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arxiv: 2606.25745 · v1 · pith:TIAYUM2Anew · submitted 2026-06-24 · 📊 stat.ML · cs.LG

Gaussian Mean Field Variational Inference can Overestimate Predictive Variance

James Odgers , Ben Riegler , Siddharth Swaroop , Vincent Fortuin This is my paper

Pith reviewed 2026-06-25 19:46 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords mean field variational inferencepredictive varianceBayesian linear regressionvariational inferencecold posterior effectGaussian modelsuncertainty estimation
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The pith

Mean-field variational inference overestimates predictive variance on in-distribution test points in Bayesian linear regression.

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

The paper establishes that in conjugate Bayesian linear regression, mean field variational inference underestimates variance in parameter space yet overestimates predictive variance for test points drawn from the training distribution. This occurs because underestimation in some directions necessarily forces overestimation in others, especially those aligned with concentrated training data. A sympathetic reader would care because the result challenges the standard view of MFVI as variance-underestimating and explains why temperature adjustments can improve its predictive behavior.

Core claim

By analyzing conjugate Bayesian linear regression, the authors show that the MFVI posterior underestimates variance in parameter space but overestimates predictive variance compared to the exact posterior, with the overestimation occurring in directions where training data concentrates. This leads to the result that for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. They also identify a pathological case where MFVI fails to reduce predictive variance relative to the prior on in-distribution data and connect the effect to the cold posterior phenomenon by showing that temperature scaling yields predictions closer to t

What carries the argument

The directional decomposition of predictive variance under the mean-field Gaussian approximation versus the exact conjugate posterior in Bayesian linear regression.

If this is right

  • For test points drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior.
  • If MFVI underestimates predictive variance in some directions it necessarily overestimates in others, with overestimation concentrated where training data lies.
  • Varying the temperature in the MFVI objective can correct the overestimation and produce predictive distributions closer to the exact posterior.
  • A pathological case exists in which MFVI fails to reduce predictive variance below the prior level on in-distribution data.

Where Pith is reading between the lines

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

  • The directional bias may partly explain why colder posteriors often improve calibration in variational inference applications.
  • The finding suggests evaluating variational methods on predictive quantities rather than parameter-space variance alone when assessing uncertainty quality.
  • Similar overestimation effects could arise in non-conjugate or non-Gaussian settings where exact posteriors cannot be computed for comparison.

Load-bearing premise

The central results rely on conjugate Gaussian priors and likelihoods in Bayesian linear regression that permit closed-form exact posteriors and predictive distributions.

What would settle it

Compute the expected predictive variance for a test point sampled from the training distribution under both MFVI and the exact posterior in a simple Bayesian linear regression model and check whether the MFVI value is larger.

Figures

Figures reproduced from arXiv: 2606.25745 by Ben Riegler, James Odgers, Siddharth Swaroop, Vincent Fortuin.

Figure 1
Figure 1. Figure 1: Although the MFVI posterior underestimates variance on average, it overestimates variance along the direction in which the data lie. Here we show this on a simple 2D linear regression, where the input data is restricted to lie on the subspace x1 = x2 (Figure 1a). We see that, for most of the input space, MFVI predictive uncertainty is less than the exact posterior predictive (blue region). At the same time… view at source ↗
Figure 4
Figure 4. Figure 4: Test NLL as a function of temperature for OOD test data orthogonal to the training subspace, averaged over 10,000 repetitions, for (a) P = 2 and (b) P = 1024. For P = 2, the pattern is reversed compared to the ID case: warm posteriors (T > 1) improve performance. For P = 1024, MFVI at T = 1 already closely matches the exact posterior, so the optimal temperature is approximately T ≈ 1. posterior is informed… view at source ↗
Figure 3
Figure 3. Figure 3: Test NLL as a function of temperature for ID test data, averaged over 10,000 repetitions, for (a) P = 2 and (b) P = 1024. Horizontal lines indicate the NLL of the exact posterior and the MAP solution. In both cases, an optimal temperature exists where the T-MFVI predictive matches the exact posterior. In high dimensions, this optimal temperature is much lower, the improvement from tuning T is much larger, … view at source ↗
Figure 5
Figure 5. Figure 5: Predictions and credible intervals for fixed basis function regression with (a) Q = 16 and (b) Q = 1024 basis functions. MFVI with T = 1 significantly overestimates the variance of the exact posterior on in-distribution data, and lower temperatures correct this. to the single data direction, and MFVI averages over all of them, the MFVI predictive variance in any one orthogonal direction is already close to… view at source ↗
Figure 7
Figure 7. Figure 7: Plot showing the mean and credible interval for a small (Figure 7a), and large (Figure 7b) BNN trained with IVON. Similar to the basis function regression in [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Optimal temperatures for various measures of divergence are shown across independent train-test splits. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Optimal temperatures for various measures of divergence are shown across independent train-test splits. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
read the original abstract

Mean Field Variational Inference (MFVI) is widely understood to underestimate posterior variance. By analysing conjugate Bayesian Linear Regression (BLR), we show that this characterization is incomplete: while MFVI underestimates the variance in parameter space, it can overestimate the predictive variance compared to the exact posterior. We show that if the MFVI posterior underestimates predictive variances in some directions, it necessarily overestimates them in others. Crucially, this overestimation occurs in directions where the training data concentrates. This leads to the surprising result that, for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. We demonstrate a pathological case of this effect, where the MFVI posterior fails to reduce predictive variance compared to the prior on in distribution data. We connect these results to the Cold Posterior Effect, arguing that varying the temperature can correct this overestimation, yielding predictions closer to those of the exact posterior. We validate our theory on synthetic and real-world regression tasks.

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 paper analyzes Gaussian mean-field variational inference (MFVI) for conjugate Bayesian linear regression (BLR). It shows that MFVI underestimates posterior variance in parameter space yet can overestimate predictive variance for test points drawn from the training distribution. The analysis establishes that underestimation in some directions necessarily implies overestimation in others (particularly those aligned with the data Gram matrix), yielding the result that the expected predictive variance under MFVI exceeds that of the exact posterior. The work identifies a pathological case where MFVI predictive variance fails to contract relative to the prior on in-distribution data, connects the phenomenon to the cold posterior effect via temperature scaling, and validates the claims on synthetic and real regression tasks.

Significance. If the central claims hold, the result is significant because it supplies a precise, closed-form counterexample to the standard characterization of MFVI as uniformly underestimating variance. The conjugate BLR setting permits direct comparison of the MFVI diagonal covariance (reciprocals of the diagonal of the posterior precision) against the exact posterior covariance via quadratic forms x^T D x versus x^T Sigma x, reducing to trace comparisons with the empirical Gram matrix. This directional trade-off and the explicit link to temperature correction constitute a substantive refinement of variational inference theory with immediate implications for predictive calibration.

minor comments (3)
  1. [§3] §3 (or the section deriving the predictive variance): the reduction from the quadratic-form comparison to trace(D X^T X) versus trace(Sigma X^T X) is central; an explicit intermediate equation would make the step from the positive-semidefinite ordering to the expected-variance inequality fully transparent.
  2. [Pathological case section] The pathological case (where MFVI predictive variance equals the prior variance on in-distribution points) is load-bearing for the overestimation claim; a short appendix deriving the exact condition on the prior precision and Gram matrix would strengthen reproducibility.
  3. [Notation introduction] Notation for the mean-field covariance (denoted D in the skeptic summary) should be introduced with an equation number at first use and cross-referenced when the trace comparison is stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as the recommendation for minor revision. No major comments were provided in the report, so we have no specific points to address point-by-point. We will make any minor editorial or formatting changes requested by the editor or in a subsequent round if applicable.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result follows from explicit closed-form expressions for both the exact posterior and the optimal MFVI Gaussian in conjugate Bayesian linear regression. The MFVI covariance is defined directly as the diagonal of the inverse posterior precision matrix, and the predictive-variance comparison is obtained by evaluating the resulting quadratic forms on the empirical data Gram matrix; this algebraic identity holds without fitted parameters, self-citations, or ansatzes that presuppose the target inequality. The argument is therefore self-contained and does not reduce any claimed prediction to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis depends on the existence of closed-form exact posteriors under Gaussian conjugacy; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Conjugate Gaussian prior and likelihood in Bayesian linear regression permit closed-form computation of both the exact posterior and the exact predictive distribution.
    This is the setting in which the under/overestimation comparison is performed.

pith-pipeline@v0.9.1-grok · 5709 in / 1250 out tokens · 17924 ms · 2026-06-25T19:46:09.293921+00:00 · methodology

discussion (0)

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Reference graph

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