Modeling probability and additive summation for detection across multiple mechanisms under the assumptions of signal detection theory
Why this work is in the frame
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Bibliographic record
Abstract
Many studies have investigated how multiple stimuli combine to reach threshold. There are broadly speaking two ways this can occur: additive summation (AS) where inputs from the different stimuli add together in a single mechanism, or probability summation (PS) where different stimuli are detected independently by separate mechanisms. PS is traditionally modeled under high threshold theory (HTT); however, tests have shown that HTT is incorrect and that signal detection theory (SDT) is the better framework for modeling summation. Modeling the equivalent of PS under SDT is, however, relatively complicated, leading many investigators to use Monte Carlo simulations for the predictions. We derive formulas that employ numerical integration to predict the proportion correct for detecting multiple stimuli assuming PS under SDT, for the situations in which stimuli are either equal or unequal in strength. Both formulas are general purpose, calculating performance for forced-choice tasks with M alternatives, n stimuli, in Q monitored mechanisms, each subject to a non-linear transducer with exponent τ. We show how the probability (and additive) summation formulas can be used to simulate psychometric functions, which when fitted with Weibull functions make signature predictions for how thresholds and psychometric function slopes vary as a function of τ, n, and Q. We also show how one can fit the formulas directly to real psychometric functions using data from a binocular summation experiment, and show how one can obtain estimates of τ and test whether binocular summation conforms more to PS or AS. The methods described here can be readily applied using software functions newly added to the Palamedes toolbox.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it