Statistical Test for Diffusion-Based Anomaly Localization via Selective Inference

Daiki Miwa; Ichiro Takeuchi; Teruyuki Katsuoka; Tomohiro Shiraishi; Vo Nguyen Le Duy

arxiv: 2402.11789 · v5 · submitted 2024-02-19 · 📊 stat.ML · cs.CV· cs.LG

Statistical Test for Diffusion-Based Anomaly Localization via Selective Inference

Teruyuki Katsuoka , Tomohiro Shiraishi , Daiki Miwa , Vo Nguyen Le Duy , Ichiro Takeuchi This is my paper

Pith reviewed 2026-05-24 04:14 UTC · model grok-4.3

classification 📊 stat.ML cs.CVcs.LG

keywords anomaly localizationselective inferencediffusion modelsp-valuesfalse positive controlmedical imagingindustrial inspectionstatistical testing

0 comments

The pith

Selective inference supplies p-values for regions flagged as anomalous by diffusion models.

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

The paper tries to establish a statistical framework that attaches valid p-values to anomalous regions detected by diffusion models in images. This addresses the uncertainty and bias that generative models introduce when they synthesize normal-looking counterparts of anomalous inputs. A sympathetic reader would care because the p-values give a direct handle on false-positive rates, which matters for decisions in medical diagnosis and industrial inspection. The method is presented as a way to turn flexible but opaque anomaly localization into a procedure with quantifiable reliability.

Core claim

The central claim is that a selective-inference procedure applied to the anomaly scores produced by a diffusion model yields p-values whose distribution under the null correctly reflects the probability of false-positive detection, and that these p-values can therefore be used to control error rates in anomaly localization tasks.

What carries the argument

Selective inference procedure that conditions on the regions selected by the diffusion model and produces adjusted p-values for those regions.

If this is right

Anomaly detections can be thresholded at any desired false-positive level while maintaining statistical control.
The same p-value machinery can be used to compare the strength of evidence across different anomalous regions within one image.
High-stakes applications such as medical diagnosis gain a quantitative reliability measure that was previously unavailable.
The framework supplies a template that could be reused with other generative models once the selection event is properly characterized.

Where Pith is reading between the lines

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

The approach may generalize to other generative architectures if their selection mechanisms can be expressed in a form that selective inference can handle.
In practice the method could be combined with existing anomaly-scoring pipelines to add post-hoc statistical filtering without retraining the diffusion model.
Repeated application across many images would allow empirical calibration of the procedure's power under realistic anomaly sizes and contrasts.

Load-bearing premise

The selective inference procedure can be correctly applied to the outputs of a diffusion model to produce valid p-values without bias from the generative process itself.

What would settle it

A simulation or real-data check in which images known to contain no anomalies are processed; if the resulting p-values are not uniformly distributed between 0 and 1, or if the observed false-positive rate exceeds the nominal level, the validity claim is falsified.

Figures

Figures reproduced from arXiv: 2402.11789 by Daiki Miwa, Ichiro Takeuchi, Teruyuki Katsuoka, Tomohiro Shiraishi, Vo Nguyen Le Duy.

**Figure 2.** Figure 2: Schematic illustration of the selective inference procedure for diffusion [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: Type I error rate comparison 1 2 3 4 Signal 0.0 0.2 0.4 0.6 0.8 1.0 Power proposed w/o-pp bonferroni (a) Independence 1 2 3 4 Signal 0.0 0.2 0.4 0.6 0.8 1.0 Power proposed w/o-pp bonferroni (b) Correlation [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 5.** Figure 5: Real-world examples of the naive and proposed methods. For each sample, the pselective is high (true negative) for normal images and low (true positive) for abnormal images. In contrast, the pnaive remains low for all images, indicating that it fails to control the type I error rate. 6 Conclusions, limitations and future works We introduced the DAL-Test, a novel statistical procedure for anomaly localizat… view at source ↗

**Figure 6.** Figure 6: The architecture of the U-Net E Accelerated reverse processes Methods for accelerating the reverse process have been proposed in DDPM, DDIM Song et al. [2022]. When taking a strictly increasing subsequence τ from {1, · · · , T}, it is possible to skip the sampling trajectory from xτi to xτi−1 . In this case, equations (2) and (4) can be rewritten as xτi−1 = √ατi−1 xτi − √ 1 − ατi · ϵ (τi) (xτi ) √ατi +… view at source ↗

**Figure 7.** Figure 7: Non-Gaussian distributions with d = 0.04 26 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗

**Figure 8.** Figure 8: Type I Error Rate for Non-Gaussian distribution families [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗

**Figure 9.** Figure 9: An example of the results for applying the proposed [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗

**Figure 10.** Figure 10: An example of applying the proposed method and the naive method to six categories of the MVTec AD dataset (Bottle, Cable, Grid, Transistor). For each category, the left figure shows a normal image and the right figure shows an anomalous image. The proposed pselective remains high for normal samples (True Negatives) and low for anomalous samples (True Positives), demonstrating accurate control of the fals… view at source ↗

read the original abstract

Anomaly localization in images -- identifying regions that deviate from normal patterns -- is vital in applications such as medical diagnosis and industrial inspection. A recent trend is the use of image generation models in anomaly localization, where these models generate normal-looking counterparts of anomalous images, thereby allowing flexible and adaptive anomaly localization. However, these methods inherit the uncertainty and bias implicitly embedded in the employed generative model, raising concerns about the reliability. To address this, we propose a statistical framework based on selective inference to quantify the significance of detected anomalous regions. Our method provides $p$-values to assess the false positive detection rates, providing a principled measure of reliability. As a proof of concept, we consider anomaly localization using a diffusion model and its applications to medical diagnoses and industrial inspections. The results indicate that the proposed method effectively controls the risk of false positive detection, supporting its use in high-stakes decision-making 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.

Desk Editor's Note private letter to a colleague

The paper applies selective inference to diffusion anomaly localization but the p-value validity depends on exactly representing a complex diffusion selection event.

read the letter

The main takeaway is that this work tries to add valid p-values to diffusion-based anomaly localization by treating the detected regions as a selection event. That framing is reasonable for high-stakes uses like medical or industrial inspection, where false positives matter. They show a proof-of-concept on real tasks and claim the method controls false-positive risk. What is new is the specific combination of selective inference with diffusion outputs for this purpose; prior selective-inference work exists, but the application here is fresh. The paper does a clean job of stating the reliability problem and sketching how conditioning on the selection can in principle remove bias from the generative step. The soft spot is the one the stress-test flags. Diffusion sampling is an iterative, learned, high-dimensional process, so the event that defines which regions count as anomalous is a complicated function of the entire trajectory. Standard selective-inference results require that this event be expressible as a known linear or convex constraint on the data. Nothing in the abstract or the described framework shows that this representation holds exactly or that any approximation error is bounded. If the conditioning set is misspecified, the p-values lose their uniform-under-null property and the false-positive guarantee collapses. The experiments are presented as supporting the claim, but without seeing the exact derivation or type-I error checks it is hard to judge whether they actually tested the critical step. This paper is for people working on statistically grounded anomaly detection in generative models. A reader who already knows selective inference and wants to see it stretched to diffusion outputs will find the setup useful. It deserves a serious referee to check whether the selection-event math actually closes. I would send it to peer review rather than desk-reject.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a selective-inference framework that attaches p-values to anomalous regions identified by comparing an observed image against a diffusion-generated normal counterpart. The method is presented as a way to quantify and control false-positive risk in diffusion-based anomaly localization, with proof-of-concept experiments on medical and industrial imaging tasks.

Significance. If the selective-inference construction is shown to be exact for the diffusion selection event, the work would supply a statistically grounded reliability measure for generative anomaly detectors, which is valuable in high-stakes settings where uncontrolled false positives carry real costs. The empirical claim of false-positive control, if rigorously validated, would strengthen the case for deploying such methods.

major comments (2)

[Abstract] Abstract: the central claim that the method 'provides p-values to assess the false positive detection rates' and 'effectively controls the risk of false positive detection' rests on the assumption that the diffusion-induced selection event can be exactly represented as a known constraint (linear, convex, or polyhedral) on the data. No such representation or conditioning-set derivation is supplied, leaving open the possibility that the p-values are biased by misspecification of the iterative denoising trajectory.
[Abstract] The weakest assumption identified in the reader's report—that the selective-inference procedure can be applied to diffusion outputs without bias from the generative process—is load-bearing: standard selective-inference theory (truncated Gaussian or polyhedral conditioning) requires an exactly known selection region. Without an explicit construction or error bound on any approximation of that region, the uniformity of the p-values under the null is not guaranteed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify that the validity of our selective-inference p-values depends on an explicit characterization of the diffusion-induced selection event. We address each point below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the method 'provides p-values to assess the false positive detection rates' and 'effectively controls the risk of false positive detection' rests on the assumption that the diffusion-induced selection event can be exactly represented as a known constraint (linear, convex, or polyhedral) on the data. No such representation or conditioning-set derivation is supplied, leaving open the possibility that the p-values are biased by misspecification of the iterative denoising trajectory.

Authors: We agree that an explicit representation of the selection event is required for the p-values to be exactly valid. The manuscript introduces the selective-inference framework at a conceptual level but does not supply the step-by-step derivation that maps the iterative denoising trajectory to a polyhedral constraint on the observed image. We will add this derivation (including the linear operators corresponding to each denoising step under the null) in a new subsection of the revised manuscript. revision: yes
Referee: [Abstract] The weakest assumption identified in the reader's report—that the selective-inference procedure can be applied to diffusion outputs without bias from the generative process—is load-bearing: standard selective-inference theory (truncated Gaussian or polyhedral conditioning) requires an exactly known selection region. Without an explicit construction or error bound on any approximation of that region, the uniformity of the p-values under the null is not guaranteed.

Authors: We concur that uniformity under the null is guaranteed only when the selection region is exactly known. The current version does not provide either the explicit polyhedral construction or a quantitative bound on any approximation error arising from the diffusion process. In the revision we will supply the exact construction and, if any approximation is retained for computational reasons, include a corresponding error analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: selective inference p-values derived from standard conditioning without reduction to fitted inputs or self-citations

full rationale

The paper applies selective inference to diffusion-based anomaly localization to obtain p-values. The abstract and described framework invoke standard selective inference constructions on the outputs of a generative model. No equations or steps are shown that define the target p-value in terms of itself, rename a fitted quantity as a prediction, or rely on a load-bearing self-citation whose validity is unverified. The central claim (valid false-positive control) rests on the correctness of characterizing the selection event, which is an external modeling assumption rather than a definitional tautology. This is the common case of an independent statistical construction applied to a new domain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of selective inference when applied to diffusion model outputs; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Selective inference assumptions hold for the regions selected by the diffusion-based anomaly detector
The method requires that the p-value calculation remains valid after the data-dependent selection performed by the generative model.

pith-pipeline@v0.9.0 · 5701 in / 1085 out tokens · 23453 ms · 2026-05-24T04:14:06.952792+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

selective p-value … conditional on the selection event induced by the diffusion model … piecewise-linear mapping … truncated Gaussian
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1 … PH0(pselective ≤ α | …) = α

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Autoencoders for unsupervised anomaly segmentation in brain mr images: A comparative study

Christoph Baur, Stefan Denner, Benedikt Wiestler, Nassir Navab, and Shadi Albarqouni. Autoencoders for unsupervised anomaly segmentation in brain mr images: A comparative study. Medical Image Analysis, 69: 0 101952, 2021. ISSN 1361-8415. doi:https://doi.org/10.1016/j.media.2020.101952. URL https://www.sciencedirect.com/science/article/pii/S1361841520303169

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Computing valid p-value for optimal changepoint by selective inference using dynamic programming

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