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Bias or Optimality? Disentangling Bayesian Inference and Learning Biases in Human Decision-Making

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Pith's one-line read An objective Bayesian learner, when fit with standard Q-learning, appears to show the same positivity and confirmation biases found in human data.

desk verdict A smart and important critique of how confirmation bias is inferred from Q-learning fits, with a sound mathematical core that needs empirical support and a cleaner treatment of the update rule settings. read the letter →

arxiv 2505.08049 v1 pith:JCFVBY3Q submitted 2025-05-12 cs.AI

Prakhar Godara This is my paper · ORCID
classification cs.AI MSC 68T0562C1091E40
keywords confirmationbiaspositivityQ-learningBayesianinferencetwo-armedBernoullibanditlearningratesactionswitchingmasterequation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the positivity and confirmation biases reported in human two-armed bandit experiments may be measurement artifacts of the fitting model rather than features of human learning. The reason is that an ideal Bayesian learner, when its choices are fit with the standard Q-learning model, appears to have asymmetric learning rates even though its updates are symmetric. Bayesian inference maps exactly onto Q-learning with learning rates $\alpha_+ = \alpha_- = 1/(t+3)$, which are equal but decrease with experience. Using a master-equation analysis of action-switching probabilities, the paper shows that unbiased decreasing learning rates and genuinely biased constant learning rates produce the same behavioral signature. If the argument is right, accounting for decreasing learning rates should shrink the estimated biases in existing data, and the paper's proposed new-arm experiment can separate the two accounts.

What carries the argument

The central object is the exact mapping from Bayesian belief updating to Q-learning, condensed in the identity $\alpha_\pm = 1/(t+3)$: after $t$ pulls of an arm, the Bayesian agent's effective learning rate is the same for positive and negative prediction errors, so the learner is unbiased but its updates shrink over time. The argument is carried by a master equation, a bookkeeping equation for how the probability distribution over Q-values changes in time, closed at second order by Taylor-expanding the softmax choice rule. The resulting moment equations show that the action-switching probability $\langle K\rangle_t$ depends on the separation $\Delta_t = \langle(Q_1-Q_2)^2\rangle/2$, which evolves as $\Delta_{t+1} = (1-\alpha_t)^2 \Delta_t + p(1-p)\alpha_t^2$; the first term is a drift that shrinks the separation, the second is environmental noise that grows it, and decreasing $\alpha_t$ sustains a low switching rate in the same way that confirmation bias does.

What would settle it

Simulate a Bayesian agent in a counterfactual two-armed bandit where both arms update with $\alpha_\pm=1/(t+3)$, fit the standard constant asymmetric Q-learning model to its choices, and check whether the fitted asymmetry grows with trial count; if it does not, the paper's artifact mechanism fails.

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Extended reading notes

Core claim

The paper establishes that Bayesian inference in a two-armed Bernoulli bandit is equivalent to a Q-learning agent with symmetric, decreasing learning rates: $\alpha_\pm = 1/(t+3)$. When an objective Bayesian agent's choices are fit with the standard constant-learning-rate Q-learning model, the fitted parameters recover positivity bias ($\alpha^c_+ > \alpha^c_-$) and confirmation bias ($\alpha^c_+ > \alpha^c_-$, $\alpha^u_- > \alpha^u_+$), exactly the patterns reported in human data. The mechanism is a shared reduction in action-switching probability: confirmation bias separates the estimated values of the arms, while decreasing learning rates let early random fluctuations separate the estimates and then lock them in place. The paper therefore concludes that apparent confirmation bias in human behavior can be an artifact of assuming constant learning rates, and it proposes a post-task third-arm choice whose predicted probabilities differ between the Bayesian and biased Q-learning accounts.

Load-bearing premise

The paper's central equivalence assumes the unchosen arm's estimate also learns at rate $1/(t+3)$, but the update rule used to derive that rate leaves the unchosen arm's estimate unchanged, so the bridge from the derived case to the fitted case is an assumption.

Editorial extensions

If this is right

  • Re-fitting existing TABB data with a model whose learning rates decrease over time should reduce or remove the fitted positivity and confirmation biases, if the artifact account is correct.
  • The two mechanisms predict different terminal estimates: Bayesian inference pulls both arms' Q-values toward the true reward probabilities, whereas biased constant-rate Q-learning pushes them apart.
  • A test with a newly introduced third arm, whose reward rate lies between the two predicted terminal values, can separate the accounts because the Bayesian and biased Q-learning models predict different probabilities of choosing it.
  • Claims that confirmation bias improves cumulative reward in bandit tasks may actually be capturing the benefit of decreasing learning rates, since both mechanisms lower switching without requiring a true bias.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the same artifact could arise in any paradigm where a constant-learning-rate model is fit to data from an adaptive process with shrinking updates, such as exponential recency or Bayesian-filter-style learning.
  • A testable extension is to introduce a change-point after learning has settled: a biased learner should stay overconfident, while a Bayesian learner should re-adapt, giving a behavioral separator beyond the third-arm probe.
  • The moment-closure analysis could be applied to other generative model pairs to ask systematically which combinations of temporal learning-rate profiles and true biases are statistically distinguishable from constant-rate Q-learning.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper argues that the positivity and confirmation biases reported in human two-armed Bernoulli bandit (TABB) experiments can arise as fitting artifacts: even if an agent updates beliefs through objective Bayesian inference, fitting the standard Q-learning model with constant, asymmetric learning rates recovers both biases. The authors map Bayesian inference onto Q-learning with symmetric, decreasing learning rates, use master equations and a moment-closure approximation to show that both confirmation bias and decreasing learning rates reduce action-switching probabilities, fit Q-learning and Bayesian models to the data of Palminteri et al. (2017), and propose an experimental protocol to distinguish true biases from decreasing-learning-rate artifacts.

Significance. If the central claim is correct, the paper would substantially change the interpretation of a large empirical literature that infers positivity and confirmation biases from TABB data using constant-learning-rate Q-learning models. The master-equation treatment of action-switching dynamics is a useful theoretical contribution, and the proposed new-arm experiment in Sec. 3.2 gives concrete, falsifiable predictions that could separate the two accounts. The analytical derivations, especially Eqs. (28)-(30), are internally plausible and are validated against SDE simulations in Fig. 2. However, the load-bearing mapping in Eq. (7) has a setting inconsistency between the no-counterfactual update rule used in its derivation and the counterfactual setting that the paper says is its focus, and the empirical demonstration that fitting recovers biases is not described with sufficient detail to be checked.

major comments (4)
  1. [§2, Eq. (7)] The derivation of α± = 1/(t+3) uses the no-counterfactual update rule Eq. (3), where only the chosen arm's posterior changes, but the paper states that its focus is the counterfactual setting, in which both chosen and unchosen arms are updated. In the no-counterfactual setting the unchosen arm's effective learning rate is zero, not 1/(t+3), and for the chosen arm the denominator should be that arm's own observation count, not the global trial count. In the counterfactual setting, Eq. (3) is the wrong generative rule because both arms receive an observed reward each trial. The paper must state which setting Eq. (7) belongs to, derive the learning rates for both arms in the counterfactual setting, and reconcile Eq. (2), whose constraint α1+β1+α2+β2=t is incompatible with Eq. (7) when both arms are observed on every trial.
  2. [§2.2, Eqs. (29)-(30)] The master-equation analysis substitutes α_t = 1/(t+3) for both the chosen and unchosen arms, but this substitution is valid only under specific observation assumptions that are not stated. If the intended setting is counterfactual feedback, then the per-arm observation count is t and the substitution is plausible, but only after a correct counterfactual belief update is written down; if the intended setting is no-counterfactual feedback, the unchosen arm's learning rate is 0 throughout. Because Eqs. (29)-(30) are the theoretical basis for the claim that decreasing learning rates mimic confirmation bias, this ambiguity is load-bearing and must be resolved before the explanation can be accepted.
  3. [§3.1 and Fig. 1(a)] The abstract and Fig. 1(a) make the empirical claim that fitting the standard Q-learning model with asymmetric learning rates to data generated by a Bayesian agent recovers both positivity and confirmation biases, but the manuscript gives no simulation protocol: it does not report the number of simulated agents, the number of trials, the fitting procedure, the parameter ranges or initializations, the optimization algorithm, or the variability of the recovered learning rates across simulations. Without this information the central empirical demonstration cannot be reproduced or assessed, and the reader cannot tell whether the recovered asymmetries are robust or an artifact of a particular fitting schedule.
  4. [§2.1, Eq. (20)] The moment-closure approximation truncates the Taylor expansion at second order and assumes that the softmax nonlinearity is well approximated by its local behavior around the mean of the Q-distribution. The paper validates the approximation against SDE simulations for constant learning rates in Fig. 2, but it does not provide the same validation for the time-varying schedule α_t = 1/(t+3) over the trial horizons used in the human data. Since Eq. (30) is derived under this approximation, a direct comparison of the approximate and simulated ⟨K⟩_t for the Bayesian schedule would strengthen the paper's main mechanistic claim.
minor comments (5)
  1. [Eq. (2)] The notation α_i, β_i for beta-distribution counts conflicts with the learning-rate notation α^v_± used throughout; this is confusing in equations such as Eq. (7), where α appears in both roles.
  2. [Fig. 1(b)] The caption uses K_t before the text defines the ensemble-average action-switching probability; define ⟨K⟩_t in the caption or earlier in the text.
  3. [§3.2] The proposed new-arm experiment is described only at a conceptual level; the manuscript should specify how p_3 is chosen for each participant, how many trials are used, how the new arm is trained, and what decision rule will be used to prefer the Bayesian over the Q-learning account.
  4. [Eq. (13)] The shorthand Q^{+,v}_i and Q^{-,v}_i is used before it is defined; define these quantities immediately before Eq. (13).
  5. [Table 1] The table reports mean BIC and NLL across subjects but does not state the number of subjects, the standard error of the mean, or whether the BIC comparison is robust to the choice of fitting procedure; adding these details would strengthen the model-comparison section.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed equivalence between Bayesian inference and Q-learning with decreasing symmetric learning rates is a direct algebraic mapping, and the master-equation analysis is self-contained.

full rationale

The paper's central claim is that Bayesian inference on a two-armed Bernoulli bandit can be rewritten as Q-learning with symmetric, decreasing learning rates, and that this temporal structure can produce the same reduced action-switching signature as confirmation bias when a constant-learning-rate Q-learning model is fit to the data. The derivation chain is self-contained and does not fit the target conclusion. Eq. 3 defines the Bayesian count update, Eq. 6 rewrites the posterior-mean update in Q-learning form, and Eq. 7 computes the resulting learning rate algebraically; no fitted parameter is renamed as a prediction. The master-equation analysis in Section 2 then independently derives moment dynamics (Eqs. 28-30) by substituting the derived alpha_t, and the analytic results are checked against SDE simulations. The model comparison with human data from Palminteri et al. (2017) is an empirical application, not a definitional input. Although the extension of Eq. 7 to the unchosen arm and to the counterfactual setting is mathematically questionable, because the unchosen arm's count does not advance under Eq. 3 and the paper uses the global trial count t rather than the per-arm count, this is a correctness concern about whether the equivalence holds in the intended setting, not a circular step, since the conclusion is not assumed in the premises. The paper also explicitly limits the equivalence to Bernoulli bandits, further indicating that the derivation is not being used to smuggle in the result.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The theoretical derivation of the Q-learning representation uses standard Bayesian machinery and no fitted constants. The empirical demonstration and model comparison depend on standard RL model parameters (learning rates, inverse temperature), and the analytical tractability relies on symmetric environments and a second-order moment closure. No new entities are introduced.

free parameters (3)
  • Inverse temperature beta of the Bayesian and Q-learning agents = not reported in the text
    Controls action stochasticity in all simulations and likelihood fits; the central demonstration that fits recover biases depends on this choice, but the value is not given.
  • Learning-rate asymmetries alpha_c+, alpha_c-, alpha_u+, alpha_u- in the fitted Q-learning model = not reported in the text
    These are the fitted parameters whose inequalities define positivity/confirmation bias in Table 1 and Fig. 1(a); the paper does not report the best-fit values or their uncertainty.
  • Demonstration schedule parameters alpha1, alpha2, tau_c in Fig. 3 = alpha1 = 0.1, alpha2 = 0.01, tau_c = 25 (also varied)
    Chosen by hand to illustrate that a stepwise decreasing learning rate produces low switching; this is a proof-of-concept, not an estimate.
assumptions (5)
  • standard math Beta-Bernoulli conjugacy and uniform prior: the agent's belief about each arm's reward probability is a Beta distribution updated by counting successes and failures.
    Used in Sec. 1 to write the belief state b_t and the update rule y(a, r) in Eq. 3.
  • domain assumption In the counterfactual-feedback TABB task, the value of future information is independent of the current action, so the Bayes-optimal policy is the immediate greedy choice of Eq. 5.
    Invoked in Sec. 1 to replace the Bellman equation (Eq. 4) with Eq. 5; requires that the agent observes both arms' outcomes regardless of which arm is chosen.
  • domain assumption Human choice probabilities follow a softmax with fixed inverse temperature beta over Q-values.
    Used in the Q-learning model (Eq. 1) and in all likelihood fits (Sec. 3.1); this is the standard behavioral model but is assumed, not tested.
  • ad hoc to paper The analytical derivations in Sec. 2 assume a symmetric environment p1 = p2 = p to isolate the effect of biases on action switching.
    The paper states this simplification in Sec. 2; it restricts the derived switching-rate relationships to symmetric reward statistics, though simulations indicate the conclusions extend.
  • ad hoc to paper The moment-closure approximation: the softmax nonlinearity is Taylor-expanded to second order around the distribution mean, truncating higher-order cumulants (Eqs. 19-20).
    Needed to close the moment equations; the approximation is uncontrolled analytically but is checked against direct SDE/master-equation simulation in Fig. 2.

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Cite this review

Pith. "Pith review of Bias or Optimality? Disentangling Bayesian Inference and Learning Biases in Human Decision-Making." pith.science (2026). https://pith.science/paper/JCFVBY3Q

@misc{pith2026250508049,
  author       = {Pith},
  title        = {Pith review of: Bias or Optimality? Disentangling Bayesian Inference and Learning Biases in Human Decision-Making},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JCFVBY3Q}},
  note         = {Machine review of arXiv:2505.08049}
}
read the original abstract

Recent studies claim that human behavior in a two-armed Bernoulli bandit (TABB) task is described by positivity and confirmation biases, implying that humans do not integrate new information objectively. However, we find that even if the agent updates its belief via objective Bayesian inference, fitting the standard Q-learning model with asymmetric learning rates still recovers both biases. Bayesian inference cast as an effective Q-learning algorithm has symmetric, though decreasing, learning rates. We explain this by analyzing the stochastic dynamics of these learning systems using master equations. We find that both confirmation bias and unbiased but decreasing learning rates yield the same behavioral signatures. Finally, we propose experimental protocols to disentangle true cognitive biases from artifacts of decreasing learning rates.

Figures

Figures reproduced from arXiv: 2505.08049 by the authors.

Figure 1
Figure 1. (a) Ensemble averaged best fit learning rates when fit to human data Palminteri et al. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Steady state ∆∗ as a function of confirmation bias x, for different values of (a) β and (b) p values. (c) shows the average action switching probability ⟨K⟩t as a function of t, for different values of confirmation bias. ⟨Q 2 1 ⟩t+1 =  (D (2) 11 + D (1) 11 ) 2 ⟨Q 2 1 ⟩t + (D (2) 1 + D (1) 1 ) 2 ⟨Q1⟩t + D(1) +  (D (2) 11 − D (1) 11 )⟨Q1⟩t β 2 ∆t + (D (2) 1 − D (1) 1 ) β 4 ∆t + (D(2) − D(1)) 2  , (28c) Here we wri… view at source ↗
Figure 3
Figure 3. (a) The minimum (across time t) value of the average action switching rate ⟨K⟩min as a function of τc for different values of α1. (b) αt and the corresponding action switching rates ⟨K⟩t as a function for time for τc = 25 and α1, α2 = 0.1, 0.01. 3 Consequences on human behavior We have so far explored that both confirmation bias and Bayesian inference can lead to similar behavioral signatures. The confounding effect… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: (a) Ensemble average action switching rates for human data (blue), best fit Bayesian agents [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

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