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arxiv: 2606.00309 · v1 · pith:Y42X5WOVnew · submitted 2026-05-29 · 💻 cs.LG · stat.ML

Large-scale Uncertainty Quantification for Latent Variable Models Using Subsampling Markov Chain Monte Carlo

Xiaoyu Wang , Jonathan H. Huggins This is my paper

Pith reviewed 2026-06-28 23:19 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords SGLD-Gibbslatent variable modelsuncertainty quantificationasymptotic limitsjump-diffusion processeshyperparameter tuningstochastic gradient Langevin dynamicsMarkov chain Monte Carlo
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The pith

A joint jump-diffusion limit for SGLD-Gibbs reveals how to tune hyperparameters for statistically meaningful uncertainty quantification in latent variable models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the lack of principled hyperparameter tuning for SGLD-Gibbs, a scalable method for approximate Bayesian inference in latent variable models. It derives a joint asymptotic limit under space-time rescaling, where global parameters approach a diffusion limit and latent variables approach a jump process due to intermittent Gibbs updates. This joint structure shows how latent-variable randomness influences the stationary distribution of the global parameters. The results are used to propose specific tuning guidance that ensures the uncertainty quantification is statistically meaningful. Numerical experiments demonstrate improved parameter estimates, uncertainty quantification, and predictive performance compared to stochastic variational inference.

Core claim

Under appropriate space-time rescaling, the global parameters in SGLD-Gibbs converge to a diffusion-type limit while each latent variable converges to a jump process. This joint jump-diffusion structure reveals the contribution of latent-variable randomness to the stationary distribution of the global parameters, enabling explicit guidance on hyperparameter tuning for meaningful uncertainty quantification.

What carries the argument

The joint asymptotic jump-diffusion limit under space-time rescaling, which characterizes the interaction between global parameters and latent variables in the scaled process.

If this is right

  • Global parameters converge to a diffusion-type limit.
  • Each latent variable converges to a jump process reflecting intermittent Gibbs updates.
  • The joint structure allows explicit characterization of how latent randomness affects global parameter stationary distribution.
  • Explicit hyperparameter tuning guidance ensures statistically meaningful uncertainty quantification.
  • SGLD-Gibbs with the guidance outperforms stochastic variational inference in estimates and predictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The tuning rules may extend to other subsampling MCMC methods that combine continuous and discrete updates.
  • Similar scaling limits could inform tuning in non-latent variable models with intermittent sampling.
  • Empirical validation of the jump-diffusion limits in high-dimensional settings would strengthen applicability to large-scale problems.
  • The framework suggests ways to adjust update frequencies for better mixing in latent variable inference.

Load-bearing premise

The space-time rescaling must be chosen so that the intermittent Gibbs updates produce a jump-process limit for the latent variables whose contribution to the global-parameter stationary distribution can be explicitly characterized.

What would settle it

A simulation or analysis showing that the proposed tuning rules fail to produce calibrated uncertainty estimates when the space-time rescaling does not yield the predicted jump-diffusion behavior.

Figures

Figures reproduced from arXiv: 2606.00309 by Jonathan H. Huggins, Xiaoyu Wang.

Figure 1
Figure 1. Figure 1: Validation of the joint scaling limit on a synthetic Gaussian mixture model. in Appendix C. 4.1. Synthetic Gaussian Mixture Model We generate synthetic data with sample size n = 30,000 with observations xi ∈ R 8 . The data are drawn from a finite Gaussian mixture with 6 clusters. We run SGLD–Gibbs using a minibatch size b = 50 and consider Gibbs updates with S ∈ {1, 10} samples. For SGLD, we follow the san… view at source ↗
Figure 2
Figure 2. Figure 2: Synthetic Gaussian GMM: uncertainty quantification and posterior accuracy 0.0 0.2 0.4 0.6 0.8 1.0 Uniform quantile 0.0 0.2 0.4 0.6 0.8 1.0 Empirical P ( π > π ? ) (sorted) Uniform reference SGRLD SVI a. Rank-uniformity diagnostic over topic-word probabilities. SGRLD–Gibbs yields empirical ranks closer to the uniform reference line, indicating better-calibrated uncertainty. 0.30 0.35 0 100 π1,28 0.275 0.300… view at source ↗
Figure 3
Figure 3. Figure 3: Synthetic LDA: uncertainty calibration and posterior accuracy [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

Stochastic gradient Langevin dynamics combined with Gibbs updates (SGLD--Gibbs) provides a highly scalable approach to approximate Bayesian inference in latent variable models. However, it remains unclear how to tune the algorithm's hyperparameters in a principled manner to ensure the uncertainty estimates are statistically meaningful. In this work, we address this gap in tuning guidance by developing a statistical scaling limit theory for SGLD--Gibbs. We derive a joint asymptotic limit for the global parameters and latent variables under appropriate space-time rescaling. We show that global parameters converge to a diffusion-type limit, while each latent variable converges to a jump process, reflecting the use of intermittent Gibbs updates. This joint jump-diffusion structure reveals how latent-variable randomness contributes to the stationary distribution of the global parameters. We leverage our results to propose explicit guidance on hyperparameter tuning for SGLD--Gibbs that ensures meaningful uncertainty quantification. Numerical experiments show that SGLD--Gibbs with our tuning guidance leads to better parameter estimates, uncertainty quantification, and predictive performance than stochastic variational inference.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a statistical scaling-limit theory for the SGLD-Gibbs algorithm in latent-variable models. Under a specific space-time rescaling, the global parameters are shown to converge to a diffusion process while each latent variable converges to a jump process induced by the intermittent Gibbs steps. The resulting joint jump-diffusion structure is used to characterize the stationary distribution of the global parameters and to extract explicit hyperparameter tuning rules that aim to produce calibrated uncertainty estimates. Numerical experiments are reported to demonstrate improved parameter recovery, uncertainty quantification, and predictive performance relative to stochastic variational inference.

Significance. If the limit theorem and the extraction of tuning rules are valid, the work supplies principled, theoretically grounded guidance for hyperparameter selection in a scalable hybrid MCMC method for latent-variable models. This directly addresses a practical gap in ensuring statistically meaningful uncertainty quantification at large scale and could influence the design of sampling-based inference procedures in machine learning.

minor comments (3)
  1. [Introduction] The precise statement of the space-time rescaling (including the scaling exponents for time, step size, and Gibbs frequency) should be stated explicitly in the introduction or early in §3 so that readers can immediately connect the abstract claim to the theorem.
  2. [Experiments] In the numerical experiments, the reported metrics for uncertainty quantification (e.g., coverage or calibration plots) would benefit from an explicit comparison against the theoretical stationary variance derived from the limit; currently the connection is only qualitative.
  3. [Theorem statement] A short remark on the regularity conditions required for the jump-diffusion limit (e.g., Lipschitz assumptions on the potential or boundedness of the latent-variable conditional) would help readers assess applicability to common latent-variable models.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work, recognition of its significance for principled hyperparameter tuning in scalable MCMC for latent-variable models, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central contribution is a derivation of a joint jump-diffusion scaling limit for SGLD-Gibbs under space-time rescaling, followed by extraction of hyperparameter tuning rules from the resulting stationary distribution. This is a standard mathematical analysis of MCMC dynamics (global parameters to diffusion, latent variables to jump process) whose output is an external statistical property (calibrated uncertainty quantification) rather than a quantity defined by the same procedure. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described derivation chain; the tuning guidance is downstream of the limit rather than presupposed by it. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of the joint scaling limit under the stated rescaling; the abstract invokes standard conditions for such limits but does not list explicit free parameters or new entities.

axioms (1)
  • domain assumption The model and algorithm satisfy the regularity conditions required for the joint space-time scaling limit to exist and separate into diffusion and jump components.
    Invoked when the abstract states that global parameters converge to a diffusion limit and latent variables to a jump process under appropriate rescaling.

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discussion (0)

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