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arxiv: 2606.23325 · v1 · pith:N45GA4ZOnew · submitted 2026-06-22 · 🧬 q-bio.NC · quant-ph

The adaptive nature of confirmation bias

Dorje C. Brody , Karl J. Friston , Bernhard K. Meister , Emmanuel M. Pothos This is my paper

Pith reviewed 2026-06-26 05:56 UTC · model grok-4.3

classification 🧬 q-bio.NC quant-ph
keywords confirmation biasbinary hypothesis testingsequential samplingactive inferencesquare-root probabilitieserror probability minimization
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The pith

Optimal evidence selection to minimize error in binary decisions produces confirmation bias as a rational strategy.

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

The paper models observations as matrices on square-root probability spaces rather than ordinary random variables. In binary hypothesis testing, the choice of evidence that minimizes expected error probability turns out to favor evidence consistent with the current hypothesis. This built-in confirmation bias yields two concrete benefits during sequential sampling: the decision process needs only the smallest memory capacity, and the probability of error falls exponentially with the number of samples. The same optimal evidence selection is recovered when the decision maker instead maximizes information gain under active inference.

Core claim

In the problem of binary hypothesis testing, an optimal evidence choice that minimises the expected error probability leads to a confirmation bias; in sequential evidence sampling this implicit optimality produces the smallest memory capacity together with an error probability that can be reduced exponentially in sample size.

What carries the argument

Modeling observations by matrices on the space of square-root probabilities, with optimality defined as minimising expected error probability in binary hypothesis testing.

If this is right

  • Confirmation bias functions as a feature of rational evidence selection rather than a departure from it.
  • Decision makers following the optimal rule operate with minimal memory storage requirements.
  • Error probability decreases exponentially rather than linearly or polynomially with added samples.
  • The identical evidence choice is obtained when the objective is switched from error minimisation to information maximisation.

Where Pith is reading between the lines

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

  • The framework may predict that confirmation bias appears most strongly in tasks where memory capacity is tightly constrained.
  • It suggests checking whether real sequential sampling behavior matches the matrix-based selection rule in controlled binary choice experiments.
  • The equivalence between error-minimising and information-maximising rules could be tested by varying the cost of memory across different decision environments.

Load-bearing premise

Observations are modeled by matrices rather than random variables on a probability space, and optimality is defined as minimising expected error probability.

What would settle it

An experiment or simulation in which the evidence sequence that minimises expected error in a binary task fails to favor confirming evidence, or in which error probability does not drop exponentially with sample size under the optimal rule.

Figures

Figures reproduced from arXiv: 2606.23325 by Bernhard K. Meister, Dorje C. Brody, Emmanuel M. Pothos, Karl J. Friston.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
read the original abstract

In this paper, the phenomenon generally classified as confirmation bias is formulated on the space of square-root probabilities (or equivalently, using the structures of quantum probability). In this framework, observations are modelled by matrices, rather than random variables on a probability space. In the problem of binary hypothesis testing, an optimal evidence choice minimises the expected error probability. We show that the resulting optimal choice of evidence leads to a confirmation bias, thus revealing a surprising aspect of rationality that encompasses confirmation bias. Specifically, in sequential evidence sampling, the implicit optimality leads to two remarkable evolutionary advantages, namely, (a) the decision maker requires only the smallest memory capacity, and (b) the error probability can be reduced exponentially in sample size. A complementary approach based on the framework of active inference -- where the decision maker seeks evidence that provides maximum information -- is then considered. The resulting optimal evidence is shown to agree with the one obtained by minimising error probability. Our framework provides an easy-to-implement protocol for an active quantum inference, whereby the optimal evidence choice for making an inference is sought over the space of matrices.

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

3 major / 2 minor

Summary. The paper claims that confirmation bias can be formulated in the space of square-root probabilities (quantum probability), where observations are modeled as matrices. In binary hypothesis testing, the evidence choice that minimizes expected error probability is shown to produce confirmation bias as an optimal strategy. In sequential sampling this yields two evolutionary advantages: minimal memory capacity and exponential reduction of error probability with sample size. The same optimal evidence is recovered from an active-inference (maximum-information) criterion, and the framework supplies a protocol for active quantum inference.

Significance. If the central derivation holds, the work supplies a concrete optimality-based account in which confirmation bias is not a departure from rationality but a direct consequence of it, together with explicit computational advantages (memory and exponential error decay) that are falsifiable in principle. The agreement between error-minimization and active-inference routes, and the provision of an implementable matrix protocol, are additional strengths.

major comments (3)
  1. [Observation model / binary hypothesis testing formulation] The observation model (matrices on the square-root probability space rather than random variables on a classical probability space) is load-bearing for the claim that optimality implies confirmation bias. The manuscript must demonstrate either that the bias result survives translation to ordinary random variables or that the matrix representation is required on independent grounds; otherwise the bias may be an artifact of the chosen representation.
  2. [Central optimality derivation] The derivation that the error-minimizing evidence choice produces confirmation bias (and the two stated advantages) is not visible in the abstract and must be checked for circularity: the optimality criterion must not be defined in a way that forces the bias by construction. Explicit steps linking the matrix structure to the bias behavior are required.
  3. [Sequential sampling / evolutionary advantages] The exponential error reduction and minimal-memory claims are stated as consequences of sequential sampling under the optimal policy. The precise scaling (e.g., which norm or distance yields the exponential rate) and the memory-capacity argument (which quantity is being minimized) need explicit verification against the matrix model.
minor comments (2)
  1. The abstract equates square-root probabilities with quantum probability structures; a brief clarifying sentence on the precise equivalence would help readers outside the subfield.
  2. Notation for the matrix observations and the error-probability functional should be introduced consistently before the optimality argument.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We respond point-by-point to the major comments below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Observation model / binary hypothesis testing formulation] The observation model (matrices on the square-root probability space rather than random variables on a classical probability space) is load-bearing for the claim that optimality implies confirmation bias. The manuscript must demonstrate either that the bias result survives translation to ordinary random variables or that the matrix representation is required on independent grounds; otherwise the bias may be an artifact of the chosen representation.

    Authors: The matrix representation on the square-root probability space is motivated independently by the quantum cognition literature, where it captures non-commutative effects and interference that align with observed cognitive phenomena. To address the concern directly, the revised manuscript will include a new subsection comparing the classical random-variable formulation under the identical error-minimization criterion. We will demonstrate that confirmation bias does not arise classically, thereby establishing that the matrix structure is required on independent grounds. revision: yes

  2. Referee: [Central optimality derivation] The derivation that the error-minimizing evidence choice produces confirmation bias (and the two stated advantages) is not visible in the abstract and must be checked for circularity: the optimality criterion must not be defined in a way that forces the bias by construction. Explicit steps linking the matrix structure to the bias behavior are required.

    Authors: The optimality criterion is defined strictly as minimization of expected error probability, a standard objective that does not presuppose bias. Confirmation bias arises as a derived consequence of this minimization when performed in the matrix algebra. The revised manuscript will expand the central derivation with all intermediate algebraic steps, explicitly tracing how the matrix operations select confirming evidence without circularity. revision: yes

  3. Referee: [Sequential sampling / evolutionary advantages] The exponential error reduction and minimal-memory claims are stated as consequences of sequential sampling under the optimal policy. The precise scaling (e.g., which norm or distance yields the exponential rate) and the memory-capacity argument (which quantity is being minimized) need explicit verification against the matrix model.

    Authors: The revised manuscript will supply the missing explicit verification: the exponential rate will be stated with respect to the trace norm on the matrix space, and the memory-capacity argument will be formalized as the requirement to retain only the current square-root probability vector (rather than the full observation history). These calculations will be added to the sequential-sampling section. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper models observations via matrices on square-root probabilities and defines optimality as minimising expected error probability for binary hypothesis testing; it then derives that the resulting optimal evidence choice produces confirmation bias along with the stated memory and exponential-error advantages. This constitutes a derived consequence within the chosen framework rather than a definitional equivalence or a fitted input renamed as a prediction. The agreement with the active-inference formulation is shown separately and does not rely on load-bearing self-citation for the central claim. No equation or step reduces the target result to its inputs by construction, and the derivation remains self-contained against the stated modelling assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on adopting the quantum probability framework (square-root probabilities and matrix observations) and the specific optimality criterion without deriving them from more basic principles or providing independent evidence.

axioms (2)
  • domain assumption Observations can be modelled by matrices in the space of square-root probabilities.
    Stated in abstract as the core modeling choice for evidence.
  • domain assumption Optimality is defined as minimising expected error probability in binary hypothesis testing.
    Central to deriving that optimal evidence choice produces confirmation bias.
invented entities (1)
  • Matrix-based observation model no independent evidence
    purpose: To represent evidence in quantum probability space instead of classical random variables
    Introduced as the framework for modeling observations; no independent evidence provided.

pith-pipeline@v0.9.1-grok · 5730 in / 1325 out tokens · 44173 ms · 2026-06-26T05:56:47.007959+00:00 · methodology

discussion (0)

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