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arxiv: 2605.20345 · v1 · pith:OOIQUZVJnew · submitted 2026-05-19 · 📊 stat.ML · cs.LG

Corrected Integrated Laplace Approximation for Bayesian Inference in Latent Gaussian Models

Jinlin Lai , Charles C. Margossian , Daniel R. Sheldon This is my paper

Pith reviewed 2026-05-21 07:18 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords latent Gaussian modelsintegrated Laplace approximationimportance samplingBayesian inferencequasi-Monte CarloHamiltonian Monte CarloGaussian processes
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The pith

Importance sampling corrects the integrated Laplace approximation so the posterior converges to the true one in latent Gaussian models.

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

Latent Gaussian models require marginalizing out latent variables for Bayesian inference, but non-Gaussian likelihoods make exact marginalization impossible. The integrated Laplace approximation offers an efficient but sometimes inaccurate substitute that can produce posteriors differing noticeably from the correct distribution. This paper introduces an importance sampling scheme that removes the approximation error, with the corrected posterior approaching the true posterior as the number of samples increases. The scheme incorporates pseudo-marginalization and quasi-Monte Carlo techniques and is implemented in an automatic differentiation framework so that gradient-based methods such as Hamiltonian Monte Carlo remain available for hyperparameter inference.

Core claim

The authors propose an importance sampling scheme to correct the error introduced by the integrated Laplace approximation. By increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior. This idea is realized with various techniques, including pseudo-marginalization, quasi-Monte Carlo and randomized quasi-Monte Carlo. The methods are implemented in an automatic differentiation framework to support gradient-based algorithms when doing inference on the hyperparameters, specifically considering Hamiltonian Monte Carlo.

What carries the argument

The importance sampling correction to the integrated Laplace approximation (ILA), which adjusts the marginal likelihood estimate so that the resulting posterior converges to the exact posterior as sample size grows.

If this is right

  • The corrected posterior can be used directly in downstream tasks with substantially lower error than plain ILA.
  • Hyperparameter inference proceeds with standard gradient-based algorithms such as Hamiltonian Monte Carlo.
  • The same correction framework applies to Gaussian processes, spatial models, and mixed-effect models.
  • Convergence to the true posterior is obtained simply by increasing the importance sample size without altering the base Laplace approximation.

Where Pith is reading between the lines

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

  • The sampling correction could be paired with other marginalization schemes to handle hierarchical models outside the current latent-Gaussian class.
  • Adaptive or learned proposal distributions might reduce the sample count needed to reach a target accuracy level.
  • The automatic-differentiation implementation suggests straightforward extension to larger-scale problems where gradient information is already available.

Load-bearing premise

A suitable proposal distribution exists for the importance sampler such that the correction is both effective and computationally tractable across the range of latent Gaussian models considered.

What would settle it

Compute the exact posterior for a simple latent Gaussian model with non-Gaussian likelihood and verify whether the discrepancy with the importance-sampling-corrected ILA posterior shrinks toward zero as the number of importance samples is increased.

Figures

Figures reproduced from arXiv: 2605.20345 by Charles C. Margossian, Daniel R. Sheldon, Jinlin Lai.

Figure 1
Figure 1. Figure 1: Error of estimating the means of parameters as a function of time in seconds for the [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Running time to collect 100,000 samples and average ESS/min of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Error of estimating the means of parameters as a function of time in seconds for the sparse [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparing the sampled posterior against the ground-truth from NUTS. We demonstrate [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Error of estimating E[Lbase] and E[log σβ] as a function of time in seconds for the Epil2 model (Lbase is transformed from L in the unconstrained space). Results are averaged from 5 independent runs. Ground-truth is estimated from NUTS with non-centered parameterization. References [1] Raj Agrawal, Brian Trippe, Jonathan Huggins, and Tamara Broderick. The kernel interaction trick: Fast Bayesian discovery o… view at source ↗
Figure 6
Figure 6. Figure 6: Error of estimating the means of parameters as a function of time in seconds for the [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Error of estimating E[f80] and E[log α] as a function of time in seconds for the synthesized Gaussian process with Poisson likelihood. Results are averaged from 5 independent runs. 0 2500 5000 7500 10000 time 0.000 0.001 0.002 0.003 0.004 Error for lo g caux QMC (n=4) 0 2500 5000 7500 10000 time QMC (n=16) 0 2500 5000 7500 10000 time QMC (n=64) 0 2500 5000 7500 10000 time 0.000 0.001 0.002 0.003 0.004 0.00… view at source ↗
Figure 8
Figure 8. Figure 8: Error of estimating E[log caux] and E[log χ] as a function of time in seconds for the sparse kernel interaction model. Results are averaged from 5 independent runs. 0 2500 5000 7500 10000 time 0.000 0.002 0.004 0.006 Error for 0 QMC (n=4) 0 2500 5000 7500 10000 time QMC (n=16) 0 2500 5000 7500 10000 time QMC (n=64) 0 2500 5000 7500 10000 time 0.00 0.05 0.10 0.15 0.20 Error for lo g T1 QMC (n=4) 0 2500 5000… view at source ↗
Figure 9
Figure 9. Figure 9: Error of estimating E[β0] and E[log T1] as a function of time in seconds for the mixed￾effects model. Results are averaged from 5 independent runs. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of πˆ RQMC(U|θ, y) with different θ and n for the simple example model, using 5,000 evaluation points in [0, 1]. The distribution is more uniform as we increase n. Also, πˆ RQMC(U|θ, y) is not continuous. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
read the original abstract

Latent Gaussian models (LGMs) are a popular class of Bayesian hierarchical models that include Gaussian processes, as well as certain spatial models and mixed-effect models. Efficient Bayesian inference of LGMs often requires marginalizing out the latent variables. For LGMs with a non-Gaussian likelihood, exact marginalization is not possible and a popular approach is to do approximate marginalization with an integrated Laplace approximation (ILA). Using ILA produces an approximate posterior which, in some settings, can differ significantly from the correct posterior, which impacts downstream applications. We propose an importance sampling scheme to correct the error introduced by ILA. By increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior. This idea is realized with various techniques, including pseudo-marginalization, quasi-Monte Carlo and randomized quasi-Monte Carlo. We implement our methods in an automatic differentiation framework to support gradient-based algorithms when doing inference on the hyperparameters. For the latter, we specifically consider the use of Hamiltonian Monte Carlo. We demonstrate the benefits of reduced error in various applied models.

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

2 major / 2 minor

Summary. The paper proposes an importance sampling scheme to correct errors from the integrated Laplace approximation (ILA) when marginalizing latent variables in latent Gaussian models (LGMs) with non-Gaussian likelihoods. The central claim is that the corrected posterior converges to the exact posterior as the number of importance samples increases. Realizations include pseudo-marginal, quasi-Monte Carlo (QMC), and randomized QMC methods, with implementation in an automatic differentiation framework to enable gradient-based hyperparameter inference via Hamiltonian Monte Carlo (HMC). Benefits are demonstrated on applied models including Gaussian processes and spatial models.

Significance. If a low-variance proposal can be constructed that remains tractable, the correction would meaningfully reduce ILA-induced bias in LGMs where approximation error affects downstream tasks. The AD/HMC integration and QMC variants are practical strengths that could improve efficiency over naive sampling corrections.

major comments (2)
  1. [Abstract] Abstract: The convergence claim ('by increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior') is load-bearing but rests on the unstated assumption that a proposal distribution q exists whose support contains the target and whose importance weights have controlled variance. In high-dimensional LGMs the posterior is typically far from Gaussian; any proposal derived from the Laplace approximation will have poor overlap exactly where ILA error is largest, undermining both unbiasedness (for pseudo-marginal HMC) and computational feasibility.
  2. [Methods] Methods (proposal construction): The manuscript lists pseudo-marginal, QMC and RQMC realizations but does not specify how the importance proposal is built or adapted to the hyperparameters. Without this detail it is impossible to verify that the scheme remains both unbiased and tractable across the range of latent dimensions and likelihoods considered.
minor comments (2)
  1. [Methods] Notation for the corrected posterior and the importance weights should be introduced with a single consistent equation early in the methods to avoid later ambiguity.
  2. [Experiments] The empirical demonstrations would benefit from explicit reporting of effective sample size or variance of the importance weights to substantiate the practical convergence rate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below, clarifying the assumptions underlying our convergence claim and providing additional details on proposal construction. We have revised the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The convergence claim ('by increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior') is load-bearing but rests on the unstated assumption that a proposal distribution q exists whose support contains the target and whose importance weights have controlled variance. In high-dimensional LGMs the posterior is typically far from Gaussian; any proposal derived from the Laplace approximation will have poor overlap exactly where ILA error is largest, undermining both unbiasedness (for pseudo-marginal HMC) and computational feasibility.

    Authors: We agree that the stated convergence relies on standard importance sampling assumptions: the proposal q must have support containing that of the target posterior and the importance weights must have finite variance for practical convergence. These conditions were implicit but will now be stated explicitly in the revised abstract and methods. Regarding high-dimensional LGMs, we acknowledge that a Laplace-derived proposal can have limited overlap where the ILA error is largest, which may lead to high variance and affect computational feasibility for pseudo-marginal HMC. However, the estimator remains unbiased (and thus the corrected posterior converges) as the number of samples tends to infinity whenever the support condition holds, independent of dimension. We will add a new paragraph discussing proposal quality, variance control strategies (including the QMC variants already in the paper), and practical limitations in high dimensions. revision: yes

  2. Referee: [Methods] Methods (proposal construction): The manuscript lists pseudo-marginal, QMC and RQMC realizations but does not specify how the importance proposal is built or adapted to the hyperparameters. Without this detail it is impossible to verify that the scheme remains both unbiased and tractable across the range of latent dimensions and likelihoods considered.

    Authors: We thank the referee for highlighting this omission. The original manuscript emphasized the general framework and its realizations but did not provide a self-contained description of proposal construction. In the revised manuscript we will insert a dedicated subsection that (i) specifies the default proposal as the Gaussian approximation obtained from the Laplace step, (ii) describes how the proposal is re-centered and re-scaled when hyperparameters change during outer-loop inference, and (iii) outlines simple adaptation heuristics (e.g., moment matching or low-rank updates) that preserve unbiasedness while remaining tractable. These additions will allow readers to verify the conditions for unbiasedness and computational feasibility across the latent dimensions and likelihoods considered in the experiments. revision: yes

Circularity Check

0 steps flagged

No circularity: importance sampling correction is a standard, independently motivated technique

full rationale

The paper proposes using importance sampling (including pseudo-marginal, QMC, and RQMC variants) to correct the known approximation error of ILA in LGMs, with the claim that the corrected posterior converges to the exact one as the number of samples grows. This follows directly from the standard properties of importance sampling and does not reduce to any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The derivation chain is self-contained and relies on external statistical results rather than circular internal constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of Laplace approximation and importance sampling in Bayesian hierarchical models; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Integrated Laplace approximation introduces a correctable error for non-Gaussian likelihoods in latent Gaussian models.
    This premise underpins the need for and the design of the importance sampling correction.

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