Abstract
Psychophysics is the quantitative study of the relationship between physical stimulus magnitude and perceived magnitude in sensory systems [1, 2]. It is the measurement foundation under the HOF-cognition account of machine cognition: the perceptual principles it established are applied here as real, measurable engineering tools, not as figures of speech. Signal Detection Theory (SDT) [3] supplies the decision layer — its sensitivity/criterion decomposition maps directly onto the accept/reject/retry gate an LLM-agent system runs when it decides whether a candidate output is signal or noise. Weber–Fechner scaling [1] and Stevens' power law [2] supply the response-scale layer — the compressive, relative-change laws that set adaptive thresholds and diminishing-returns resource schedules. Bayesian models of cognition [4] supply the inference framing, treating a decision under uncertainty as belief updated by evidence. Each is an applied psychophysical instrument with a defined role in the design; this note states which principle governs which engineering decision.
1. Introduction
The HOF-cognition corpus treats a software system as a system that perceives, decides, and adapts — and it grounds that treatment in the same laws that quantify biological perception. Psychophysics measures how a reported sensation changes as a physical stimulus changes: the brightness of a light, the loudness of a tone, the weight of a held object [1, 2]. Those laws are transferable because they are measurement relations, not anatomy. An LLM-agent retry loop has stimuli (candidate outputs), a decision (accept or reject), and an adaptation regime (how sensitivity moves as signal strength changes) — the same structure the perceptual laws quantify, applied to a computational observer.
This note fixes the mapping precisely. Signal Detection Theory governs the accept/reject decision. Weber–Fechner and Stevens govern the scale on which a change counts as noticeable and on which added resource yields diminishing return. Bayesian inference supplies the belief-updating layer beneath both. Each principle is load-bearing at a named stage of the design.
2. Related Work
Signal Detection Theory. Green and Swets [3] formalized the separation between an observer's sensitivity (how distinguishable signal-plus-noise is from noise alone, summarized by d′) and the observer's decision criterion (β, the willingness to answer "signal"), and the Receiver Operating Characteristic (ROC) that relates hit rate to false-alarm rate as the criterion sweeps. SDT is a decision theory, applicable wherever a system must commit to "present/absent" under uncertainty — which is why it transfers cleanly to a software accept/reject/retry gate.
Weber–Fechner and Stevens. Fechner's logarithmic law [1] and Stevens' power law [2] quantify how perceived magnitude grows with physical stimulus magnitude: sublinear, compressive, driven by relative rather than absolute change. These are the response-scale laws the design applies to set where an adaptive threshold sits and how quickly returns on additional resource compress.
Bayesian cognition. Griffiths, Kemp, and Tenenbaum [4] review the program of modeling reasoning as approximate Bayesian inference — combining priors with evidence to produce graded belief. This supplies the layer beneath the decision: a candidate's quality is a posterior, updated by validator evidence.
Compositional framing. The higher-order-function vocabulary is standard functional programming: composition and function-valued abstraction are the subject of Backus's functional-style algebra [5], Hughes's account of why higher-order functions aid modularity [6], and Strachey's treatment of functions as first-class values [7]. The perceptual principles above are composed as functions over this substrate.
3. Signal Detection Theory as a Validation/Retry Gate
The accept/reject decision is where the perceptual measurement is most directly load-bearing. Cast in SDT terms [3]:
- The stimulus is a candidate output (a retrieved item, a tool result, a generated answer).
- Signal vs. noise is "acceptably grounded/correct" vs. "hallucinated or off-target."
- Sensitivity (d′) is how well the validation procedure separates good candidates from bad ones — a property of the validator, improvable by better checks.
- Criterion (β) is the acceptance threshold — tunable to trade false accepts (shipping a wrong answer) against false rejects (needless, costly retries).
- The ROC is the family of operating points that threshold choice sweeps out.
The mapping is exact because SDT is a theory of decisions under uncertainty, defined independently of the observer's substrate. It gives a principled account of where to set a retry gate and how to reason about its two error modes. This note presents the design mapping, not a measured operating point: any specific d′, false-alarm rate, or ROC value would be [ILLUSTRATIVE] until produced by a run, and the gate's numbers are reported where the system is measured, not here.
4. Weber–Fechner and Stevens: Adaptive Thresholds and Diminishing Returns
A system should react to relative change and should grow less sensitive to further change once a signal is already strong. That is Weber–Fechner compression [1] and Stevens' power law [2] applied directly: the just-noticeable-difference sets the threshold at which a change is worth acting on, and the compressive response curve sets how added resource yields diminishing return. The laws quantify a real scaling relation — sublinear growth of noticeable change — and the design uses that relation to schedule adaptive thresholds and resource budgets. The response scale is the tool; the JND is where it is read off.
5. Bayesian Framing
Treating agent decisions under uncertainty as approximate inference [4] places a probabilistic layer beneath the SDT gate: priors over candidate quality, updated by validator evidence, yield a graded accept/reject posterior. SDT is the decision layer over that inferred posterior; the two compose. This is a modeling stance, stated as such, and does not by itself assert that the system realizes optimal Bayesian inference.
Evidence & Scope
This is a conceptual mapping between the psychophysics literature and the HOF-cognition design: it fixes which perceptual measurement principle governs which engineering decision — SDT [3] the accept/reject/retry gate, Weber–Fechner [1] and Stevens [2] the adaptive-threshold and diminishing-returns scales, Bayesian inference [4] the belief layer beneath them, all composed as higher-order functions [5, 6, 7]. Measured values — d′, false-alarm rate, ROC — are reported where a system is instrumented and run; any figure stated here would be [ILLUSTRATIVE]. The principles are applied instruments, and this note states the instrument each decision uses.
References
- Gustav Theodor Fechner (1860). Elemente der Psychophysik. Breitkopf und Härtel.
- S. S. Stevens (1957). On the Psychophysical Law. Psychological Review.
- David M. Green & John A. Swets (1966). Signal Detection Theory and Psychophysics. Wiley.
- Thomas L. Griffiths et al. (2008). Bayesian Models of Cognition. The Cambridge Handbook of Computational Psychology.
- John Backus (1978). Can Programming Be Liberated from the von Neumann Style? A Functional Style and Its Algebra of Programs. Communications of the ACM. [1977 ACM Turing Award Lecture]
- John Hughes (1989). Why Functional Programming Matters. The Computer Journal.
- Christopher Strachey (2000). Fundamental Concepts in Programming Languages. Higher-Order and Symbolic Computation. [Reprint of 1967 lecture notes]