convergence rates:

• Out-of-Order Block Gibbs: Nonstandard reordering of updates may yield chains with incorrect invariant distribution; however, Jin & Hobert (2021) prove that the geometric convergence rate is unaffected and can be leveraged for indirect analysis, provided irreducibility and positivity (but not invariance) remain (Jin et al., 2021).
• Dimension-Independence: In models with sparse conditional structure, MALA-within-Gibbs achieves acceptance and convergence rates independent of overall dimension when blocking reflects conditional independence, but trade-offs exist between block size and total cost (Tong et al., 2019).

Advanced variants such as hybrid-scan Gibbs and sandwich-step schemes promote uniform geometric ergodicity, with explicit drift bounds and operator norm reductions supported by theoretical analysis (Backlund et al., 2018).

Ultimately, blocking choices are highly model-dependent; practical implementation requires awareness of parameter dependencies, computational architecture, and convergence diagnostics.

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References:

• "Analysis of the Polya-Gamma block Gibbs sampler for Bayesian logistic linear mixed models" (Wang et al., 2017)
• "Gibbs Sampling for a Bayesian Hierarchical General Linear Model" (0712.3056)
• "Convergence analysis of the block Gibbs sampler for Bayesian probit linear mixed models with improper priors" (Wang et al., 2017)
• "A Scalable Blocked Gibbs Sampling Algorithm For Gaussian And Poisson Regression Models" (Johnson et al., 2016)
• "Blocking Collapsed Gibbs Sampler for Latent Dirichlet Allocation Models" (Zhang et al., 2016)
• "A positive-definiteness-assured block Gibbs sampler for Bayesian graphical models with shrinkage priors" (Oya et al., 2020)
• "Gibbs Sampler for Matrix Generalized Inverse Gaussian Distributions" (Hamura et al., 2023)
• "Blocked Gibbs sampler for hierarchical Dirichlet processes" (Das et al., 2023)
• "The MCMC split sampler: A block Gibbs sampling scheme for latent Gaussian models" (Geirsson et al., 2015)
• "A Hybrid Scan Gibbs Sampler for Bayesian Models with Latent Variables" (Backlund et al., 2018)
• "On the convergence rate of the 'out-of-order' block Gibbs sampler" (Jin et al., 2021)
• "MALA-within-Gibbs samplers for high-dimensional distributions with sparse conditional structure" (Tong et al., 2019)
• "Gibbs Sampling the Posterior of Neural Networks" (Piccioli et al., 2023)
• "A Distributed Block-Split Gibbs Sampler with Hypergraph Structure for High-Dimensional Inverse Problems" (Thouvenin et al., 2022)