Auxiliary Variable Gibbs Sampler

• Auxiliary variable Gibbs samplers are MCMC techniques that introduce latent variables to simplify and speed up conditional updates in complex models.
• They employ methods like recycling and uniformization to enhance variance reduction and ensure tractable inference in high-dimensional and stochastic process models.
• These samplers achieve robust mixing and increased effective sample size, making them essential for scalable Bayesian inference in challenging model settings.

An auxiliary variable Gibbs sampler is a Markov chain Monte Carlo (MCMC) method that augments the original sampling problem with auxiliary or latent variables, enabling more efficient or feasible updates in otherwise intractable high-dimensional or structured probability models. By expanding the state space, these samplers often facilitate exact or improved-variance conditional updates, enable the use of sub-sampling or latent representation schemes, and underlie several modern MCMC and stochastic simulation frameworks across Bayesian inference, stochastic processes, and machine learning.

1. Canonical Principle: Auxiliary Variable Augmentation in Gibbs Sampling

Classical Gibbs sampling targets a joint density $\pi(x_{1:D})$ by sequentially updating each variable from its full conditional. When direct sampling is intractable or inefficient, the introduction of auxiliary variables $u$ transforms the original target into an augmented joint,

$$\pi^+(\mathbf{x}, u) = \pi(\mathbf{x})\,q(u|\mathbf{x}),$$

where $q(u|\mathbf{x})$ is chosen for computational tractability or algorithmic benefits. The Markov chain alternates between sampling $u \mid \mathbf{x}$ and $\mathbf{x} \mid u$, often simplifying one or both steps or allowing more informed proposals. This overarching principle underpins a diverse array of auxiliary-variable MCMC algorithms, including block-splitting, uniformization for stochastic processes, minibatching with control variates, and path augmentation for diffusion models (Titsias et al., 2016, Zhang et al., 2017, Zhang et al., 2019, Wang et al., 2019).

2. Key Methodological Variants

Sampler Variant	Augmentation Mechanism	Main Application Domains
Recycling Gibbs (RG)	Retain all auxiliary samples from internal MCMC per block	General Bayesian hierarchical models
Uniformization-based Gibbs	Augment with latent Poisson grids of event times	Continuous-time Markov jump processes, CTBNs
Auxiliary-gradient MCMC	Auxiliary noise for improved proposal adaptation	Latent Gaussian/mixed models
Poisson-minibatching Gibbs	Add Poisson-auxiliary counts to control minibatch stochasticity	High-dim factor graphs, scalable inference
Path-augmentation for SDEs	Random Poisson grid, thinning for exact path sampling	Diffusion processes under noisy or partial obs
Ancestor-extended Particle Gibbs	Full particle system and ancestor lineage augmentation	State-space smoothing for general nonlinear models

Each method leverages the auxiliary structure to reformulate the conditional distributions in a way that either:

• enables analytic, efficient, or parallel updates,
• grants variance reduction by recycling otherwise-discarded samples,
• or allows exactness via reweighting or careful latent representation (Martino et al., 2016, Corenflos et al., 2023, Chopin et al., 2013).

3. Recycling Gibbs Sampler: The Auxiliary-Variable Trick and Variance Reduction

The Recycling Gibbs (RG, or MRG for multiple recycling) sampler is an archetype of auxiliary-variable Gibbs sampling. Beginning with a standard sampler where each block coordinate is updated via $M$ internal MCMC draws (e.g., Metropolis or ARMS), classical implementations utilize only the final draw, discarding $M-1$ auxiliary samples per block. RG systematically recycles all these auxiliary draws for estimation:

• For $T$ iterations and $D$ coordinates, if $M$ internal samples are drawn per coordinate, RG accumulates $TDM$ effective samples instead of $TD$ while incurring identical likelihood evaluation cost.
• The estimator

$$\widehat{I}_{\mathrm{RG}} = \frac{1}{TDM} \sum_{t=1}^T \sum_{d=1}^D \sum_{m=1}^M f(X_{d,m}^{(t)})$$

is unbiased and consistent with variance

$$\operatorname{Var}[\widehat{I}_{\mathrm{RG}}] \approx \frac{1}{TDM} \operatorname{Var}_\pi[f(X)],$$

yielding an $M$-fold variance reduction versus standard Gibbs for the same computational budget.

Empirical results confirm the RG sampler’s efficiency gains in Gaussian mean/covariance estimation, multimodal posteriors, hyperparameter learning for GPs, and probabilistic graphical model structure discovery. In high-dimensional or multimodal targets, MRG attains up to 2–4× reduction in mean squared error relative to standard MCMC (Martino et al., 2016).

4. Uniformization, Path and Grid Augmentation in Stochastic Process Models

In continuous-time Markov jump processes (MJPs), exact path inference is facilitated via uniformization: introducing a Poisson grid of candidate-jump (virtual) event times and recasting the path-update as an HMM inference problem. The Rao–Teh sampler alternates between:

• Sampling virtual jumps from a Poisson process,
• Resampling the full path conditional on augmented time grid via forward-backward algorithms, ensuring linear or quadratic cost per iteration and circumventing expensive matrix exponentiations (Rao et al., 2012).

Extensions to exact SDE inference deploy a similar Poisson-grid approach: latent Poisson points are generated at dominating rates, thinned according to drift-induced weights, and conditioned paths are imputed via Brownian bridges between observed and latent times (Wang et al., 2019). These schemes render the otherwise intractable continuous sampling problem amenable to block Gibbs updates, producing unbiased samples without discretization bias.

5. Auxiliary Variable Gibbs in Scalable and High-dimensional Inference

Advanced auxiliary-variable schemes address high computational cost and slow mixing in large-scale systems:

• Poisson-Minibatching Gibbs Sampling: Introduces Poisson-distributed auxiliary count variables per factor in a graphical model. Conditioned on the augmented latent counts, block updates of variables can be performed using only a (typically small) random subset of model factors, achieving unbiased updates at cost $\mathcal{O}(L^2)$ with mixing time preserved up to a constant factor in the spectral gap (Zhang et al., 2019).
• Auxiliary Kalman and Particle Gibbs for Dynamical Systems: Artificial Gaussian observations (auxiliaries) are introduced to latent dynamical models, enabling tractable Kalman-based or particle-based conditional updates with parallel-in-time architecture. These methods attain robust mixing in high-dimensional spaces and logarithmic scaling in wall-clock time when deployed on appropriate hardware (Corenflos et al., 2023).

6. Architectural and Theoretical Foundations

Auxiliary-variable Gibbs samplers frequently yield improved ergodicity and variance properties:

• Uniformization-based and Rao–Teh samplers admit geometric ergodicity under mild conditions. Symmetrized MH augmentation to these schemes enhances mixing, with empirical 3–10× improvements in effective sample size versus standard Gibbs and superior performance over particle MCMC (Zhang et al., 2017).
• The recycling of auxiliary samples, as in the RG and blockwise methods, preserves chain invariance and reversibility, often producing an estimator with reduced variance and increased effective sample size—especially when inner proposals are conditionally nearly independent.
• In particle Gibbs, viewing the entire particle ancestry as auxiliary variables enables uniform ergodicity with coupling rate exponentially fast in the number of particles, and allows the implementation of backward sampling steps for variance reduction (Chopin et al., 2013).

7. Practical Guidelines and Limitations

• The auxiliary sample size (or number of latent variables) should be matched to the mixing time of the inner sampler; too-small auxiliary blocks slow adaptation, while oversizing induces computational redundancy without further variance reduction (Martino et al., 2016).
• While these methods are general, their advantage can be diminished when the auxiliary-to-target mapping is highly correlated or memory/storage constraints arise.
• Efficient implementation (e.g., spectral decompositions for latent Gaussian models, parallelization across time for dynamic systems) is often crucial to fully realize the gains in both computational and statistical efficiency.

Auxiliary variable Gibbs samplers represent an essential paradigm for modern MCMC, broadening the range of feasible Bayesian inference and enabling scalable, efficient, and often exact computation in otherwise computationally intractable or high-dimensional models (Martino et al., 2016, Titsias et al., 2016, Rao et al., 2012, Wang et al., 2019, Zhang et al., 2019, Corenflos et al., 2023, Chopin et al., 2013, Zhang et al., 2017).