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# Measurement invariance explains the universal law of generalization for psychological perception

Edited by Günter P. Wagner, Yale University, New Haven, CT, and approved August 15, 2018 (received for review June 7, 2018)

## Significance

When an animal is presented with two stimuli, it may consider them similar or different. Similarity often expresses a generalized notion of a category, such as two circles with different sizes, shadings, and colors both being circles. In many studies, perception of similarity declines exponentially with the measure of separation, a pattern often called the universal law of generalization. This article shows that the universal exponential law can be explained by simple properties any reasonable perceptual scale must have. A shift of the scale by a constant amount, or a stretch by a constant amount, should not change the animal’s ability to perceive generalities or differences. Those invariant measurement properties by themselves explain why perceived generalization follows an exponential pattern.

## Abstract

The universal law of generalization describes how animals discriminate between alternative sensory stimuli. On an appropriate perceptual scale, the probability that an organism perceives two stimuli as similar typically declines exponentially with the difference on the perceptual scale. Exceptions often follow a Gaussian probability pattern rather than an exponential pattern. Previous explanations have been based on underlying theoretical frameworks such as information theory, Kolmogorov complexity, or empirical multidimensional scaling. This article shows that the few inevitable invariances that must apply to any reasonable perceptual scale provide a sufficient explanation for the universal exponential law of generalization. In particular, reasonable measurement scales of perception must be invariant to shift by a constant value, which by itself leads to the exponential form. Similarly, reasonable measurement scales of perception must be invariant to multiplication, or stretch, by a constant value, which leads to the conservation of the slope of discrimination with perceptual difference. In some cases, an additional assumption about exchangeability or rotation of underlying perceptual dimensions leads to a Gaussian pattern of discrimination, which can be understood as a special case of the more general exponential form. The three measurement invariances of shift, stretch, and rotation provide a sufficient explanation for the universally observed patterns of perceptual generalization. All of the additional assumptions and language associated with information, complexity, and empirical scaling are superfluous with regard to the broad patterns of perception.

The probability that an organism perceives two stimuli as similar typically decays exponentially with separation between the stimuli. The exponential decay in perceptual similarity is often referred to as the universal law of generalization (1, 2).

“Generalization” arises because perceived similarity may describe recognition of a general category. For example, two circles may have different sizes, colors, and shadings. Perceived similarity arises from the generalized perception of “circle” as a category.

“Universal law” arises because many empirical observations fit the pattern for diverse sensory modalities across different species. Typical exceptions take on a Gaussian probability pattern for perceived separation (3).

Both theory and empirical analysis depend on the definition of the perceptual scale. How does one translate the perceived differences between two circles with different properties into a quantitative measurement scale?

There are many different suggestions in the literature for how to define a perceptual scale. Each of those suggestions develop very specific notions of measurement based, for example, on information theory, Kolmogorov complexity theory, or multidimensional scaling descriptions derived from observations (1, 2, 4).

I focus on the minimal properties that any reasonable perceptual measurement scale must have rather than on detailed assumptions motivated by external theories of information, complexity, or empirical scaling. I express the minimal properties as simple invariances.

I show that a few inevitable invariances of any reasonable perceptual scale determine the exponential form for the universal law of generalization in perception. All of the other details of information, complexity, and empirical scaling are superfluous with respect to understanding why the universal law of generalization has the exponential form.

I also show that, when the separation between stimuli depends on various underlying perceptional dimensions, it sometimes makes sense to assume that the perceptual scale will also obey exchangeability or rotational invariance. When that additional invariance holds, the universal law takes on the Gaussian form, which I show to be a special case of the general exponential form.

## Basic Problem and Notation

Chater and Vitányi (ref. 2, p. 346) state the law as “the probability of perceiving similarity or analogy between two items, a and b, is a negative exponential function of the distance

Let the notation

The goal here is to understand how the perceived similarity of b to a, observed as

## Invariant Properties of Measurement

There are many different suggestions in the literature for how to define a perceptual scale,

First, suppose we wish to analyze the perception of temperature for event b, given that event a is at the freezing point for water. If we choose to measure the temperature on the Celsius scale, then

Second, suppose we wish to measure the perception of separation between two potentially dangerous prey items, such as noxious butterflies (4, 8). We begin by exposing a noxious butterfly, a, to a predator. After the predator tastes butterfly a, we then expose butterfly b to the same predator. For the exposure to b, we measure the tendency for the predator to attack the potential prey item. Data may include the directions of movements relative to the butterfly, attacks per minute, or the probability of attack over repeated experiments. We now wish to find a scale,

However we choose that scale, it makes sense to suppose that the information in

For example, we may wish to set **1**. Here, similarity associates with the probability of avoidance response. We may also wish to express our scale standardized with respect to a unit response,

## Affine and Rotational Invariance

In other words, the way in which we measure perceptual distance between two stimuli should be independent of a shift and stretch of the scale by constant values. Formally, the scale should be shift invariant with respect to any constant, α, such that

Thus, the scale

In some cases, it makes sense to assume that the perceptual scale should also obey rotational invariance, such that the Pythagorean partition

Rotational invariance partitions a conserved quantity into additive components, for which the order may be exchanged without altering the invariant quantity. When rotational invariance holds, the universal law takes on a Gaussian form, which we will see to be a special case of the general exponential form.

The following sections develop the three invariances of shift, stretch, and rotation. I show that essentially all of the common properties of perceptual generalization follow from these invariances. The analysis here briefly summarizes the detailed development described in Frank (9). The novelty in this article concerns the simple understanding of widely observed psychological patterns.

## Shift Invariance Implies the Exponential Form

To simplify notation, denote the perceptual scale by

From this equality for total probability, which holds for any shift α by adjustment of the constant,

The assumption that a perceptual scale must be shift invariant is, by itself, sufficient to explain the exponential form of the universal law of generalization.

## The Exponential Form Implies Shift Invariance

The previous section showed that if the perceptual scale, x, is shift invariant, then the exponential form of the universal law of generalization follows. This section shows that if the universal law of generalization takes on the exponential form, then the underlying perceptual scale must be shift invariant. Thus, shift invariance is necessary and sufficient for the exponential form. Any assumptions about the perceptual scale beyond shift invariance must be superfluous with respect to the exponential form.

Begin with the assumption of the exponential form in Eq. **7** and write the consequence of a shift of the scale x by α as

## Stretch Invariance and Rate of Perceptual Change

If we assume that the perceptual scale is defined for positive values,

It makes sense to assume that the average discrimination would not change if we arbitrarily multiplied our numerical scale for perception, x, by a constant, β. The conservation of average value and stretch invariance are equivalent, because

The constant

## Rotational Invariance and Gaussian Patterns

The scale, x, measures the perceptual difference between two entities or events. In some cases, the total difference, x, depends on the perceived differences along several distinct underlying dimensions. With two underlying dimensions, we may write

I now show that rotational invariance leads to the Gaussian pattern as a special case of the general exponential form. In the exponential form derived in earlier sections,

A radial vector intersects a fraction of the total probability density in the circumferential path in proportion to

We can also write the Gaussian in terms of the standard perceptual scale, x, as

## Discussion

Any reasonable perceptual scale must satisfy the simple affine invariances of shift and stretch. I have shown that those invariances are sufficient to explain the exponential form of the universal law of generalization. I have also shown that an additional common invariance of rotation explains why some observed patterns of generalization follow a Gaussian rather than an exponential pattern. The Gaussian pattern is, in fact, a special case of exponential scaling, when the scale is a squared Euclidean distance over several underlying dimensions.

Previous explanations also generate the exponential pattern of the universal law (1, 2, 4). The reason those explanations succeed is that they include assumptions about shift invariance, which by itself generates an exponential pattern. All of the other assumptions and language associated with those prior explanations are superfluous with respect to the exponential form. Conclusions about rate of change in discrimination typically associate with an assumption about stretch invariance or, equivalently, conservation of average value.

It is certainly true that additional assumptions will lead to more precise predictions, which may then be tested to rule out particular mechanisms. But those additional assumptions and tests do not directly bear on the general exponential form itself.

I do not know of explicit prior explanations that unify the Gaussian pattern with the universal exponential law. Such explanations, if they exist, will generally reduce to the assumption of rotational invariance. Again, additional assumptions or arguments about particular underlying mechanisms are superfluous with regard to the general pattern.

It is, of course, interesting to consider what underlying perceptual mechanisms lead to the universal law. However, almost certainly, there is no single mechanism that could explain such a widely observed pattern. General patterns require general explanations that apply broadly. The simple invariances of meaningful measurement scales provide that general explanation for the observed patterns of perceptual scaling.

## Acknowledgments

National Science Foundation Grant DEB–1251035 and the Donald Bren Foundation support my research. I completed this work while on sabbatical in the Theoretical Biology group of the Institute for Integrative Biology at Eidgenössische Technische Hochschule (ETH) Zürich.

## Footnotes

- ↵
^{1}Email: safrank{at}uci.edu.

Author contributions: S.A.F. designed research, performed research, and wrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.

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*The New Palgrave Dictionary of Economics, eds*Durlauf SN, Blume LE (Palgrave Macmillan, Basingstoke, UK). - ↵
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*The New Palgrave Dictionary of Economics, eds*Durlauf SN, Blume LE (Palgrave Macmillan, Basingstoke, UK). - ↵
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