Two base rates, two weights: base-rate neglect has a second axis

Abstract

Base-rate neglect is usually treated as one mistake: giving the prior too little weight. Turning co-occurrences into a judgment takes two base-rate corrections, not one: weighting in the prior (how common the outcome is) and dividing out the cue's own frequency. Under-correcting the prior is classical base-rate neglect. Under-correcting the cue is the cue-density effect, long studied but not usually framed as base-rate neglect's cue-side analogue. We write both corrections as two weights in one Bayesian equation. The dual-factor heuristic, a leading model of covariation judgment, sits where both equal one-half. Response format fixes which weight a study can measure. A graded rating carries the cue-frequency weight directly. A forced choice between two outcomes cancels it, measuring a separate odds-space weight that agrees with it only at full correction. The cancellation is exact, so same-cue choice proportions carry no information about the cue-frequency weight, not even its direction. A choice between two cues keeps the term, and differing cue frequencies are necessary and sufficient for a softmax choice to depend on it. The same weight appears in six learning-and-memory models (in odds space for signal detection), some exactly, others as a limiting or constructed correspondence. When an experiment keeps only the four cells of a cue x outcome contingency table, each of the six model families projects onto one scalar coordinate, which alone cannot separate the accounts free to sit anywhere along it. On a suitable design the two weights are separately estimable: an experimenter can move either weight's regressor without moving the other's. The nested common-weight model excludes a two-coefficient separation by construction. This paper lays out the framework and the rating experiment that tests the separation. A companion paper fits the odds-space cue weight to a colour-flavour dataset.

arXiv comment49 pages, 14 figures. v4: voice/clarity revision. Claim fixes: signal-detection face scoped to odds space, correlation-sign test corrected, prior-intrusion example, iff scoped to softmax, estimator bias within 0.03, lift not called causal strength, Prop. B1 restated, Sec. 6 and 7 readings. Prior art: dual-factor heuristic credited, yes/no scope stated. Code: https://osf.io/9qvbr/