Measurement Invariance, Entropy, and Probability

We show that the natural scaling of measurement for a particular problem defines the most likely probability distribution of observations taken from that measurement scale. Our approach extends the method of maximum entropy to use measurement scale as a type of information constraint. We argue that a very common measurement scale is linear at small magnitudes grading into logarithmic at large magnitudes, leading to observations that often follow Student's probability distribution which has a Gaussian shape for small fluctuations from the mean and a power law shape for large fluctuations from the mean. An inverse scaling often arises in which measures naturally grade from logarithmic to linear as one moves from small to large magnitudes, leading to observations that often follow a gamma probability distribution. A gamma distribution has a power law shape for small magnitudes and an exponential shape for large magnitudes. The two measurement scales are natural inverses connected by the Laplace integral transform. This inversion connects the two major scaling patterns commonly found in nature. We also show that superstatistics is a special case of an integral transform, and thus can be understood as a particular way in which to change the scale of measurement. Incorporating information about measurement scale into maximum entropy provides a general approach to the relations between measurement, information and probability.


Introduction
Suppose you have a ruler that is about the length of your hand. With that ruler, you can measure the size of all the visible objects in your office. That scaling of objects in your office with the length of the ruler means that those objects have a natural linear scaling in relation to your ruler. Now consider the distances from your office to various galaxies. Your ruler is of no use, because you cannot distinguish whether a particular galaxy moves farther away by one ruler unit. Instead, for two galaxies, you can measure the ratio of distances from your office to each galaxy. You might, for example, find that one galaxy is twice as far as another, or, in general, that a galaxy is some percentage farther away than another.
Percentage changes define a ratio scale of measure, which has natural units in logarithmic measure [1]. For example, a doubling of distance always adds log (2) to the logarithm of the distance, no matter what the initial distance.
Measurement naturally grades from linear at local magnitudes to logarithmic at distant magnitudes when compared to some local reference scale. The transition between linear and logarithmic varies between problems. Measures from some phenomena remain primarily in the linear domain, such as measures of height and weight in humans. Measures for other phenomena remain primarily in the logarithmic domain, such as cosmological distances. Other phenomena scale between the linear and logarithmic domains, such as fluctuations in the price of financial assets [2] or the distribution of income and wealth [3].
The second section of this article shows how the characteristic scaling of measurement constrains the most likely probability distribution of observations. We use the standard method of maximum entropy to find the most likely probability distribution [4]. But, rather than follow the traditional approach of starting with the information in the summary statistics of observations, such as the mean or variance, we begin with the information in the characteristic scale of measurement. We argue that measurement sets the fundamental nature of information and shapes the probability distribution of observations. We present a novel extension of the method of maximum entropy to incorporate information about the scale of measurement.
The third section emphasizes the naturalness of the measurement scale that grades from linear at small magnitudes to logarithmic at large magnitudes. This linear to logarithmic scaling leads to observations that often follow a linear-log exponential or Student's probability distribution. A linear-log exponential distribution is an exponential shape for small magnitudes and a power law shape for large magnitudes. Student's distribution is a Gaussian shape for small fluctuations from the mean and a power law for large fluctuations from the mean. The shapes correspond to linear scaling at small magnitudes and logarithmic scaling at large magnitudes. Many naturally observed patterns follow these distributions. The particular form depends on whether the measurement scale for a problem is primarily linear, primarily logarithmic, or grades from linear to logarithmic.
The fourth section inverts the natural linear to logarithmic scaling for magnitudes. Because magnitudes often scale from linear to logarithmic as one moves from small to large magnitudes, inverse measures often scale from logarithmic to linear as one moves from small to large magnitudes. This logarithmic to linear scaling leads to observations that often follow a gamma probability distribution.
A gamma distribution is a power law shape for small magnitudes and an exponential shape for large magnitudes, corresponding to logarithmic scaling at small values and linear scaling at large values. The gamma distribution includes as special cases the exponential distribution, the power law distribution, and the chi-square distribution, subsuming many commonly observed patterns.
The fifth section demonstrates that the Laplace integral transform provides the formal connection between the inverse measurement scales. The Laplace transform, like its analytic continuation the Fourier transform, changes a magnitude with dimension d on one scale into an inverse magnitude with dimension 1/d on the other scale. This inversion explains the close association between the linear to logarithmic scaling as magnitudes increase and the inverse scale that grades from logarithmic to linear as magnitudes increase. We discuss the general role of integral transforms in changing the scale of measurement. Superstatistics is the averaging of a probability distribution with a variable parameter over a probability distribution for the variable parameter [5]. We show that superstatistics is a special case of an integral transform, and thus can be understood as a particular way in which to change the scale of measurement.
In the sixth section, we relate our study of measurement invariance for continuous variables to previous methods of maximum entropy for discrete variables. We also distinguish the general definition of measurement scale by information invariance from our particular argument about the commonness of linear-log scales.
In the discussion, we contrast our emphasis on the primacy of measurement with alternative approaches to understanding measurement, randomness, and probability. One common approach changes the definition of randomness and entropy to incorporate a change in measurement scale [6]. We argue that our method makes more sense, because we directly incorporate the change in measurement scale as a kind of information, rather than alter the definition of randomness and entropy to match each change in measurement scale. It is measurement that changes empirically between problems rather than the abstract meaning of randomness and information. Although we focus on the duality between linear to logarithmic scaling and its inverse logarithmic to linear scaling, our general approach applies to any type of measure invariance and measurement scale.

Measurement, Information Invariance, and Probability
We derive most likely probability distributions. Our method follows the maximum entropy approach [4,7,8]. That approach assumes that the most likely distribution has the maximum amount of randomness, or entropy, subject to the constraint that the distribution must capture all of the information available to us. For example, if we know the average value of a sample of observations, and we know that all values from the underlying probability distribution are positive, then all candidate probability distributions must have only positive values and have a mean value that agrees with the average of the empirically observed values. By maximum entropy, the most random distribution constrained to have positive values and a fixed mean is the exponential distribution.
We express the available information by constraints. Typical constraints include the average or variance of observations. But we must use all available information, which may include information about the scale of measurement itself. Previous studies have discussed how the scale of measurement provides information. However, that aspect of maximum entropy has not been fully developed [4,9].
Our goal is to develop the central role of measurement scaling in shaping the commonly observed probability distributions.
In the following sections, we show how to use information about measurement invariances and associated measurement scales to find most likely probability distributions.

Maximum entropy
The method of maximum entropy defines the most likely probability distribution as the distribution that maximizes a measure of entropy (randomness) subject to various information constraints. We write the quantity to be maximized as where E measures entropy, the C i are the constraints to be satisfied, and α and the λ i are the Lagrange multipliers to be found by satisfying the constraints. Let C 0 = p y dy − 1 be the constraint that the probabilities must total one, where p y is the probability distribution function of y. The other constraints are usually written as , where the f i (y) are various transformed measurements of y. Angle brackets denote mean values. A mean value is either the average of some function applied to each of a sample of observed values, or an a priori assumption about the average value of some function with respect to a candidate set of probability laws. If f i (y) = y i , then y i are the moments of the distribution-either the moments estimated from observations or a priori values of the moments set by assumption. The moments are often regarded as "normal" constraints, although from a mathematical point of view, any properly formed constraint can be used.
Here, we confine ourselves to a single constraint of measurement. We express that constraint with a more general notation, C 1 = p y T[f (y)]dy − T[f (y)] , where T() is a transformation. We could, of course, express the constraining function for y directly through f (y). However, we wish to distinguish between an initial function f (y) that can be regarded as a normal measurement, in any sense in which one chooses to interpret the meaning of normal, and a transformation of normal measurements denoted by T() that arises from information about the measurement scale.
The maximum entropy distribution is obtained by solving the set of equations where one checks the candidate solution for a maximum and obtains α and λ by satisfying the constraint on total probability and the constraint on T[f (y)] . We assume that we can treat entropy measures as the continuous limit of the discrete case.
In the standard approach, we define entropy by Shannon information which yields the solution of Equation (2) as where k and λ satisfy the two constraints.

Measurement and transformation
Maximum entropy, in order to be a useful method, must capture all of the available information about a particular problem. One form of information concerns transformations to the measurement scale that leave the most likely probability distribution unchanged. Suppose, for example, that we obtain the same information from measurements of x and transformed measurements, G(x). Put another way, if one has access only to measurements on the G(x) scale, one has the same information that would be obtained if the measurements were reported on the x scale. We say that the measurements x and G(x) are equivalent with respect to information, or that the transformation x → G(x) is an invariance [1,10,11].
To capture this information invariance in maximum entropy, we must express our measurements on a transformed scale. In particular, we must choose the transformation, T(), for expressing measurements so that for some arbitrary constants γ and δ. Putting this definition of T(x) into Equation (4) shows that we get the same maximum entropy solution whether we use the direct scale x or the alternative measurement scale, G(x), because the k and λ constants will adjust to the constants γ and δ so that the distribution remains unchanged. Given the transformation T(x), the derivative of that transformation expresses the information invariance in terms of measurement invariance. In particular, we have the following invariance of the measurement scale under a change dx We may also examine m x = T ′ (x) = dT(x)/dx to obtain the change in measurement scale required to preserve the information invariance between x and G(x). If we know the measurement invariance, G(x), we can find the correct transformation from Equation (5). If we know the transformation T(x), we can find G(x) by inverting Equation (5) to obtain Alternatively, we may deduce the transformation T(x) by examining the form of a given probability distribution and using Equation (4) to find the associated transformation. In summary, x and G(x) provide invariant information, and the transformation of measurements T(x) captures that information invariance in terms of measurement invariance.

Example: ratio and scale invariance
Suppose the information we obtain from positive-valued measurements depends only on the ratio of measurements, y 2 /y 1 . In this particular case, all measurements with the same ratio map to the same value, so we say that the measurement scale has ratio invariance. Pure ratio measurements also have scale invariance, because ratios do not depend on the magnitude or scale of the observations.
We express the invariances that characterize a measurement scale by the transformations that leave the information in the measurements unchanged [1,10,11]. If we obtain values x and use the measurement scale from the transformation T(x) = log(x), the information in x is the same as in

which means that the information in the measurement scale given by T(x) is invariant under the transformation G(x).
We can express the invariance in a way that captures how measurement relates to information and probability. The transformation T(x) = log(x) shrinks increments on the uniform scaling of x so that each equally spaced increment on the original uniform scale shrinks to length 1/x on the transformed scale. We can in general quantify the deformation in incremental scaling by the derivative of the transformation T(x) with respect to x. In the case of the logarithmic measurement scale with ratio invariance, the measure invariance in Equation (6) is x showing in another way that the logarithmic measure T(x) is invariant under the transformation G(x). With regard to probability or information, we can think of the logarithmic scale with ratio invariance as having an expected density of probability per increment in proportion to 1/x, so that the expected density of observations at scale x decreases in proportion to 1/x. Roughly, we may also say that the information value of an increment decreases in proportion to 1/x. For example, the increment length of our hand is an informative measure for the visible objects near us, but provides essentially no information on a cosmological scale.
If we have measurements f (y) = y, and we transform those measurements in a way consistent with a ratio and scale invariance of information, then we have the transformed measures T[f (y)] = log(y). The constraint for maximum entropy corresponds to log(y) , which is logarithm of the geometric mean of the observations on the direct scale y. Given that constraint, the maximum entropy distribution is a power law For y ≥ 1, we can solve for the constants k and λ, yielding p y = δy −(1+δ) , with δ = 1/ log(y) .

Measurement
In the previous section, we obtained ratio and scale invariance with a measure m x = T ′ (x) ∝ 1/x. In this section, we consider the more general measure At small values of x, the measure becomes linear, m x ∝ 1, and at large values of x, the measure becomes ratio invariant (logarithmic), m x ∝ 1/x. This measure has scale dependence with ratio invariance at large scales, because the measure changes with the magnitude (scale) of x, becoming ratio invariant at large values of x. The parameter b controls the scale at which the measure grades between linear and logarithmic.
Given m x = T ′ (x), we can integrate this deformation of measurement to obtain the associated scale of measurement as where we have expressed the proportionality constant as 1/a and we have dropped the constant of integration. The expression log(1 + bx) is just a logarithmic measurement scale for positive values in relation to a fixed origin at x = 0, because log(1) = 0. The standard logarithmic expression, log(x), has an implicit origin for positive values at x = 1, which is only appropriate for purely ratio invariant problems with no notion of an origin to set the scale of magnitudes. In most empirical problems, there is some information about the scaling of magnitudes. Thus, log(1 + bx) is more often the natural measurement scale. Next, we seek an expression G(x) to describe the information invariance in the measurement scale, such that the information in x and in G(x) is the same. The expression in Equation (6), , sets the condition for information invariance, leading to On the measurement scale T(x), the information in x is the same as in G(x), because We now use x = f (y) to account for initial normal measures that may be taken in any way we choose. Typically, we use direct values, f (y) = y, or squared values, f (y) = y 2 , corresponding to initial measures related to the first and second moments-the average and variance. For now, we use f (y) to hold the place of whatever direct values we will use. Later, we consider the interpretations of the first and second moments.

Probability
The constraint for maximum entropy corresponds to T[f (y)] = log[1 + bf (y)] 1 a , a value that approximately corresponds to an interpolation between the linear mean and the geometric mean of f (y). Given that constraint, the maximum entropy distribution from Equation (4) is where α = λ/a acts as a single parameter chosen to satisfy the constraint, and b is a parameter derived from the measurement invariance that expresses the natural scale of measurement for a particular problem. From Equation (10), we can express simple results when in either the purely linear or purely logarithmic regime. For small values of bf (y) we can write p y ∝ e −αbf (y) . For large values of bf (y) we can write p y ∝ f (y) −α , where we absorb b −α into the proportionality constant. Thus, the probability distribution grades from exponential in f (y) at small magnitudes to a power law in f (y) at large magnitudes, corresponding to the grading of the linear to logarithmic measurement scale.

Transition between linear and logarithmic scales
We mentioned that one can obtain the parameter α in Equation (10) directly from the constraint T[f (y)] , which can be calculated directly from observed values of the process or set by assumption. What about the parameter b that sets the grading between the linear and logarithmic regimes?
When we are in the logarithmic regime at large values of bf (y), probabilities scale as p y ∝ f (y) −α independently of b. Thus, with respect to b, we only need to know the magnitude of observations above which ratio invariance and logarithmic scaling become reasonable descriptions of the measurement scale.
In the linear regime, p y ∝ e −αbf (y) , thus b only arises as a constant multiplier of α and so can be subsumed into a single combined parameter β = αb estimated from the single constraint. However, it is useful to consider the meaning of b in the linear regime to provide guidance for how to interpret b in the mixed regime in which we need the full expression in Equation (10).
When f (y) = y, the linear regime yields an exponential distribution p y ∝ e −αby . In this case, b weights the intensity or rate of the process α that sets the scaling of the distribution.
When f (y) = y 2 , the linear regime yields a Gaussian distribution p y ∝ e −αby 2 , where 2αb is the reciprocal of the variance that defines the precision of measurements-the amount of information a measurement provides about the location of the average value. In this case, b weights the precision of measurement. The greater the value of b, the more information per increment on the measurement scale.

Linear-log exponential distribution
When f (y) = y, we obtain from Equation (10) what we will call the linear-log exponential distribution for y > 0. This distribution is often called the generalized type II Pareto distribution or the Lomax distribution [12]. Small values of by lead to an exponential shape, p y ∝ e −αby . Large values of by lead to power law tails, p y ∝ y −α . The parameter b determines the grading from the exponential to the power law. Small values of b extend the exponential to higher values of y, whereas large values of b move the extent of the power law shape toward smaller values of y. Many natural phenomena follow a linear-log exponential distribution [6].

Student's distribution
When f (y) = y 2 , we obtain from Equation (10) Student's distribution Here, we assume that y expresses deviations from the average. Small deviations lead to a Gaussian shape around the mean, p y ∝ e −αby 2 . Large deviations lead to power law tails, p y ∝ f (y) −α . The parameter b determines the grading from the Gaussian to the power law. Small values of b expand the Gaussian shape far from the mean, whereas large values of b move the extent of the power law shape closer to the central value at the average. Many natural phenomena expressed as deviations from a central value follow Student's distribution [6]. The ubiquity of both Student's distribution and the linear-log exponential distribution arises from the fact that the grading between linear measurement scaling at small magnitudes and logarithmic measurement scaling at large magnitudes is inevitably widespread. Many cases will be primarily in the linear regime and so be mostly exponential or Gaussian except in the extreme tails. Many other cases will be primarily in the logarithmic regime and so be mostly power law except in the regime of small deviations near the origin or the central location. Other cases will produce measurements across both scales and their transition.

The Inverse Logarithmic to Linear Measurement Scale
We have argued that the linear to logarithmic measurement scale is likely to be common. Magnitudes such as time or distance naturally grade from linear at small scales to logarithmic at large scales.
Many problems measure inverse dimensions, such as the reciprocals of time or distance. If magnitudes of time or space naturally grade from linear to logarithmic as scale increases from small to large, then how do the reciprocals scale? In this section, we argue that the inverse scale naturally grades from logarithmic to linear as scale increases from small to large.
We first describe the logarithmic to linear measurement scale and its consequences for probability. We then show the sense in which the logarithmic to linear scale is the natural inverse of the linear to logarithmic scale.

Measurement
The transformation corresponds to the change in measurement scale m x = 1 + b/x. As x becomes small, the measurement scaling m x → 1/x becomes the ratio-invariant logarithmic scale. As x increases, the measurement scaling m x → 1 becomes the uniform measure associated with the standard linear scale. Thus, the scaling m x = 1 + b/x interpolates between logarithmic and linear measurements, with the weighting of the two scales shifting from logarithmic to linear as x increases from small to large values.

Probability
The constraint for maximum entropy corresponds to T[f (y)] = f (y) + b log[f (y)] , a value that interpolates between the linear mean and the geometric mean of f (y). Given that constraint, the maximum entropy distribution is p y ∝ f (y) −λb e −λf (y) , with λ chosen to satisfy the constraint.
The direct measure f (y) = y for positive values is the gamma distribution As y becomes small, the distribution approaches a power law form, p y ∝ y −λb . As y becomes large, the distribution approaches an exponential form in the tails, p y ∝ e −λy . Thus, the distribution grades from power law at small scales to exponential at large scales, corresponding to the measurement scale that grades from logarithmic to linear as magnitude increases. Larger values of b extend the power law to higher magnitudes by pushing the logarithmic to linear change in measure to higher magnitudes. The combination of power law and exponential shapes in the gamma distribution is the direct inverse of the linear-log exponential distribution given in Equation (11).
The squared values f (y) = y 2 , which we interpret as squared deviations from the average value, lead to p y ∝ y −λb e −λy 2 /2 (14) where the exponent of two on the first power law component is subsumed in the other parameters. This distribution is a power law at small scales with Gaussian tails at large scales, providing the inverse of Student's distribution in Equation (12). This distribution is a form of the generalized gamma distribution [12], which we call the gamma-Gauss distribution. This distribution may, for example, arise as the sum of truncated power laws or Lévy flights [9].

Integral Transforms and Superstatistics
The previous sections showed that linear to logarithmic scaling has a simple relation to its inverse of logarithmic to linear scaling. That simple relation suggests that the two inverse scales can be connected by some sort of transformation of measure. We will now show the connection.
Suppose we start with a particular measurement scale given by T(x) and its associated probability distribution given by Consider a second measurement scale T(σ) with associated probability distribution What sort of transformation relates the two measurement scales?
The integral transforms often provide a way to connect two measurement scales. For example, we could write This expression is called an integral transform of p σ with respect to the transformation kernel g x|σ . If we interpret g x|σ as a probability distribution of x given a parameter σ, and p σ as a probability distribution over the variable parameter σ, then the expression for p x is called a superstatistic: the probability distribution, p x , that arises when one starts with a different distribution, g x|σ , and averages that distribution over a variable parameter with distribution p σ [5]. It is often useful to think of a superstatistic as an integral transform that transforms the measurement scale. In particular, we can expand Equation (15) as which shows that the transformation kernel g x|σ changes the measurement scale from T(σ) to T(x). It is not necessary to think of g x|σ as a probability distribution-the essential role of g x|σ concerns a change in measurement scale.
The Laplace transform provides the connection between our inverse linear-logarithmic measurement scales. To begin, expand the right side of Equation (16) using the Laplace transform kernel g x|σ = e −σx , and use the inverse logarithmic to linear measurement scale, T(σ) = σ + b log(σ). Integrating from zero to infinity yields e −αT(x) ∝ (1 + x/λ) bλ−1 with the requirement that bλ < 1. From this, we have T(x) ∝ log(1 + x/λ), which is the linear to logarithmic scale. Thus, the Laplace transform inverts the logarithmic to linear scale into the linear to logarithmic scale. The inverse Laplace transform converts in the other direction. If we use x = y, then the transform relates the linear-log exponential distribution of Equation (11) to the gamma distribution of Equation (13). If we use x = y 2 , then the transform relates Student's distribution of Equation (12) to the gamma-Gauss distribution of Equation (14).
The Laplace transform inverts the measurement scales. This inversion is consistent with a common property of Laplace transforms, in which the transform inverts a measure with dimension d to a measure with dimension 1/d. One sometimes interprets the inversion as a change from a direct measure to a rate or frequency. Here, it is only the inversion of dimension that is significant. The inversion arises because, in the transformation kernel g x|σ = e −σx , the exponent σx is typically non-dimensional, so that the dimensions of σ and x are reciprocals of each other. The transformation takes a distribution in σ given by p σ and returns a distribution in x given by p x . Thus, the transformation typically inverts the dimension.

Discrete versus continuous variables
We used measure invariance to analyze maximum entropy for continuous variables. We did not discuss discrete variables, because measure invariance applied to discrete variables has been widely and correctly used in maximum entropy [4,9,13]. In the Discussion, we describe why previous attempts to apply invariance to continuous variables did not work in general. That failure motivated our current study.
Here, we briefly review measure invariance in the discrete case for comparison with our analysis of continuous variables. We use the particular example of N Bernoulli trials with a sample measure of the number of successes y = 0, 1, . . . , N. Frank [9] describes the measure invariance for this case: "How many different ways can we can obtain y = 0 successes in N trials? Just one: a series of failures on every trial. How many different ways can we obtain y = 1 success? There are N different ways: a success on the first trial and failures on the others; a success on the second trial, and failures on the others; and so on. The uniform solution by maximum entropy tells us that each different combination is equally likely. Because each value of y maps to a different number of combinations, we must make a correction for the fact that measurements on y are distinct from measurements on the equally likely combinations. In particular, we must formulate a measure. . . that accounts for how the uniformly distributed basis of combinations translates into variable values of the number of successes, y. Put another way, y is invariant to changes in the order of outcomes given a fixed number of successes. That invariance captures a lack of information that must be included in our analysis." In this particular discrete case, transformations of order do not change our information about the total number of successes. Our measurement scale expresses that invariance, and that invariance is in turn captured in the maximum entropy distribution.
The nature of invariance is easy to see in the discrete case by combinatorics. The difficulty in past work has been in figuring out exactly how to capture the same notion of invariance in the continuous case. We showed that the answer is perhaps as simple as it could be: use the transformations that do not change information in the context of a particular problem. Jaynes [4] hinted at this approach, but did not develop and apply the idea in a general way.

General measure invariance versus particular linear-log scales
Our analysis followed two distinct lines of argument. First, we presented the general expression for invariance as a form of information in maximum entropy. We developed that expression particularly for the case of continuous variables. The general expression sets the conditions that define measurement scales and the relation between measurement and probability. But the general expression does not tell us what particular measurement scale will arise in any problem.
Our second line of argument claimed that various types of grading between linear and logarithmic measures arise very commonly in natural problems. Our argument for commonness is primarily inductive and partially subjective. On the inductive side, the associated probability distributions seem to be those most commonly observed in nature. On the subjective side, the apparently simplest assumptions about invariance lead to what we called the common gradings between linear and logarithmic scales. We do not know of any way to prove commonness or naturalness. For now, we are content that the general mathematical arguments lead in a simple way to those probability distributions that appear to arise commonly in nature.
A different view of what is common or what is simple would of course lead to different information invariances, measurement scales, and probability distributions. In that case, our general mathematical methods would provide the tools by which to analyze the alternative view.

Discussion
We developed four topics. First, we provided a new extension to the method of maximum entropy in which we use the measurement scale as a primary type of information constraint. Second, we argued that a measurement scale that grades from linear to logarithmic as magnitude increases is likely to be very common. The linear-log exponential and Student's distributions follow immediately from this measurement scale. Third, we showed that the inverse measure that grades from logarithmic at small scales to linear at large scales leads to the gamma and gamma-Gauss distributions. Fourth, we demonstrated that the two measurement scales are natural inverses related by the Laplace integral transform. Superstatistics are a special case of integral transforms and can be understood as changes in measurement scale.
In this discussion, we focus on measurement invariance, alternative definitions of entropy, and maximum entropy methods.
Jaynes [4] summarized the problem of incorporating measurement invariance as a form of information in maximum entropy. The standard conclusion is that one should use relative entropy to account for measurement invariance. In our notation, for a measurement scale T(y) with measure deformation m y = T ′ (y), the form of relative entropy is the Kullback-Leibler divergence E = − p y log p y m y dy in which the m y is proportional to a prior probability distribution that incorporates the information from the measurement scale and leads to the analysis of maximum relative entropy. This approach works in cases where the measure change, m y , is directly related to a change in the probability measure. Such changes in probability measure typically arise in combinatorial problems, such as a type of measurement that cannot distinguish between the order of elements in sets. For continuous deformations of the measurement scale, using m y as a relative scaling for probability does not always give the correct answer. In particular, if one uses the constraint f (y) and the measure m y in the above definition of relative entropy, the maximum relative entropy gives the probability distribution p y ∝ m y e −λf (y) which is often not the correct result. Instead, the correct result follows from the method we gave, in which the information from measurement invariance is incorporated by transforming the constraint as T[f (y)] , yielding the maximum entropy solution It is possible to change the definition of entropy, such that maximum entropy applied to the transformed measure of entropy plus the direct constraint f (y) gives the correct answer [6]. The resulting probability distributions are of course the same when transforming the constraint or using an appropriate matching transformation of the entropy measure. We discuss the mathematical relation between the alternative transformations in a later paper.
We prefer the transformation of the constraint, because that approach directly shows how information about measurement alters scale and is incorporated into maximum entropy. By contrast, changing the definition of entropy requires each measurement scale to have its own particular definition of entropy and information. Measurement is inherently an empirical phenomenon that is particular to each type of problem and so naturally should be changed as the nature of the problem changes. The abstract notions of entropy and information are not inherently empirical factors that change among problems, so it seems perverse to change the definition of randomness with each transformation of measurement.
The Tsallis and Rényi entropy measures are transformations of Shannon entropy that incorporate the scaling of measurement from linear to logarithmic as magnitude increases [6]. Those forms of entropy therefore could be used as a common alternative to Shannon entropy whenever measurements scale linearly to logarithmically. Although mathematically correct, such entropies change the definition of randomness to hide the particular underlying transformation of measurement. That approach makes it very difficult to understand how alternative measurement scales alter the expected types of probability distributions.

Conclusions
Linear and logarithmic measurements seem to be the most common natural scales. However, as magnitudes change, measurement often grades between the linear and logarithmic scales. That transition between scales is often overlooked. We showed how a measurement scale that grades from linear to logarithmic as magnitude increases leads to some of the most common patterns of nature expressed as the linear-log exponential distribution and Student's distribution. Those distributions include the exponential, power law, and Gaussian distributions as special cases, and also include hybrids between those distribution that must commonly arise when measurements span the linear and logarithmic regimes.
We showed that a measure grading from logarithmic to linear as magnitude increases is a natural inverse scale. That measurement scale leads to the gamma and gamma-Gauss distributions. Those distributions are also composed of exponential, power law, and Gaussian components. However, those distributions have the power law forms at small magnitudes corresponding to the logarithmic measure at small magnitudes, whereas the inverse scale has the power law components at large magnitudes corresponding to the logarithmic measure at large magnitudes.
The two measurement scales are natural inverses connected by the Laplace transform. That transform inverts the dimension, so that a magnitude of dimension d on one scale becomes an inverse magnitude of dimension 1/d on the other scale. Inversion connects the two major scaling patterns commonly found in nature. Our methods of incorporating information about measurement scale into maximum entropy also apply to other forms of measurement scaling and invariance, providing a general method to study the relations between measurement, information, and probability.