arxiv: 2604.18089 · v1 · submitted 2026-04-20 · 💻 cs.LG · stat.ML

Recognition: unknown

Towards E-Value Based Stopping Rules for Bayesian Deep Ensembles

David R\"ugamer, Emanuel Sommer, Julius Kobialka, Rickmer Schulte, Sarah Deubner

Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:39 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords Bayesian deep ensemblesE-valuesstopping rulesMCMCsequential hypothesis testinguncertainty quantificationdeep learning

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The pith

E-value sequential tests give a principled early stopping rule for MCMC sampling in Bayesian deep ensembles.

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

The paper develops a way to decide when MCMC sampling in Bayesian deep ensembles has stopped adding value. Bayesian deep ensembles improve uncertainty estimates by combining optimized deep ensembles with additional MCMC chains, yet full sampling remains computationally heavy. The authors treat the addition of each new chain as a sequential hypothesis test that uses E-values to check whether further samples still improve predictions over the fixed deep-ensemble baseline. Because the test is anytime-valid, sampling can be halted as soon as the null hypothesis of no further gain can be rejected. Experiments across several tasks show that the rule typically terminates after only a fraction of the usual full-chain budget while retaining the observed gains.

Core claim

We propose a stopping rule based on E-values. We formulate the ensemble construction as a sequential anytime-valid hypothesis test, providing a principled way to decide whether or not to reject the null hypothesis that MCMC offers no improvement over a strong baseline, to early stop the sampling. Empirically, we study this approach for diverse settings. Our results demonstrate the efficacy of our approach and reveal that only a fraction of the full-chain budget is often required.

What carries the argument

An E-value based sequential anytime-valid hypothesis test that rejects the null of no improvement from additional MCMC samples over the initial deep-ensemble baseline.

If this is right

Only a fraction of the usual full-chain sampling budget is typically needed.
The procedure supplies a statistically valid, fixed-budget-independent criterion for halting MCMC.
The same rule can be applied across varied network architectures and data sets without retuning.
Early stopping preserves the uncertainty-quantification gains that MCMC is known to deliver over plain deep ensembles.

Where Pith is reading between the lines

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

The same E-value construction could be reused to monitor other sequential Monte-Carlo or variational procedures whose cost grows with iteration count.
If the test is combined with cheaper surrogate models for the null, overall wall-clock time for Bayesian neural-network training could drop further.
Deployment pipelines that already cache deep-ensemble checkpoints could insert the E-value monitor with negligible extra code.

Load-bearing premise

The E-value sequential test stays valid and correctly detects when extra MCMC samples stop improving the ensemble beyond the deep-ensemble baseline in the neural-network regimes examined.

What would settle it

An experiment in which the rule stops sampling and the resulting ensemble performs no better than the deep-ensemble baseline, even though continuing the full MCMC run would have produced a clear improvement.

Figures

Figures reproduced from arXiv: 2604.18089 by David R\"ugamer, Emanuel Sommer, Julius Kobialka, Rickmer Schulte, Sarah Deubner.

**Figure 2.** Figure 2: Evolution of E-values for ResNet-7 chains [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Chainwise and ensemble hold-out test performance of the E-value induced minimal BDEs in comparison [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Chainwise and ensemble hold-out test performance of the E-value induced minimal BDEs in comparison [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Chainwise and ensemble hold-out test performance of the E-value induced minimal BDEs in comparison [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Chainwise and ensemble hold-out test performance of the E-value induced minimal BDEs in comparison [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Chainwise and ensemble hold-out test performance of the E-value induced minimal BDEs in comparison [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: Chainwise and ensemble hold-out test performance of the E-value induced minimal BDEs in comparison [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

read the original abstract

Bayesian Deep Ensembles (BDEs) represent a powerful approach for uncertainty quantification in deep learning, combining the robustness of Deep Ensembles (DEs) with flexible multi-chain MCMC. While DEs are affordable in most deep learning settings, (long) sampling of Bayesian neural networks can be prohibitively costly. Yet, adding sampling after optimizing the DEs has been shown to yield significant improvements. This leaves a critical practical question: How long should the sequential sampling process continue to yield significant improvements over the initial optimized DE baseline? To tackle this question, we propose a stopping rule based on E-values. We formulate the ensemble construction as a sequential anytime-valid hypothesis test, providing a principled way to decide whether or not to reject the null hypothesis that MCMC offers no improvement over a strong baseline, to early stop the sampling. Empirically, we study this approach for diverse settings. Our results demonstrate the efficacy of our approach and reveal that only a fraction of the full-chain budget is often required.

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.

Desk Editor's Note private letter to a colleague

The paper applies E-values to create early stopping for MCMC in Bayesian deep ensembles, but the supermartingale property for the test statistic is not clearly established.

read the letter

The core idea is to stop MCMC sampling early in Bayesian deep ensembles once E-values show that extra chains are unlikely to beat a fixed deep ensemble baseline. They cast the decision as a sequential hypothesis test and report that only a fraction of the usual sampling budget is often needed in their experiments. That addresses a real pain point: full MCMC on neural nets is expensive, yet some added sampling helps uncertainty estimates. The empirical results across a few settings give a sense of the potential savings, which is the practical hook here. The approach draws on established E-value machinery for anytime-valid tests, so the framing itself is not invented from scratch. What stands out is the direct application to the ensemble construction process. The main weakness is that the validity claim depends on the improvement statistic forming a supermartingale under the null of no gain. With neural nets, metrics like accuracy or log-likelihood on a validation set can have dependencies from shared optimization and data reuse that break the required conditional expectation property. The paper does not lay out an explicit martingale construction or verification for the chosen statistic, so the anytime-valid guarantee rests on an assumption rather than a shown fact. Experiments are useful as a starting point but cannot substitute for that check. Readers working on efficient Bayesian deep learning or sequential testing would get the most from this. It is worth sending to peer review so referees can examine the test construction and run additional checks on the martingale condition.

Referee Report

2 major / 2 minor

Summary. The paper proposes an E-value-based stopping rule for Bayesian Deep Ensembles (BDEs), framing the sequential addition of MCMC samples as an anytime-valid hypothesis test. The null hypothesis is that further MCMC sampling yields no improvement over a fixed deep ensemble (DE) baseline; the procedure rejects the null (and continues sampling) while the cumulative E-value exceeds a threshold, with the goal of early-stopping once gains plateau. Empirical studies across diverse settings indicate that only a fraction of the full-chain MCMC budget is typically required.

Significance. If the sequential test is valid, the method supplies a statistically principled, computationally efficient way to allocate MCMC resources in BDE training, addressing the practical cost barrier of long-chain sampling while preserving the performance gains that MCMC can provide over DEs alone. This could make uncertainty-aware Bayesian deep learning more accessible in resource-constrained settings.

major comments (2)

[§3] §3 (E-value construction): The central claim that the procedure yields an anytime-valid test requires that the chosen improvement statistic (predictive performance delta on a validation set) produces a supermartingale under the null of no MCMC gain. No explicit martingale construction, conditional-expectation argument, or proof is supplied showing that E[increment | filtration] ≤ 1 holds when the statistic is a non-linear functional of the posterior predictive and the DE baseline is itself optimized on overlapping data. This property is load-bearing for the validity of early stopping.
[§4] §4 (empirical validation): The reported experiments demonstrate early stopping but do not include a direct check (e.g., type-I error rate under a synthetic null where MCMC truly adds nothing) that the E-value process remains valid in the neural-network regime. Without such a diagnostic, it is unclear whether the observed savings reflect genuine anytime-validity or merely empirical behavior on the studied tasks.

minor comments (2)

[§2] Notation for the E-value process and the filtration should be introduced explicitly in §2 before the hypothesis-test formulation; current usage in the abstract and §3 is informal.
[Abstract] The abstract claims “only a fraction of the full-chain budget is often required” but does not report the precise fractions or variance across runs; a table summarizing budget savings per dataset would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our work. We address the major concerns point by point below and outline the revisions we plan to make.

read point-by-point responses

Referee: [§3] §3 (E-value construction): The central claim that the procedure yields an anytime-valid test requires that the chosen improvement statistic (predictive performance delta on a validation set) produces a supermartingale under the null of no MCMC gain. No explicit martingale construction, conditional-expectation argument, or proof is supplied showing that E[increment | filtration] ≤ 1 holds when the statistic is a non-linear functional of the posterior predictive and the DE baseline is itself optimized on overlapping data. This property is load-bearing for the validity of early stopping.

Authors: We appreciate the referee pointing out the need for a more explicit justification of the supermartingale property. In our construction, the E-value is defined using the ratio of the likelihood under the alternative (improved model) to the null, but we acknowledge that for the specific statistic involving non-linear predictive performance on validation data with overlapping optimization, a detailed conditional expectation argument is missing. We will revise §3 to include a formal proof sketch showing that under the null of no improvement, the expected increment of the E-value is at most 1, leveraging the fact that the baseline DE is fixed and the MCMC samples are drawn from the posterior. This will ensure the anytime-validity is rigorously established. revision: yes
Referee: [§4] §4 (empirical validation): The reported experiments demonstrate early stopping but do not include a direct check (e.g., type-I error rate under a synthetic null where MCMC truly adds nothing) that the E-value process remains valid in the neural-network regime. Without such a diagnostic, it is unclear whether the observed savings reflect genuine anytime-validity or merely empirical behavior on the studied tasks.

Authors: We agree that a direct validation of the type-I error control under a controlled null scenario would provide stronger evidence for the method's validity. In the revised manuscript, we will add a synthetic experiment where we simulate a setting in which additional MCMC samples do not improve upon the DE baseline (e.g., by using a fixed model or a null posterior), and report the empirical type-I error rate of the stopping procedure to confirm it does not exceed the nominal level. revision: yes

Circularity Check

0 steps flagged

No circularity: E-value stopping rule applies external theory without self-referential reduction

full rationale

The paper formulates ensemble construction as a sequential hypothesis test using E-values to decide early stopping for MCMC sampling in Bayesian Deep Ensembles. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The central claim relies on the external validity of E-value theory for anytime-valid testing rather than deriving the supermartingale property from the paper's own fitted quantities or prior self-citations. The skeptic concern about the test statistic's martingale property under the null is a question of assumption validity and external verification, not an internal circular reduction by construction. The derivation chain remains self-contained against the stated inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of E-value based sequential hypothesis testing from prior literature and the assumption that the null hypothesis of no improvement is appropriately defined for this setting.

axioms (1)

standard math E-values provide valid anytime-valid sequential tests for the null hypothesis that MCMC sampling offers no improvement
Invoked when formulating the ensemble construction as a sequential hypothesis test.

pith-pipeline@v0.9.0 · 5481 in / 1145 out tokens · 44543 ms · 2026-05-10T05:39:40.982009+00:00 · methodology

discussion (0)

Reference graph

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