Gibbs-Like Sampling Procedures

• Gibbs-like sampling procedures are a class of MCMC algorithms that extend classical Gibbs sampling by using conditional and blockwise updates to efficiently explore complex, high-dimensional distributions.
• They incorporate auxiliary-variable methods, minibatch updates, and parallel architectures to enhance convergence, mixing, and computational performance.
• Recent innovations include hybrid, asynchronous, and likelihood-free adaptations, which have been effectively applied in Bayesian models, deep learning, and molecular dynamics.

Gibbs-like sampling procedures refer to a broad class of Markov Chain Monte Carlo (MCMC) algorithms that generalize the classical Gibbs sampling scheme through conditional or blockwise updates, the introduction of auxiliary variables, modifications of scan order, or architectural adaptations for large-scale and high-dimensional applications. These methods retain the hallmark structure of the Gibbs sampler—alternating direct or approximate updates for conditionals of the target distribution—while extending its applicability, efficiency, and mixing behavior across diverse statistical modeling contexts.

1. Core Principles of Gibbs-like Sampling

A Gibbs-like sampling procedure constructs a Markov chain whose stationary distribution is the target posterior or Gibbs distribution, typically on a high-dimensional or combinatorial state space. The classic Gibbs sampler forms the backbone, wherein each iteration updates a subset of variables (blocks or coordinates) by sampling exactly from their full conditional distribution given all other components. In foundational work, this mechanism is formalized as coordinate-wise conditional resampling, with variants including random scan (random coordinate updates), systematic scan (fixed sweep order), and hybrid schemes that blend these strategies (He et al., 2016, Backlund et al., 2018).

Extensions to the basic mechanism include:

• Blocked and auxiliary-variable versions to permit conditionals over aggregated or higher-dimensional subsets.
• The introduction of additional variables (e.g., diffusion paths, latent assignments, or random mini-batch indicators) allowing the construction of general joint samplers whose marginal chains target the correct law.
• Architectural or algorithmic adaptations (asynchronous execution, parallel or minibatch updates) facilitating high-throughput or distributed inference.

Rigorous guarantees of stationarity, ergodicity, and convergence arise from the reversibility or contraction properties of the joint transition kernel with respect to the target distribution (Terenin et al., 2015, Laddha et al., 2020).

2. Auxiliary-variable and Augmented Gibbs Schemes

Auxiliary-variable Gibbs-like samplers (sometimes termed data-augmentation, extended-state, or block-auxiliary methods) introduce new variables with tractable conditionals to facilitate sampling of otherwise intractable or badly mixing target distributions.

Prominent examples include:

• Diffusive Gibbs Sampling (DiGS): Incorporates an auxiliary noisy variable $\tilde{x}$, sampled from a Gaussian convolution of the original variable $x$, to connect separated modes. DiGS alternates between “noise” (sampling $\tilde{x} \mid x$) and “denoise” (sampling $x \mid \tilde{x}$) steps. By employing a Metropolis-within-Gibbs correction in the denoising phase, the chain becomes irreducible across disconnected or low-density regions. DiGS delivers marked improvements in mixing for multimodal targets, outperforming parallel tempering and HMC in empirical studies across mixtures of Gaussians, Bayesian neural networks, and molecular dynamics systems (Chen et al., 5 Feb 2024).
• Ordered Allocation Sampler (OAS): Reformulates mixture model inference by expressing allocations and mixture weights in the (randomized) order of appearance. Conditional updates of allocations, weights, and parameters avoid the need for truncation in infinite mixtures and enhance label identifiability and mixing properties. OAS efficiently handles both infinite and random-finite mixtures with adaptive blocked moves (Blasi et al., 2021).
• Negative Binomial Poisson–Kingman Sampler: Employs a two-variable representation—parameterizing by cluster counts and a global auxiliary variable $V$—for Bayesian nonparametric models based on negative binomial processes. This augmentation leads to efficient Gibbs updates for both the auxiliary and allocation variables, and is applied to population genetics simulations and general genealogical inference under nonstandard NRMIs (Griffiths et al., 18 Feb 2024).

3. Architectural Adaptations and Parallel Gibbs Sampling

Gibbs-like procedures have been adapted for parallel and distributed computation, necessary for high-throughput inference on large models.

• Asynchronous Gibbs Sampling: Eliminates sequential dependency by allowing multiple workers to update variables (or blocks) concurrently and asynchronously. The standard “approximate” version, in which updates are applied without synchronization, can diverge when dependencies are strong. By embedding a Metropolis–Hastings correction for remote updates, the “exact asynchronous” scheme is shown to converge under uniform spectral gap or Dobrushin conditions (Terenin et al., 2015). This framework is validated in hierarchical Bayesian models (Gaussian processes, mixed effects) and exposes the risk of divergence when the underlying precision is not diagonally dominant.
• Short-chain, Multi-simulation Gibbs Architecture: In multi-target tracking problems modeled using labeled random finite sets (e.g., δ-GLMB filters), a parallel architecture runs many short, independent Gibbs chains with adaptive early stopping criteria (stall/stale heuristics). This approach achieves near-identical accuracy to long MCMC chains at drastically reduced compute times and naturally enables distributed processing (Trezza et al., 2023).

4. Minibatch and Subsampled Gibbs Methods

For high-dimensional graphical models, the computational bottleneck of standard Gibbs sampling (per-step cost scaling with factor graph degree) is addressed by minibatching local neighborhoods or energy contributions.

• Minibatch Gibbs: Subsamples factors or data to form unbiased or approximately unbiased estimates of local energies or conditionals. Several variants—MIN-Gibbs, MGPMH (Minibatch-Gibbs-Proposal Metropolis–Hastings), and DoubleMIN-Gibbs—employ Poisson or multinomial-type randomization to maintain detailed balance. Under controlled estimator variance, these methods provably preserve the stationary target distribution and degrade the spectral gap by at most a fixed constant, achieving computational speedups when the squared sum of local energies is small relative to the graph degree (Sa et al., 2018).
• Poisson-minibatching Gibbs: Introduces an auxiliary Poisson variable per factor; factors with zero draws are omitted in each update, yielding expected $O(\lambda+L)$ active factors per step. The chain is reversible and preserves the stationary law; convergence rates are related to those of full-batch Gibbs up to an exponential (but controllable) factor (Zhang et al., 2019). These advances support unbiased minibatch MCMC on both discrete and continuous graphical models.

5. Scan Order, Hybrid, and Interdependent Gibbs Variants

Beyond classical scan orders, a number of scan and dependency innovations provide either improved theoretical convergence, better empirical mixing, or studiously altered stationary laws.

• Systematic vs. Random Scan: While systematic scan is standard for hardware efficiency, random scan yields better-understood mixing behavior. The difference in mixing times between scan orders is at worst polynomial in the number of variables under broad conditions, but scan order can matter dramatically in adversarially structured models (He et al., 2016).
• Hybrid Scan Gibbs: Alternates full latent-variable updates with randomly chosen blockwise parameter updates, preserving reversibility and facilitating geometric ergodicity proofs. The hybrid-scan approach is uniquely amenable to “sandwich” augmentation—intercalating additional MCMC moves (e.g., scale updates) into the scan order—which always contracts the operator norm and improves the asymptotic variance under mild conditions (Backlund et al., 2018).
• Interdependent (Multipath) Gibbs: Couples multiple chains via a shared parameter (e.g., topic matrix or HMM parameters). The joint marginal law for the shared parameter becomes proportional to the $m$th power of the likelihood, concentrating the sampler on high-likelihood regions and overcoming identifiability problems. The method yields state-of-the-art recovery in LDA and HMM models (Kozdoba et al., 2018).

6. Nonstandard Conditional and Likelihood-free Gibbs Procedures

Departures from exact conditional sampling and standard likelihood models motivate further generalizations.

• ABC-Gibbs: Implements each coordinate update by a conditional Approximate Bayesian Computation accept/reject, targeting marginalized posteriors over low-dimensional summary statistics. While not a proper Gibbs chain unless summaries are compatible, under sufficient continuity and contractivity conditions, ABC-Gibbs converges to a well-defined pseudo-posterior that approximates the true posterior as tolerances vanish, greatly improving computational efficiency in high-dimensional likelihood-free inference (Clarté et al., 2019).
• Particle Gibbs: Embeds a particle filter’s state and ancestor index structure in an extended Gibbs-type target, enabling efficient hidden Markov model inference. Inclusion of backward sampling or low-variance resampling schemes yields chains with strictly smaller asymptotic variance than standard particle Gibbs, as evidenced in uniform ergodicity and empirical mixing analyses (Chopin et al., 2013).

7. Applications, Performance, and Theoretical Guarantees

Gibbs-like procedures have been systematically applied and analyzed across:

• Spin system and Ising model simulation with improved local sampling efficiencies and robustness to high degrees or near-criticality (Liu et al., 15 Feb 2025).
• Deep learning, Bayesian neural networks, and molecular dynamics, enabling effective inference in multi-modal and high-dimensional spaces (Chen et al., 5 Feb 2024).
• Bayesian nonparametric clustering, mixture modeling, and population genetics, efficiently handling infinite mixture laws and genealogical tree queries (Blasi et al., 2021, Griffiths et al., 18 Feb 2024).
• Combinatorial tracking, sensor assignment, and random finite set filtering, achieving real-time inference at scale (Trezza et al., 2023).

Theoretical developments yield quantitative mixing bounds, contractivity characterizations, and explicit error controls (for instance in minibatch spectral gap slowdowns (Sa et al., 2018, Zhang et al., 2019) and asynchronous parallel convergence (Terenin et al., 2015)). For convex bodies and log-concave targets, coordinate Gibbs/CHAR mixing times are polynomial in problem dimension and diameter, though suboptimal in high aspect-ratio regimes (Laddha et al., 2020).

Gibbs-like methods remain a core toolkit for scalable, flexible inference and learning, with ongoing advances focusing on improved mixing in complex geometries, robustness to model misspecification, and efficient parallel/distributed implementation.