Perceptual Power Laws

Quantifying how physical stimulus intensity maps onto sensation is the founding problem of experimental psychology. Three laws — Weber’s, Fechner’s and Stevens’ — plus the just-noticeable difference (JND) they are built on, still organise how vision scientists, psychophysicists and animal-behaviour researchers design stimuli and interpret thresholds. This page sets out what each law claims, what it assumes, where it fails, and how the picture has changed since 2017: the JND has been demoted from sensory quantum to statistical convention; Fechner’s law has been shown to be underdetermined by discrimination data; Weber’s and Stevens’ laws have been shown to be compatible rather than rivals; and Weber’s law itself now has causal, distribution-level evidence for an efficient-coding origin — while appearing, in modified form, inside large language models.

The three laws at a glance

Law	Form	What it relates	Empirical status
Weber	ΔI / I = k	discrimination threshold to background intensity	robust over a wide mid-range; fails near absolute threshold, at high intensity, and under skewed stimulus statistics
Fechner	S = k ln(I / I₀)	sensation magnitude to intensity, by integrating Weber’s law	not entailed by Weber’s law; one of an infinite family of internal codes consistent with it
Stevens	S = k I^a	sensation magnitude to intensity, from direct scaling	descriptive fit with modality-specific exponents; measurement-theoretic assumptions contested; compatible with Weber’s law once noise scales with the mean

The three are not three competing measurements of one quantity. Weber’s law concerns discriminability (a threshold), whereas Fechner’s and Stevens’ laws concern appearance (a magnitude). Much of the historical dispute, and much of the recent resolution, turns on that distinction.¹

1. The just-noticeable difference (JND)

The JND, or difference limen, is the smallest change in a stimulus that can be reliably detected against a reference. The absolute threshold is the minimum detectable stimulus; the differential threshold is the minimum detectable change. In practice the JND is read off a psychometric function — proportion of correct responses plotted against ΔI — at a conventional criterion: 75% correct in two-alternative forced choice, or 50% in yes/no after correction for guessing.

Why it is not a sensory quantum

Classical psychophysics treated the JND as a unit of sensation: a fixed increment of the mind, below which nothing is registered. Signal detection theory (SDT), developed from the 1950s, removed that interpretation. Performance is continuous and limited by noise; sensitivity is indexed by d′ = separation / noise, while the observed threshold also depends on the observer’s criterion and lapse rate. There is no ΔI below which an observer is at chance — only a ΔI at which the evidence is too weak relative to noise and criterion to support a response.

That conclusion has not fully displaced the intuitive picture. Sanford and Halberda (2023) found that both novices and psychologically educated students believe a literal threshold exists — that below some difference observers are completely unable to say which of two quantities is larger, for example which of two bags holds 3,000 versus 3,001 grains of sand. That belief is inconsistent with psychophysical law, and the authors argue it is part of an intuitive theory of psychology that is very hard to dispel.²

Measurement. Contemporary practice therefore treats a threshold as a fitted parameter with an explicit model: a psychometric function with guess rate, lapse rate and slope, estimated by adaptive procedures (staircases, QUEST, psi methods) or by maximum-likelihood fitting of all trials (e.g. Psignifit). A JND reported without its criterion, guess rate and lapse rate is not a well-defined quantity. The same logic carries over to confidence and metacognitive measures, where criterion and sensitivity must likewise be separated (see metacognition).

2. Weber’s law

Statement and range of validity

Weber’s law states that the smallest detectable change is proportional to the reference intensity:

ΔI / I = k

where k is the Weber fraction. Weber’s own example remains the cleanest illustration: if 105 g can just be distinguished from 100 g, the JND is 5 g and the Weber fraction is about 5%; doubling the mass doubles the JND, so 210 g is just distinguishable from 200 g.³

Weber’s law is not universal. It fails near the absolute threshold, where sensitivity is limited by intrinsic noise rather than by the signal, and often at high intensities. In audition the Weber fraction does not stay constant but declines slowly with level — the “near miss” — described across frequencies by ΔI/I = 0.463 (I/I₀)^−0.072 (Jesteadt et al. 1977).³ In vision, Weber’s law is a good approximation in the photopic range, where it amounts to constancy of luminance contrast, but breaks down in the mesopic range and is inapplicable at scotopic levels.³ A generalised form adds a constant, ΔI = kI + c, which captures the low-intensity divergence.³¹

Why it holds: four families of explanation

1. Sensory gain and adaptation. Compressive transduction, logarithmic tuning and contrast gain control reduce the neural response per unit stimulus as intensity rises, so a constant relative change produces a roughly constant neural change. This is the classical “Weber adaptation” story.⁴
2. Decision-level emergence. Wundt’s older suggestion — that Weber’s law is a law of judgement rather than of sensation — has been revived in a 2025 model in which categorical choices are computed by a circuit with global inhibition. With linearly encoded intensities, the ratio principle emerges near threshold; for easy discriminations it fails, producing the mild convexity of Weber fractions seen empirically. The same model does not, however, reproduce the large low-intensity deviations.⁵
3. Natural input statistics. Beck et al. (2024) argue that noisy neurons are not required. In natural environments, sensory measurements are scaled by global nuisance variables — contrast in vision, overall loudness in audition — which induce positive correlations among measurements; that shared scaling generates Weber behaviour as an unavoidable consequence. For numerosity, independent noise predicts a Weber fraction falling as 1/√N, whereas a shared contrast gain predicts variance growing as N², i.e. Weber’s law — and the theory correctly predicts the breakdown of Weber’s law for small numerosities.⁶
4. Efficient coding. If representational resources are allocated according to stimulus frequency, a distribution skewed toward small magnitudes yields discriminability that declines with magnitude — Weber’s law — and a distribution skewed the other way should invert the pattern. Prat-Carrabin, Yamamoto and Gershman (2026) tested this causally: with a small-dominant prior (80% of magnitudes below the midpoint) observers showed the usual Weber pattern in line length, brightness and speed; with a large-dominant prior the pattern was suppressed in all three modalities, and for line length the slope was statistically indistinguishable from zero with a negative median — an anti-Weber pattern. Adaptation also improved task performance across conditions.⁷ (Preprint.)

Where it breaks

The violations are systematic rather than noise. Carriot, Cullen and Chacron (2021) recorded from vestibular thalamocortical neurons in rhesus monkeys during self-motion and found that neural discrimination thresholds rose and then saturated at high amplitudes. Computational dissection attributed the saturation to the structure of neural variability rather than to response nonlinearities — providing a neural basis for the better-than-Weber performance consistently observed at high, naturally occurring amplitudes across modalities.⁴

The strongest recent demonstration that Weber’s law is context-dependent comes from magnitude estimation rather than discrimination. Prat-Carrabin and Gershman (2025), in two preregistered numerosity experiments, manipulated the prior distribution and the reward function. When large numerosities were more frequent or more rewarded, both a Bayesian observer and human participants became less variable at large magnitudes — “anti-Weber” behaviour. Yet the same observers’ responses were best reproduced by a logarithmic encoding of magnitude, the proposal usually invoked to explain Weber behaviour.⁸ Weber’s law, on this view, is a property of the observer’s priors and payoffs as much as of the sensory code.

3. Fechner’s law

From Weber to a logarithm

Fechner (1860) turned Weber’s rule into a candidate law of sensation. If Weber’s law holds for infinitesimal increments, dI / I = constant, and if every JND corresponds to an equal increment of sensation dS, then dS = c dI / I and integration gives:

S = k ln(I / I₀)

Equivalently, equal ratios of stimulus produce equal differences of sensation — a geometric progression in the world maps onto an arithmetic progression in the mind. This is the Weber–Fechner law. Logarithmic scales of this kind long predate and postdate psychophysics — stellar magnitudes before it, decibels after.³

What it assumes

Two assumptions carry the derivation, and neither is forced:

• that Weber’s law holds exactly, including at low intensity (it does not); and
• that all JNDs are subjectively equal — that a just-noticeable increment is a constant unit of sensation.

Fechner did not measure how perceived heaviness grows with mass; he assumed the equal-JND premise and derived the logarithm. Both assumptions were questioned almost immediately,³ and the second is in direct tension with Stevens’ results: under a power law, a constant relative stimulus increment produces a constant relative sensation increment, not a constant absolute one.

Modern status

The decisive modern point is that discrimination data cannot identify the internal code. Zhou, Duong and Simoncelli (2024) show that Weber’s law — sensitivity proportional to 1/s — is consistent with an infinite family of internal representations. Three illustrative members: a logarithmic transducer with additive Gaussian noise; a linear transducer with multiplicative noise whose SD is proportional to the stimulus; and Poisson spike counts with a transducer proportional to (log s + c)². All three produce identical Weber-law discrimination thresholds, yet imply entirely different transducers.¹

Fechner’s logarithm is therefore one solution among many, not an inference from Weber’s law. It remains empirically productive — a logarithmic encoding best accounted for the variability of numerosity estimates in the 2025 anti-Weber study⁸ — but it is a modelling choice, not a theorem. Johnson, Hsiao and Yoshioka (2002) make the historical version of the same point: MacKay showed that logarithmic and power laws are indistinguishable without specifying the underlying neural mechanisms, and Mountcastle’s programme of measuring transducer functions was an attempt to break that deadlock. Their own work on tactile roughness led them to propose that the basic law may be linearity between subjective experience and the relevant neural activity.⁹

4. Stevens’ power law

Method and exponents

Stevens (1957, 1961) rejected the logarithmic law and proposed instead that sensation grows as a power function of intensity:

S = k I^a

The exponent a is modality-specific: below 1 the function is compressive, at 1 linear, above 1 expansive. Stevens’ method was direct scaling — magnitude estimation (assign numbers in proportion to perceived magnitude), magnitude production (adjust a stimulus to a required ratio), and cross-modality matching (adjust one modality until it matches another). On log–log axes a power function is a straight line, so the exponent is the slope.

Representative exponents Stevens reported:¹⁰

Continuum	Exponent	Stimulus condition
Loudness	0.67	3000 Hz tone
Brightness	0.33	5° target in the dark
Lightness	1.2	reflectance of grey papers
Visual length	1.0	projected line
Heaviness	1.45	lifted weights
Taste	1.3	sucrose
Electric shock	3.5	current through fingers
Duration	1.1	white-noise stimuli

The span from 0.33 to 3.5 is the empirical core of the law: no single logarithmic compression can accommodate both brightness and electric shock.

Criticisms

• No direct test. Stevens fitted power functions to averaged magnitude estimates and inferred the law from the quality of the fit; the procedure does not independently test either the power form or the assumptions of magnitude estimation.¹⁰
• The ratio-scale assumption. Magnitude estimation presupposes that observers can report ratios veridically. Narens (1996) formulated testable axioms for this; the strongest (that judged proportions compose multiplicatively) was rejected by Ellermeier and Faulhammer (2000) and Zimmer (2005), while a weaker commutativity property was sustained. Luce’s (2002) behavioural condition equivalent to a power psychophysical function held for just over half of respondents, with the power form a reasonable approximation for most of the rest.¹⁰
• Context and range effects. Luce (1990) noted that “by introducing contexts such as background noise in loudness judgements, the shape of the magnitude estimation functions certainly deviates sharply from a power function”; stimulus range and sequence systematically alter the exponent.¹⁰
• Averaging. Exponent estimates pooled across subjects obscure large individual differences, and hierarchical fitting is now preferred.¹⁰

Reconciliation with Fechner. Mathematically the two laws are closer than the controversy suggested. As a → 0, I^a = exp(a ln I) ≈ 1 + a ln I, so the power function tends to a logarithmic function. MacKay (1963) showed that log and power laws can be made to fit the same data, and that distinguishing them requires specifying the neural transducer.¹⁰⁹

The 2024 Fisher-information framework supplies the missing link. Discrimination constrains Fisher sensitivity, D(s) = |μ′(s)| / σ(s), the ratio of the slope of the internal representation to its noise. Weber’s law requires D(s) ∝ 1/s. If the mean representation is a power function, μ(s) = s^a, then Weber’s law holds exactly when σ(s) ∝ μ(s) — noise SD proportional to the mean — for every exponent a.¹ Fechner’s law and Stevens’ law are thus not rival descriptions of one data set but different measurements (appearance versus discriminability) of a single noisy representation. This is the sense in which proposals for unification made over more than 60 years are now realised.¹

5. The current synthesis (2017–2026)

Efficient coding as a generating principle

Wei and Stocker (2017) derived a remarkably simple relation between perceptual bias and discrimination threshold from a theory combining optimal encoding with optimal decoding, and showed it holds across a wide range of psychophysical data including contextual modulation — proposing it as a psychophysical law in its own right.¹¹ Combined with the Fisher-sensitivity result, this reframes Weber’s, Fechner’s and Stevens’ laws as consequences of how limited coding resources are allocated to a stimulus distribution, rather than as independent regularities.¹¹¹

A Bayesian treatment reaches overlapping conclusions from a different direction. Petzschner, Glasauer and Stephan (2015) show that a generic Bayesian observer — noisy likelihood plus prior — reproduces the characteristic biases of magnitude estimation across modalities: regression to the mean, range effects and sequential effects. On that account the Weber–Fechner law and Stevens’ power law can be linked through Bayes’ rule, with the form of the psychophysical function depending on the observer’s prior and on the noise model.¹²

Where the accounts disagree. The efficient-coding and natural-inference accounts are not the same claim. Prat-Carrabin et al. locate Weber’s law in the allocation of coding resources to a stimulus distribution and demonstrate it by manipulating that distribution.⁷ Beck et al. locate it in natural input statistics — global scale variables that induce correlated noise — and argue that no noisy neurons are required.⁶ They agree that Weber’s law is context-dependent and distribution-sensitive, but disagree about the locus. A related Bayesian result shows that priors and reward can produce anti-Weber behaviour even while the encoding remains logarithmic, which suggests that “Weber behaviour” and “Weber encoding” can come apart.⁸

The laws in machines

If the laws are consequences of learning skewed data under limited resources, they should appear in artificial systems trained on natural data. Two 2025–2026 results test this.

• Language models and magnitude. Cacioli (2026) applied psychophysical methods to three 7–9B instruction-tuned models (Llama, Mistral, Qwen) across numerical, temporal and spatial magnitudes. Representational similarity to a Weber-law dissimilarity matrix was high (.68–.96) across 96 model–domain–layer cells, and linear geometry was never preferred — the geometry is consistently log-compressive. But geometry did not guarantee behaviour: of the two models tested behaviourally, one produced a human-range Weber fraction (0.20) and the other did not, and both were at chance on temporal and spatial discrimination. Causal intervention implicated early layers, where geometry was weakest, rather than later layers where it was strongest. Corpus integer frequencies followed a power law (α = 0.77), satisfying the efficient-coding precondition.¹³ (Preprint.)
• Language models and time. Li et al. (2025) found that larger models spontaneously establish a subjective temporal reference point (around 2025) and show logarithmic compression of perceived distance as years recede from it — the Weber–Fechner pattern — with a subgroup of units whose activation is minimal at the reference point and implements logarithmic coding.¹⁴ (Preprint.)

The cautious reading is that log-compressive geometry is a generic consequence of learning skewed magnitude distributions, but that psychophysical laws of behaviour require the decoder and decision stage, not just the representation.

Evidence status

Claim	Status	Basis	What would settle it
The JND is a fixed sensory quantum	Rejected	SDT; persistence of the belief documented in 20232	definitional, not empirical
Weber’s law holds universally	Qualified	Fails near threshold, at high intensity, at scotopic levels34	parametric sampling across the full natural range
Fechner’s law follows from Weber’s law	Rejected as an inference	Weber underdetermines the transducer1	joint measurement of μ(s) and σ(s)
Fechner and Stevens are rival laws	Dissolved as a forced choice	Both follow from one representation when σ ∝ μ1	measuring thresholds and ratings in the same observers and modality
Efficient coding generates Weber’s law	Strong but preprints	Causal prior manipulation in three modalities7; supporting Bayesian results8	peer review; extension beyond vision; independent replication
Weber’s law reflects input statistics alone	Contested	Natural-inference account6 versus coding-allocation account7	experiments that decorrelate input statistics from resource allocation
The laws appear in artificial networks	Partial	Log geometry in LLMs, but behaviour dissociates13; Weber–Fechner timing in LLMs14	tasks measuring geometry and behaviour in the same model

6. Open questions

• Which stage owns the law? Encoding (transducer and noise), decoding (prior and loss), or the decision criterion? The 2024–2026 work suggests all three can produce or abolish Weber behaviour.
• Why these exponents? Is the modality ranking of Stevens exponents derivable from receptor transduction and natural stimulus statistics, or is it a measurement artefact of scaling tasks?
• How general is the distribution manipulation? The 2026 demonstration covers line length, brightness and speed. Audition, numerosity, time and interoception remain untested in that paradigm.
• What is a law? Weber’s, Fechner’s and Stevens’ laws are descriptive regularities with regime boundaries. Whether any is derivable from a deeper principle — efficient coding, Bayesian inference, or both — remains the live theoretical question.
• Measurement theory. If magnitude estimates are not ratio-scale, which parts of Stevens’ edifice survive? The bias-and-discriminability relation¹¹ may survive even where the exponent does not.

7. Practical consequences for experiment design

• Space stimuli geometrically when Weber holds. Equal log steps equalise discriminability; linear spacing silently confounds difficulty with magnitude, which is exactly what the 2026 prior manipulation exploited.
• Never report a JND without its psychometric specification. State criterion, guess rate, lapse rate and fitting method — or report d′.
• Fit the generalised law when spanning a wide range. ΔI = kI + c, or D(s) = w/(s+d)^β, accommodates the low- and high-intensity deviations that the pure law cannot.¹
• Counterbalance range and context in scaling tasks. Exponents depend on the stimulus range, the background and the sequence of trials.¹⁰¹²
• Fit per subject, then pool hierarchically. Averaging magnitude estimates or exponents across observers distorts both.¹⁰
• For animal touchscreen work, define difficulty by ratio, not absolute difference — and expect the effective Weber fraction to depend on the animal’s priors, reward history and criterion, not only on its sensory system.⁷⁸

Relation to other pages in the wiki

• bouba-kiki-effect — cross-modal correspondence is a scaling problem; Stevens’ cross-modality matching is the psychophysical method behind it.
• metacognition — the criterion/sensitivity distinction that dissolved the JND is the same one that separates confidence from competence (meta-d′).
• natural-scenes-dataset — the natural image statistics that efficient-coding accounts of Weber’s law presuppose.
• psychophysical-harmony — a different use of “psychophysical”: a philosophical argument about the alignment of experience and function, not the scaling laws described here.

References

• Weber, E. H. (1834). De pulsu, auditu et tactu: Annotationes anatomicae et physiologicae. Leipzig: Koehler.
• Fechner, G. T. (1860). Elemente der Psychophysik. Leipzig: Breitkopf und Härtel.
• Stevens, S. S. (1957). On the psychophysical law. Psychological Review, 64(3), 153–181. doi:10.1037/h0046162
• Stevens, S. S. (1961). To honor Fechner and repeal his law. Science, 133(3446), 80–86. doi:10.1126/science.133.3446.80
• MacKay, D. M. (1963). Psychophysics of perceived intensity: a theoretical basis for Fechner’s and Stevens’ laws. Science, 139, 1213–1216.
• Jesteadt, W., Wier, C. C., & Green, D. M. (1977). Intensity discrimination as a function of frequency and sensation level. Journal of the Acoustical Society of America, 61, 169–177.
• Carriot, J., Cullen, K. E., & Chacron, M. J. (2021). The neural basis for violations of Weber’s law in self-motion perception. PNAS, 118(36), e2025061118. doi:10.1073/pnas.2025061118

Footnotes

1. raw/papers/zhou-2024-unified-framework-magnitude-discriminability.md ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹

2. raw/papers/sanford-halberda-2023-jnd-misconception.md ↩ ↩²

3. raw/articles/wikipedia-weber-fechner-law.md ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷

4. raw/papers/carriot-2021-webers-law-violations-self-motion.md ↩ ↩² ↩³

5. raw/papers/penconek-2025-webers-law-global-inhibition.md ↩

6. raw/papers/beck-2024-webers-law-natural-inference.md ↩ ↩² ↩³

7. raw/papers/prat-carrabin-2026-efficient-coding-webers-law.md ↩ ↩² ↩³ ↩⁴ ↩⁵

8. raw/papers/prat-carrabin-gershman-2025-anti-weber-variability.md ↩ ↩² ↩³ ↩⁴ ↩⁵

9. raw/papers/johnson-2002-neural-coding-basic-law.md ↩ ↩²

10. raw/articles/wikipedia-stevens-power-law.md ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸

11. raw/papers/wei-stocker-2017-lawful-relation-bias-discriminability.md ↩ ↩² ↩³

12. raw/papers/petzschner-2015-bayesian-magnitude-estimation.md ↩ ↩²

13. raw/papers/cacioli-2026-webers-law-transformer-magnitude.md ↩ ↩²

14. raw/papers/li-2025-llm-temporal-cognition-weber-fechner.md ↩ ↩²