Confidence is detection-like in high-dimensional spaces

Kevin O'Neill; Stephen M. Fleming; Wiktoria Kozyra

arxiv: 2410.18933 · v4 · submitted 2024-10-24 · 🧬 q-bio.NC

Confidence is detection-like in high-dimensional spaces

Wiktoria Kozyra , Kevin O'Neill , Stephen M. Fleming This is my paper

Pith reviewed 2026-05-23 19:10 UTC · model grok-4.3

classification 🧬 q-bio.NC

keywords confidencemetacognitionBayesian inferencesignal detection theorypositive evidence biashigh-dimensional spacesconvolutional neural networks

0 comments

The pith

Bayesian confidence estimates become detection-like in high-dimensional spaces because normalization over many unchosen alternatives creates a nonlinearity favoring congruent evidence.

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

The paper shows that detection-like confidence, where judgments respond mainly to evidence supporting the chosen decision, arises in Bayesian models when the space of possible decisions grows large. This pattern results from normalizing the chosen option's probability against many unchosen alternatives, which introduces a nonlinearity that amplifies decision-congruent evidence. A sympathetic reader would therefore treat the effect as rational Bayesian behavior rather than a metacognitive flaw, provided participants entertain more hypotheses than the experimental design assumes. The same normalization mechanism also produces and modulates positive evidence biases inside convolutional neural networks.

Core claim

Bayesian confidence estimates also exhibit heightened sensitivity to decision-congruent evidence in higher-dimensional signal detection theoretic spaces, leading to detection-like confidence criteria. This effect is due to a nonlinearity induced by normalisation of confidence by a large number of unchosen alternatives. Our analysis suggests that detection-like confidence is rational when participants consider a greater number of hypotheses than assumed by the experimenter. Further, we show that a similar dimensionality-driven mechanism can give rise to and modulate the strength of the positive evidence biases in convolutional neural networks.

What carries the argument

Normalisation of the chosen option's posterior probability by the sum over probabilities of a large number of unchosen alternatives

If this is right

Detection-like confidence criteria emerge from standard Bayesian updating once the decision space is high-dimensional
The size of the detection-like effect scales directly with the number of unchosen alternatives
Positive evidence biases observed in convolutional neural networks arise from the same dimensionality-driven normalisation
Metacognitive judgments should be evaluated against the participant's actual hypothesis space rather than the experimenter's reduced set

Where Pith is reading between the lines

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

If experimenters systematically underestimate the hypothesis space, they will misclassify rational confidence patterns as biases
Perceptual confidence studies could test the account by explicitly varying the number of response options available to participants
The normalisation account may extend to other metacognitive domains such as value-based or memory-based confidence

Load-bearing premise

Participants consider a greater number of hypotheses than the experimenter assumes when forming confidence judgments.

What would settle it

An experiment that measures or manipulates the effective number of alternatives participants consider and checks whether this number predicts the strength of the detection-like confidence pattern.

Figures

Figures reproduced from arXiv: 2410.18933 by Kevin O'Neill, Stephen M. Fleming, Wiktoria Kozyra.

**Figure 1.** Figure 1: Discrimination and detection criteria in a 2-dimensional signal detection theory (SDT) model. Circles indicate bivariate Gaussian stimulus distributions. Reproduced from Mazor et al. [8]. Previous efforts to explain a PEB have focused on explaining why confidence criteria become detection-like. These explanations include proposals that confidence estimates are based on response-congruent evidence [10, 11]… view at source ↗

**Figure 2.** Figure 2: A-D) Impact of computing confidence in higher-dimensional SDT spaces for a 2D decision between s1 and s2 on A) confidence, B) confidence’s sensitivity to evidence for the chosen alternative, C) confidence’s sensitivity to evidence for the unchosen alternative, and D-E) the positive evidence bias (PEB), here computed as the log ratio of the two previous quantities. Confidence is increasingly detection-like … view at source ↗

**Figure 3.** Figure 3: A) Samples from two equal variance stimulus distributions (s1 and s2), colour-coded by the level of confidence attached to a decision arising from each evidence sample, for increasing dimensionality. For a decision of s1, confidence increases from blue to green. For a decision of s2, confidence increases from red to yellow. B, C) Target:non-target variance ratio computed in (B) evidence space and (C) conf… view at source ↗

**Figure 4.** Figure 4: Impact of computing confidence in higher-dimensional SDT spaces on confidence surfaces for a 3-way decision between s1, s2 and s3. Conventions as in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: A) Model architecture (modified from [14]). An image x, belonging to class y, was passed through a deep neural network (DNN) encoder f, followed by two output layers: gclass generated a decision classifying the image, and gconf generated a confidence score by predicting p(ˆy = y), the probability that the decision was correct. B, C) Impact of varying the dimensionality of the training set on a positive evi… view at source ↗

**Figure 6.** Figure 6: Influence of distractor dimensions on (A) the posterior probability of target and non-target stimuli, (B) confidence for correct and incorrect decisions, and (C) the type 2 ROC. For all values of k, posterior probabilities tend to be higher for target relative to non-target stimuli, confidence tends to be higher after a correct decision, and metad’ is above 0. However, as k increases, differences between … view at source ↗

read the original abstract

Confidence estimates are often "detection-like" - driven by positive evidence in favour of a decision. This empirical observation has been interpreted as showing that human metacognition is limited by biases or heuristics. Here, we show that Bayesian confidence estimates also exhibit heightened sensitivity to decision-congruent evidence in higher-dimensional signal detection theoretic spaces, leading to detection-like confidence criteria. This effect is due to a nonlinearity induced by normalisation of confidence by a large number of unchosen alternatives. Our analysis suggests that detection-like confidence is rational when participants consider a greater number of hypotheses than assumed by the experimenter. Further, we show that a similar dimensionality-driven mechanism can give rise to and modulate the strength of the positive evidence biases in convolutional neural networks, linking our signal detection theoretic analysis to confidence behaviour in artificial systems.

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 shows mathematically that normalizing Bayesian posteriors over many alternatives in high-dimensional SDT produces detection-like confidence, and the same mechanism appears in CNNs, but the claim that this makes the pattern rational for humans hinges on an untested assumption about hypothesis spaces.

read the letter

The new piece here is the derivation that dimensionality plus normalization over unchosen options creates a nonlinearity making confidence more responsive to positive evidence. This holds in both the Bayesian SDT setup and in the CNN simulations they run. It gives a clean account of the positive evidence bias without needing to invoke metacognitive limits or heuristics as the primary cause. That linkage between the theoretical mechanism and artificial systems is the part that stands out from prior work on confidence biases. The math looks internally consistent on the description given, and the extension to CNNs is a useful bridge. The main limitation is that the rational interpretation for human data requires participants to maintain and normalize over a hypothesis space whose size exceeds the low-dimensional task structure the experimenter assumes. No evidence or argument is offered that people actually do this rather than restricting attention to the presented alternatives; if the effective space is small, the nonlinearity vanishes and the account does not apply. The abstract leaves the exact parameter settings and simulation details implicit, so a referee would need to check whether the result is robust or sensitive to those choices. This is worth sending to review for the computational part. Readers working on metacognition models or on confidence in neural networks would get the most from it; the core derivation is worth checking even if the human rationalization remains conditional.

Referee Report

2 major / 2 minor

Summary. The paper claims that Bayesian confidence estimates in high-dimensional signal detection theoretic (SDT) spaces exhibit detection-like properties (heightened sensitivity to decision-congruent evidence) due to a nonlinearity from normalizing posteriors over a large number of unchosen alternatives. This is argued to render the empirical pattern rational when participants maintain a higher-dimensional hypothesis space than the experimenter assumes, with an analogous dimensionality-driven mechanism demonstrated in convolutional neural networks.

Significance. If the central derivation holds, the work supplies a normative, mechanism-based account for detection-like confidence that does not require positing metacognitive limitations or heuristics. The explicit link between the SDT normalization effect and positive-evidence biases in CNNs is a strength, as is the parameter-free character of the core nonlinearity once dimensionality is fixed.

major comments (2)

[Abstract] Abstract (final sentence) and the corresponding discussion of rationality: the claim that detection-like confidence 'is rational when participants consider a greater number of hypotheses than assumed by the experimenter' is load-bearing for the interpretive conclusion, yet the manuscript supplies neither argument nor evidence that participants actually normalize over a high-dimensional space rather than restricting attention to the low-dimensional alternatives presented in the task. If the effective hypothesis space matches the experimenter’s design, the claimed nonlinearity and rationalization do not follow.
[The high-dimensional SDT model] The derivation of the normalization-induced nonlinearity (the section presenting the high-dimensional SDT model): while mathematically plausible, the manuscript does not report the precise dimensionality at which the detection-like criterion emerges, nor does it include a sensitivity analysis showing how the effect scales with the number of alternatives or with the prior over unchosen options. This leaves open whether the result is robust or depends on post-hoc choices of dimensionality.

minor comments (2)

Notation for the number of alternatives and the normalization term should be introduced explicitly with an equation number on first use to aid readability.
The CNN section would benefit from a brief statement of the architecture, training objective, and how confidence was extracted so that the dimensionality analogy can be evaluated directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (final sentence) and the corresponding discussion of rationality: the claim that detection-like confidence 'is rational when participants consider a greater number of hypotheses than assumed by the experimenter' is load-bearing for the interpretive conclusion, yet the manuscript supplies neither argument nor evidence that participants actually normalize over a high-dimensional space rather than restricting attention to the low-dimensional alternatives presented in the task. If the effective hypothesis space matches the experimenter’s design, the claimed nonlinearity and rationalization do not follow.

Authors: We agree that the manuscript provides neither direct argument nor evidence that participants actually maintain or normalize over a high-dimensional hypothesis space. The paper's core contribution is a normative derivation showing that detection-like confidence emerges as a rational consequence of Bayesian posterior normalization when the hypothesis space is high-dimensional. The rationality claim is therefore conditional on that assumption rather than an assertion that participants do in fact use such spaces. We will revise the abstract and discussion to make this conditional framing explicit and to avoid any implication that the manuscript demonstrates participants' actual hypothesis-space dimensionality. revision: yes
Referee: [The high-dimensional SDT model] The derivation of the normalization-induced nonlinearity (the section presenting the high-dimensional SDT model): while mathematically plausible, the manuscript does not report the precise dimensionality at which the detection-like criterion emerges, nor does it include a sensitivity analysis showing how the effect scales with the number of alternatives or with the prior over unchosen options. This leaves open whether the result is robust or depends on post-hoc choices of dimensionality.

Authors: The referee is correct that the manuscript does not report a specific dimensionality threshold for the emergence of the detection-like effect or include a sensitivity analysis with respect to the number of alternatives or the prior over unchosen options. Although the analytic derivation indicates that the nonlinearity strengthens with increasing dimensionality, we will add this analysis during revision, including quantitative reporting of the scaling and robustness checks across different numbers of alternatives and priors. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation follows from explicit Bayesian normalization in high-dimensional SDT without reduction to inputs or self-citations

full rationale

The paper's core derivation shows that posterior normalization over a large number of alternatives in high-dimensional signal detection spaces produces a nonlinearity yielding detection-like confidence criteria. This is obtained directly from the stated Bayesian model and does not rely on fitting parameters to the target pattern, renaming known results, or load-bearing self-citations. The interpretive claim that this renders the pattern 'rational' when participants maintain a high-dimensional hypothesis space is presented as a suggestion rather than a mathematical step that collapses to the inputs by construction. No equations or sections exhibit self-definitional loops, fitted-input predictions, or ansatz smuggling. The analysis is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard Bayesian updating and signal-detection normalization; the abstract introduces no new free parameters, axioms beyond domain conventions, or invented entities.

axioms (1)

domain assumption Participants perform Bayesian inference over a high-dimensional hypothesis space when computing confidence
Invoked to explain why normalization produces the observed effect; stated in the abstract as the condition under which detection-like confidence is rational.

pith-pipeline@v0.9.0 · 5661 in / 1252 out tokens · 29505 ms · 2026-05-23T19:10:19.473355+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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How to measure metacognition

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Overprecision is a property of thinking systems

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Visual capacity in the hemianopic field following a restricted occipital ablation

Lawrence Weiskrantz, Elizabeth K Warrington, MD Sanders, and J Marshall. Visual capacity in the hemianopic field following a restricted occipital ablation. Brain, 97(1):709–728, 1974

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Is blindsight like normal, near-threshold vision? Proceedings of the National Academy of Sciences, 94(25):14190–14194, 1997

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Does unconscious perception really exist? continuing the assc20 debate

Megan AK Peters, Robert W Kentridge, Ian Phillips, and Ned Block. Does unconscious perception really exist? continuing the assc20 debate. Neuroscience of Consciousness, 2017(1):nix015, 2017. 10 Confidence is detection-like in high-dimensional spaces A Confidence can be computed using the softmax As described in the Introduction, we can use Bayes’ rule to ...

work page 2017

[1] [1]

The construction of confidence in a perceptual decision

Ariel Zylberberg, Pablo Barttfeld, and Mariano Sigman. The construction of confidence in a perceptual decision. Frontiers in Integrative Neuroscience, 6:79, September 2012

work page 2012

[2] [2]

Perceptual confidence neglects decision- incongruent evidence in the brain

Megan A K Peters, Thomas Thesen, Yoshiaki D Ko, Brian Maniscalco, Chad Carlson, Matt Davidson, Werner Doyle, Ruben Kuzniecky, Orrin Devinsky, Eric Halgren, and H C Lau. Perceptual confidence neglects decision- incongruent evidence in the brain. Nature Human Behaviour, 1(7):0139, 2017

work page 2017

[3] [3]

Does perceptual confidence facilitate cognitive control? Attention, perception & psychophysics, 77(4):1295–1306, May 2015

Ai Koizumi, Brian Maniscalco, and H C Lau. Does perceptual confidence facilitate cognitive control? Attention, perception & psychophysics, 77(4):1295–1306, May 2015. 9 Confidence is detection-like in high-dimensional spaces

work page 2015

[4] [4]

Jason Samaha, Missy Switzky, and Bradley R. Postle. Confidence boosts serial dependence in orientation esti- mation. Journal of Vision, 19(4):25, April 2019

work page 2019

[5] [5]

Superior colliculus neuronal ensemble activity signals optimal rather than subjective confidence

Brian Odegaard, Piercesare Grimaldi, Seong Hah Cho, Megan AK Peters, Hakwan Lau, and Michele A Basso. Superior colliculus neuronal ensemble activity signals optimal rather than subjective confidence. Proceedings of the National Academy of Sciences , 115(7):E1588–E1597, 2018

work page 2018

[6] [6]

Hauser, Rani Moran, Raymond J

Max Rollwage, Alisa Loosen, Tobias U. Hauser, Rani Moran, Raymond J. Dolan, and Stephen M. Fleming. Confidence drives a neural confirmation bias. Nature Communications, 11(1):2634, May 2020

work page 2020

[7] [7]

Confidence in masked orientation judgments is informed by both evidence and visibility

Manuel Rausch, Sebastian Hellmann, and Michael Zehetleitner. Confidence in masked orientation judgments is informed by both evidence and visibility. Attention, Perception & Psychophysics, 80(1):134–154, January 2018

work page 2018

[8] [8]

Paradoxical evidence weighting in confidence judgments for detection and discrimination

Matan Mazor, Lucie Charles, Roni Maimon Mor, and Stephen Fleming. Paradoxical evidence weighting in confidence judgments for detection and discrimination. Attention, Perception & Psychophysics , 85(7):2356– 2385, October 2023

work page 2023

[9] [9]

Detection theory: A user’s guide

Michael J Hautus, Neil A Macmillan, and C Douglas Creelman. Detection theory: A user’s guide . Routledge, 2021

work page 2021

[10] [10]

Heuristic use of perceptual evidence leads to dissociation between performance and metacognitive sensitivity

Brian Maniscalco, Megan A K Peters, and H C Lau. Heuristic use of perceptual evidence leads to dissociation between performance and metacognitive sensitivity. Attention, Perception & Psychophysics , 78(3):923–937, April 2016

work page 2016

[11] [11]

A decision-congruent heuristic gives superior metacognitive sensitivity under realistic variance assumptions

Kiyofumi Miyoshi and Hakwan Lau. A decision-congruent heuristic gives superior metacognitive sensitivity under realistic variance assumptions. Psychological Review, 127(5):655–671, October 2020

work page 2020

[12] [12]

Basso, Hakwan Lau, and Megan A

Brian Maniscalco, Brian Odegaard, Piercesare Grimaldi, Seong Hah Cho, Michele A. Basso, Hakwan Lau, and Megan A. K. Peters. Tuned inhibition in perceptual decision-making circuits can explain seemingly suboptimal confidence behavior. PLoS Computational Biology, 17(3):e1008779, March 2021

work page 2021

[13] [13]

Reasons for confidence

Asher Koriat, Sarah Lichtenstein, and Baruch Fischhoff. Reasons for confidence. Journal of Experimental Psychology: Human Learning and Memory , 6:107–118, 1980

work page 1980

[14] [14]

Natural statistics support a rational account of confidence biases

Webb, Taylor W and Miyoshi, Kiyofumi and So, Tsz Yan and Rajananda, Sivananda and Lau, Hakwan. Natural statistics support a rational account of confidence biases. Nature Communications, 14(1):3992, 2023

work page 2023

[15] [15]

Human observers have optimal introspective access to perceptual processes even for visually masked stimuli

Megan AK Peters and Hakwan Lau. Human observers have optimal introspective access to perceptual processes even for visually masked stimuli. Elife, 4:e09651, 2015

work page 2015

[16] [16]

Confidence reports in decision-making with multiple alternatives violate the bayesian confidence hypothesis

Hsin-Hung Li and Wei Ji Ma. Confidence reports in decision-making with multiple alternatives violate the bayesian confidence hypothesis. Nature Communications, 11(1):2004, 2020

work page 2004

[17] [17]

How to measure metacognition

Stephen M Fleming and Hakwan C Lau. How to measure metacognition. Frontiers in human neuroscience , 8:443, 2014

work page 2014

[18] [18]

Visual attention modulates the integration of goal-relevant evidence and not value

Pradyumna Sepulveda, Marius Usher, Ned Davies, Amy A Benson, Pietro Ortoleva, and Benedetto De Martino. Visual attention modulates the integration of goal-relevant evidence and not value. eLife, 9:e60705, November 2020

work page 2020

[19] [19]

Recurrent networks endowed with structural priors explain suboptimal animal behavior

Manuel Molano-Maz ´on, Yuxiu Shao, Daniel Duque, Guangyu Robert Yang, Srdjan Ostojic, and Jaime De La Rocha. Recurrent networks endowed with structural priors explain suboptimal animal behavior. Cur- rent Biology, 33(4):622–638, 2023

work page 2023

[20] [20]

Overprecision is a property of thinking systems

Don A Moore. Overprecision is a property of thinking systems. Psychological Review, 130(5):1339, 2023

work page 2023

[21] [21]

Koosha Khalvati, Roozbeh Kiani, and Rajesh P. N. Rao. Bayesian inference with incomplete knowledge explains perceptual confidence and its deviations from accuracy. Nature Communications, 12(1):5704, September 2021

work page 2021

[22] [22]

Visual capacity in the hemianopic field following a restricted occipital ablation

Lawrence Weiskrantz, Elizabeth K Warrington, MD Sanders, and J Marshall. Visual capacity in the hemianopic field following a restricted occipital ablation. Brain, 97(1):709–728, 1974

work page 1974

[23] [23]

Is blindsight like normal, near-threshold vision? Proceedings of the National Academy of Sciences, 94(25):14190–14194, 1997

Paul Azzopardi and Alan Cowey. Is blindsight like normal, near-threshold vision? Proceedings of the National Academy of Sciences, 94(25):14190–14194, 1997

work page 1997

[24] [24]

Blindsight is qualitatively degraded conscious vision

Ian Phillips. Blindsight is qualitatively degraded conscious vision. Psychological Review, 128(3):558, 2021

work page 2021

[25] [25]

Does unconscious perception really exist? continuing the assc20 debate

Megan AK Peters, Robert W Kentridge, Ian Phillips, and Ned Block. Does unconscious perception really exist? continuing the assc20 debate. Neuroscience of Consciousness, 2017(1):nix015, 2017. 10 Confidence is detection-like in high-dimensional spaces A Confidence can be computed using the softmax As described in the Introduction, we can use Bayes’ rule to ...

work page 2017