Average is Over

Accepted Manuscript Average is Over Iddo Eliazar

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S0378-4371(17)30936-6 https://doi.org/10.1016/j.physa.2017.09.044 PHYSA 18649

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Physica A

Received date : 6 June 2017 Please cite this article as: I. Eliazar, Average is Over, Physica A (2017), https://doi.org/10.1016/j.physa.2017.09.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Average is Over Iddo Eliazar June 6, 2017

Abstract The popular perception of statistical distributions is depicted by the iconic bell curve which comprises of a massive bulk of ‘middle-class’values, and two thin tails – one of small left-wing values, and one of large right-wing values. The shape of the bell curve is unimodal, and its peak represents both the mode and the mean. Thomas Friedman, the famous New York Times columnist, recently asserted that we have entered a human era in which “Average is Over ”. In this paper we present mathematical models for the phenomenon that Friedman highlighted. While the models are derived via di¤erent modeling approaches, they share a common foundation. Inherent tipping points cause the models to phase-shift from a ‘normal’ bell-shape statistical behavior to an ‘anomalous’ statistical behavior: the unimodal shape changes to an unbounded monotone shape, the mode vanishes, and the mean diverges. Hence: (i) there is an explosion of small values; (ii) large values become super-large; (iii) ‘middle-class’values are whipped out, leaving an in…nite rift between the small and the super large values; and (iv) “Average is Over” indeed.

Keywords: entropy maximization; exponential growth; tipping points; phase transitions; lognormal statistics; power-law statistics. PACS: 02.50.-r (probability theory, stochastic processes, and statistics)

E-mail: [email protected]

1

1

Introduction

In his recent book “Thank You for Being Late: An Optimist’s Guide to Thriving in the Age of Accelerations” [1], famed publicist and author Thomas Friedman asserts that “Average is Over ”. The picture underlying Friedman’s notion of average is the iconic bell curve [2], which is applied prevalently across the sciences to model the distribution of a multitude of quantities. Qualitatively, the bell curve depicts three tiers: a dominant middle-class bulk of near-average values, a thin tail of extreme-left values, and a thin tail of extreme-right values (Figure 1a). Illustrative examples of the bell curve include the distributions of test scores, and the distributions of weight and height. Friedman argues that in a world driven by accelerating change [3]-[5] the bellcurve picture is no longer valid, as in such a world the middle tier is disappearing. The backbone of the American economy in the twentieth century was highwage-middle-skill jobs, and Friedman claims that this era is over. Welcome to the twenty-…rst century, in which there are only high-wage-high-skill jobs: “Average is Over”, and in our era there are only two tiers – a low tier and a high tier. Friedman further explains how the “Average is Over” phenomenon goes far beyond the accelerating-change technology sectors, and reaches many other sectors (including his own, journalism). In this paper we present several mathematical models for the “Average is Over” phenomenon. These models establish a general and common setting in which distributions shift – via phase transitions that occur at tipping points – from a three-tier bell-shape con…guration (Figure 1b) to a two-tier monotone and unbounded con…guration (Figure 1d). The models are based on two pillars: entropy maximization and exponential growth. Randomness is inherently present everywhere. Indeed, practically all quantities we encounter display some degree of randomness, and thus should be modeled statistically. Moreover, even the statistics of given quantities are rarely known precisely. Commonly, only partial information regarding the statistics of given quantities is available. The principle of entropy maximization – which is keystone in statistical physics and in information theory – asserts that [6]-[8]: the model that should be applied for a quantity of interest is the statistical distribution that maximizes entropy with respect to the available information about the quantity. Accelerating change, which was noted above, is characterized by exponential growth: progress that advances according to Moore’s law, i.e. keeps on doubling in performance [9]-[11]. In this paper, combining together entropy maximization and exponential growth, we mathematically model the “Average is Over” phenomenon and investigate how a ‘normal ’statistical behavior (Figure 1b) can phase-shift to an ‘anomalous’ [12]-[15] statistical behavior (Figure 1d). The paper is organized as follows. We begin with a general setting that is common to all models (section 2), and thereafter present the models: log-Gibbs (section 3), gamma-Pareto (section 4), and beta-prime (section 5). We then ‘close the circle’by revisiting to the log-Gibbs model (section 6), and conclude (section 7). Two notes about notation. Throughout the paper: (i) in the context of real 2

valued random variables, E [ ] denotes the operation of mathematical expectation; (ii) in the context of probability density functions, c denotes normalization constants.

2

Foundation

As noted in the introduction, the bell curve [2] is the iconic depiction of the popular perception of statistical distributions (Figure 1a). The distinctive point of the bell curve is its peak, and this peak point manifests both the mode and the mean. Namely, the mode is the value with the highest likelihood of occurrence, and the mean is the average value. In the models to be presented hereinafter the “Average is Over” phenomenon will be characterized by the vanishing of the mode on the one hand, and by the divergence of the mean on the other hand. The models share a common setting which we describe and discuss in this section.

2.1

Setting

The quantity of interest X is a non-negative valued random variable, whose statistical distribution is modeled by a probability density function ' (x) (0 x < 1). The density ' (x) is obtained via the principle of entropy maximization [6]-[8], and it is governed by a pair of positive parameters, and . The parameter determines the vanishing of the mode xmode of the density ' (x), i.e. the point at which this density peaks: xmode = arg max0 x<1 ' (x). The parameter value = 1 is a tipping point at which a phase transition takes place: as the parameter crosses the critical value = 1 the mode xmode changes from positive to zero. Speci…cally: the mode is positive, 0 < xmode < 1, if and only if > 1; and the mode is zero, xmode = 0, if and only if 1. More speci…cally, the parameter determines the shape of the density ' (x) as follows:1 > 1 , Unimodal : in the range 0 x < xmode the density ' (x) increases monotonically from ' (0) = 0 to its peak value, and in the range xmode < x < 1 the density ' (x) decreases monotonically from its peak value to ' (1) = 0 (Figure 1b). = 1 , Monotone and bounded : in the range 0 = xmode x < 1 the density ' (x) decreases monotonically from its peak value 0 < ' (0) < 1 to ' (1) = 0 (Figure 1c). < 1 , Monotone and unbounded : in the range 0 = xmode x < 1 the density ' (x) decreases monotonically from its peak value ' (0) = 1 to ' (1) = 0 (Figure 1d). 1 In

the three shape scenarios we use the shorthand notation ' (1) = limx!1 ' (x).

3

The parameter determines the convergence / divergence Rof the mean xmean 1 of the density ' (x), i.e. the density’s average value: xmean = 0 x' (x) dx. The parameter value = 1 is a tipping point at which a phase transition takes place: as the parameter crosses the critical value = 1 the mean xmean changes from …nite to in…nite. Speci…cally: > 1 , Convergence: the mean is …nite, 0 < xmean < 1. 1 , Divergence: the mean is in…nite, xmean = 1. “Average is Over” when the mode vanishes, xmode = 0 , 1, and when the mean diverges, xmean = 1 , 1. As long as the mode is positive the density retains a qualitative bell-shape that manifests three tiers: a left-wing tail, a middle-class bulk, and a right-wing tail. As the mode vanishes the leftwing explodes and the middle-class disappears. As long as the mean is …nite the mass of the left-wing together with the mass of the middle-class balance the mass of the right-wing. As the mean diverges the right-wing tail becomes so ‘heavy’and ‘fat’that its mass can no longer be counter-balanced. In e¤ect, the parameters and are exponents. Indeed, the density ' (x) admits the following power-law tails:2 8 as x ! 0, < x 1 ' (x) (1) : 1 x as x ! 1.

The exponent governs the near-zero asymptotics (x ! 0), and thus the statistical behavior of small values. The exponent governs the near-in…nity asymptotics (x ! 1), and thus the statistical behavior of large values. Eq. (1) is in accord with the two aforementioned phase transitions: the mode phasetransition that takes place at the critical exponent value = 1, and the mean phase-transition that takes place at the critical exponent value = 1.

2.2

Discussion

The right-wing of Eq. (1) is intimately related to Zipf ’s law. To illustrate this law consider city sizes in a given country, and rank them in decreasing order: S1 being the size of the largest city, S2 being the size of the second largest city, S3 being the size of the third largest city, etc. Now, plot the log-sizes ln (Sr ) vs. the log-ranks ln (r) (r = 1; 2; 3; ). Remarkably, this log-log plot yields an approximately a¢ ne line with slope ' 1 [16]-[18]. Even more remarkably, such approximately a¢ ne log-log plots –with general negative slopes – are encountered across the sciences [19]-[21]. This empirical behavior is commonly referred to as “Zipf’s law”[22]-[23] –though it was …rst discovered in demography by Auerbach [16]-[18], in linguistics by Estoup [24]-[26], and in scienti…c productivity by Lotka [27]-[29]. 2 The sign denotes asymptotic equivalence: the ratio of the left-hand side to the righthand side converges to a positive constant (in the respective limits x ! 0 and x ! 1).

4

Now, consider a large ensemble of independent and identically distributed copies of the random variable X. When observed from a macroscopic perspective, the ensemble converges (in law) to a limiting random object [30]: a Pareto Poisson Process. The Pareto Poisson Process is a countable collection of points that are scattered randomly over the positive half-line, and whose scattering statistics are governed by a power-law intensity [30]. The points of the limiting Pareto Poisson Process display Zipf’s law with slope [31]. In particular, the tipping point = 1 corresponds to the slope 1 –which is empirically encountered in the context of city sizes [16]-[18], as well as in other self-organized complex systems [19]-[21]. The right-wing of Eq. (1) is also a hallmark of anomalous di¤ usion [32][41], as well as the hallmark of the “rich get richer” phenomenon explained by preferential attachment [42]-[50]. Anomalous di¤usion models usually set their focus on the right-wing tail, and preferential attachment models focus only on the right-wing tail. In this paper we extend the focus beyond the rightwing tail to the overall con…guration and shape. Indeed, sharing the common setting described above, the models to be presented hereinafter: (i) are based on the combination of entropy maximization and exponential growth; (ii) involve tipping points and phase transitions; (iii) address the global con…guration of the density functions under consideration; and (iv) facilitate a phase-shift of the con…guration from a three-tier bell-shape (Figure 1b) to a two-tier monotone and unbounded shape (Figure 1d).

3

Log-Gibbs model

The model of lognormal statistics plays a major role across the sciences [51]-[58]. The bedrock of the lognormal model is a quadratic structure. In this section we replace the quadratic structure by a general convex structure, and consequently ‘transcend’from the lognormal model to a log-Gibbs model. In turn, as we will demonstrate, this transcendence yields a Gibbsian explanation to the “Average is Over” phenomenon. The content of this section follows [59] and builds on it.

3.1

Gibbs densities

Consider a real quantity of interest Z. As noted in the introduction, practically any measurable quantity Z you can think of will exhibit some degree of randomness. Indeed, randomness is inherently present almost everywhere, and it ranges from the mild (e.g. the height and the weight of people) to the wild (e.g. the wealth and the number of twitter followers of people) [60]-[61]. Thus, we model the quantity Z as a real valued random variable whose statistical distribution is governed by a probability density function (z) ( 1 < z < 1). Often, the quantity Z is represented as ‡uctuating about some typical value z . The most straightforward way of measuring the ‡uctuations of the quantity Z about a typical value z is via the Mean Absolute Deviation (MAD): E[jZ z j]. The most common way of measuring the ‡uctuations of the quantity 5

2

Z about a typical value z is via the Mean Square Deviation (MSD): E[jZ z j ]. p The MAD and the MSD are both special cases of the Lp norm: E[jZ z j ]1=p , where p 1. The MAD and the MSD manifest two di¤erent underlying geometries; to visualize these geometries envision two junctions in Manhattan (above 14th street). The MAD is equivalent to the walking distance between the two junctions, the L1 norm. The root of the MSD is equivalent to the ‡ying distance between the two junctions, the L2 norm. The question regarding which measure of deviation to apply, the MAD or the MSD, is profound: Eddington advocated for the MAD [62], Fisher advocated for the MSD [63], and the debate is going on for over a century [64]-[66]. The MAD and the MSD also imply di¤erent ‘candidates’ for the typical value z . With respect to the MAD the best deterministic approximation of the quantity Z is its median: zmedian = arg min R z 1
1 exp c

1

G (z

z )

(2)

( 1 < z < 1). The parameter appearing in Eq. (2) is a positive ‘temperature’that is tuned to meet the known deviation magnitude E[G (Z z )]. The shape of the gauge function G (z) implies that the shape of the Gibbs density (z) is unimodal, and that the typical value is the mode: z = zmode . Speci…cally: in the range 1 < z < zmode the density (z) increases monotonically from ( 1) = 0 to its peak value, and in the range zmode < z < 1 the density (z) decreases monotonically from its peak value to (1) = 0.3 Moreover, if the gauge function G (z) is symmetric –as in the case of the MAD and the MSD, as well as in the case of the Lp norm – then the Gibbs density (z) is symmetric about its peak, and the typical value is also the median and the mean: z = zmedian = zmean . In particular, the max-entropy principle yields the Laplace density [68] for the MAD, the Gauss density [2] for the MSD, and the Subbotin density [69]-[72] for the Lp norm. Table 1 summarizes the details of these speci…c densities. In 3 We

used the shorthand notations

( 1) = limz!

6

1

(z) and

(1) = limz!1

(z).

p

Table 1 the deviation magnitude is set to be given by E[jZ z j ] = p , where is a positive constant manifesting the Lp norm. In the case of the MSD the resulting Gauss density is the famous bell curve [2]. In fact, any smooth gauge function G (z) yields a bell-shape Gibbs density. As noted in the introduction, a bell shape is the iconic depiction of the popular perception of statistical distributions: a dominant middle-class bulk of values that are centered around the typical value z , a thin tail of extreme-left values, and a thin tail of extreme-right values. This perception is so widespread that the Gauss bell-curve density is commonly referred to as the “normal distribution” [2].

Table 1

MAD

MSD

G (z) =

jzj

jzj

2

jzj

c=

2

p

2

cp

2

p

=

Density

2

Laplace

Gauss

Lp norm

p

p

Subbotin

Table 1: Gibbs examples. The table summarizes the details of the maxentropy Gibbs densities that emanate from the MAD (L1 norm), the MSD (L2 norm), and the general Lp norm (p 1). The deviation magnitude is p E[jZ z j ] = p , where is a positive constant manifesting the Lp norm. The rows of the table specify: (i) the gauge function G (z); (ii) the normalization constant c in terms of (with cp = 2p1=p (1 + 1=p)); (iii) the temperature parameter in terms of ; and (iv) the type of the resulting Gibbs density.

3.2

Log-Gibbs densities

In the real physical world perhaps the most ubiquitous rate of change is velocity, and: if we move with velocity v over a time period of duration t then our displacement is Z = vt. In a world governed by Moore’s law change is 7

quanti…ed by growth, and: if we grow at rate r over a time period of duration t then our overall growth is X = exp (rt). Evidently, the product rt is the counterpart of the product vt, and hence the mapping from the displacement Z to the growth factor X is given by exponentiation: X = exp (Z). Namely, the exponential mapping X = exp (Z) shifts us from the ‘linear-world’quantity Z to the ‘exponential-world’quantity X. As noted above, measurable quantities always exhibit some degree of variability. The exponential mapping X = exp (Z) implies that if the statistical distribution of the quantity Z is governed by the probability density function (z) ( 1 < z < 1) – then the statistical distribution of the quantity X is governed by the probability density function ' (x) = [ln (x)] =x (0 x < 1). In particular, if the quantity Z is governed by Gibbs density of Eq. (2) then the quantity X is governed by the following log-Gibbs density: ' (x) =

1

1 exp cx

h x i G ln s

(3)

(0 x < 1), where s = exp (z ). The various terms appearing on the righthand side of Eq. (3) –the normalization constant c, the temperature parameter , and the gauge function G (z) – are as in the Gibbs density of Eq. (2). The term s appearing on the right-hand side of Eq. (3) is a log-Gibbs scale parameter. To analyze the log-Gibbs density ' (x) we denote by F (z) = G0 (z) ( 1 < z < 1) the derivative of the gauge function G (z). From a Langevin perspective the gauge function G (z) quanti…es an underlying potential, and consequently the derivative F (z) quanti…es an underlying force [59]. We assume that the force F (z) is a continuous function. The shape of the potential G (z) implies that the force F (z) is a monotone increasing function that passes through the origin, F (0) = 0. Hence, the lower bound of the force is given by the negative limit F ( 1) = limz! 1 F (z), and the upper bound of the force is given by the positive limit F (1) = limz!1 F (z). If the force’s lower bound is in…nite, F ( 1) = 1, then the shape of the log-Gibbs density ' (x) is unimodal, and its mode is positive: 0 < xmode < 1. If the force’s upper bound is in…nite, F (1) = 1, then the mean of the logGibbs density ' (x) is …nite: 0 < xmean < 1. If the force is bounded, 1 < F ( 1) < F (1) < 1, then the setting of section 2 holds with exponents =

1

F ( 1) &

=

1

F (1) .

(4)

Hence, the exponents and are determined by the interplay between the temperature parameter and the force’s bounds. The potentials corresponding to bounded forces are asymptotically a¢ ne (in the limits z ! 1). In the context of convex potentials a¢ ne asymptotes of the potentials mark the “edge of convexity”. In terms of the force’s lower bound F ( 1) and the temperature the mode

8

of the log-Gibbs density is given by 8 < s exp F 1 ( xmode = : 0

)

if F ( 1) <

,

if F ( 1)

,

(5)

where s = exp (z ) is the log-Gibbs scale parameter. In contrast to the Gibbs mode zmode = z , the log-Gibbs mode xmode depends on the temperature parameter . Speci…cally, the log-Gibbs mode is monotone decreasing with respect to the temperature parameter: the larger the temperature , the smaller the log-Gibbs mode xmode . Eq. (5) implies that the relationship between the modes xmode and zmode is given by: xmode < exp (zmode ). The monotonicity of the exponential mapping X = exp (Z) implies that the relationship between the medians xmedian and zmedian is given by: xmedian = exp (zmedian ). The exponential mapping X = exp (Z), combined together with Jensen’s inequality [73], implies that the relationship between the means xmean and zmean is given by: xmean > exp (zmean ). Eq. (5) further implies that the following ratio is a measure of the skewness of the log-Gibbs density: 8 if F ( 1) < , < exp F 1 ( ) s xmode =1 (6) Rmode = : s 0 if F ( 1) . As s = exp (zmode ), the ratio Rmode quanti…es the rift between the modes xmode and zmode . This ratio takes values in the unit interval, 0 Rmode 1, and is monotone increasing with respect to the temperature parameter: the larger the temperature , the larger the skewness of the log-Gibbs density. The lower bound Rmode = 0 is attained at the zero-temperature limit ! 0 – which characterizes no randomness: Pr (Z = z ) = Pr (X = s) = 1. The upper bound Rmode = 1 is attained at the temperature limit ! F ( 1) – at which the log-Gibbs mode vanishes: xmode ! 0.

3.3

Discussion

We began this section with the MAD, the MSD, and the Lp norm – whose underlying potentials are symmetric. Let us close this section by reviewing the log-Gibbs model in the case of symmetric potentials, G ( z) = G (z). Note that symmetric potentials induce anti-symmetric forces, F ( z) = F (z). Hence F ( 1) = F (1), and when the forces are bounded then the exponents of Eq. (4) coincide: = . Entropy maximization led us to the Gibbs density (z) of Eq. (2), whose shape is symmetric and unimodal (Figure 1a). Also, for the Gibbs density (z) we have zmode = zmedian = zmean , (7) i.e.: the typical value z is the Gibbs mode zmode , as well as the Gibbs median zmedian and the Gibbs mean zmean . 9

Exponentiation further led us to the log-Gibbs density ' (x) of Eq. (3), whose shape is skewed and is either unimodal (Figure 1b) or monotone (Figures 1c and 1d). Also, for the log-Gibbs density ' (x) we have xmode < xmedian < xmean ,

(8)

i.e.: the log-Gibbs scale s = exp (z ) is the log-Gibbs median xmedian , and it separates apart the log-Gibbs mode xmode from the log-Gibbs mean xmean . Exponentiation manifests a shift from the ‘linear-world’ quantity Z to the ‘exponential-world’quantity X, and it results in two major rami…cations. The …rst rami…cation is the breaking of symmetry: a transition from the symmetric Gibbs density (z) to the skewed log-Gibbs density ' (x), and thus a transition from Eq. (7) to Eq. (8). The second rami…cation is the possible breaking of unimodality: a transition from the unimodal Gibbs density (z) to a monotone log-Gibbs density ' (x), depending on the interplay between the temperature parameter and the boundness of the underlying force F (z). As long as the underlying force F (z) is unbounded the log-Gibbs density ' (x) is unimodal (Figure 1b), and its mean xmean is …nite. Hence, the logGibbs density ' (x) exhibits a skewed bell-shape that admits the following socioeconomic interpretation: a dominant middle-class bulk centered around the log-Gibbs mode xmode , a poor-class tail, and a rich-class tail. Qualitatively, the unimodal log-Gibbs density ' (x) is identical to the bell curve, and hence it manifests a ‘normal’type of a statistical distribution. Matters become more intricate when the underlying force F (z) is bounded – in which case the temperature parameter plays a crucial role. For sub-critical temperatures, < F (1), the unimodal shape (Figure 1b) and the …nite mean xmean hold intact. At the critical temperature, = F (1), the shape of the log-Gibbs density ' (x) is monotone and bounded (Figure 1c), and the mean xmean is in…nite. For super-critical temperatures, > F (1), the shape of the log-Gibbs density ' (x) is monotone and unbounded (Figure 1d), and the mean xmean is in…nite. Hence, the critical temperature = F (1) is a tipping point at which a dramatic phase transition takes place. The transition from sub-critical to super-critical temperatures yields the following socioeconomic interpretation: the middle-class is whipped out, the poor-class grows vast, and the rich become super rich. Once the tipping point = F (1) is crossed, the statistical distribution governed by the log-Gibbs density ' (x) shifts from a ‘normal’type to an ‘anomalous’type. Above the tipping point = F (1) only extremes exist: you can be either poor (with high likelihood), or super-rich (with low likelihood); you cannot be ‘average’, as there is no middle-class. Hence, above the tipping point “Average is Over” indeed. Eq. (8) implies that the following ratio is a measure of the skewness of the log-Gibbs density: Rmean =

xmean xmode =1 xmean

10

xmode . xmean

(9)

The ratio Rmean quanti…es the rift between the log-Gibbs mode xmode and the log-Gibbs mean xmean . Similarly to the ratio of Eq. (6), the ratio of Eq. (9) takes values in the unit interval, 0 Rmean 1. The lower bound Rmean = 0 is attained when the mean and the mode coincide –xmean = xmode –which occurs at the zero-temperature limit ! 0. The upper bound Rmean = 1 is attained when the mode vanishes and the mean diverges –xmode = 0 and xmean = 1 – which occurs at the temperature limit ! F (1). Eqs. (7)-(9) provide the following perspective to the “Average is Over” phenomenon. We begin with the mode-median-mean coincidence of Eq. (7). Shifting from the ‘linear-world’quantity Z to the ‘exponential-world’quantity X, we arrive at the mode-median-mean separation of Eq. (8). Then, increasing the temperature , we increase the mode-median-mean rift. The temperature limit ! F (1) yields the ultimate mode-median-mean rift, 0 = xmode < xmedian < xmean = 1, at which “Average is Over”. For unbounded forces the ultimate rift is attained at in…nite temperatures, whereas for bounded forces the ultimate rift is attained at …nite temperatures. Consider now the log-Gibbs counterparts of the Gibbs examples of Table 1: Laplace, Gauss and Subbotin. The force underlying the Laplace example is discontinuous; nonetheless, the setting of section 2 applies to the Laplace example with the following caveat: at its critical exponent value = 1 = the log-Laplace density is ‡at along the range 0 x s, and hence it has no single mode. The forces underlying the Gauss example and the Subbotin example (for p > 1) are unbounded, and hence the corresponding log-Gauss and log-Subbotin densities are unimodal and possess …nite means. Table 2 summarizes the details of these speci…c log-Gibbs densities. Yet another counterpart of a symmetric Gibbs example –a generalization of the log-logistic density [74] –is detailed in the Appendix. We noted above that the Gauss distribution is commonly referred to as the “normal distribution” [2]. In turn, the corresponding log-Gauss distribution is commonly referred to as the “lognormal distribution”[51]-[58]. As stated in the opening of this section the log-Gibbs model is a generalization of the lognormal 2 model: the quadratic potential jzj underlying the lognormal model is replaced by a general convex potential G (z).

11

Table 2

MAD

MSD

Lp norm

Density

Log-Laplace

Log-Gauss

Log-Subbotin

xmode =

s

s exp

xmean =

1

1

s 2

s exp

1 2

s exp

1 4

p

1 p

1

——

1

Rmode =

Rmean =

0

2

1

exp

1 2

1

1

exp

3 4

——

exp

1 p

p

1

Table 2: Log-Gibbs examples. The table summarizes the details of the logGibbs densities that emanate from the MAD (L1 norm), the MSD (L2 norm), and the general Lp norm (p 1). The rows of the table specify: (i) the type of the resulting log-Gibbs density (the details regarding the log-Laplace example are for sub-critical temperatures < 1); (ii) the log-Gibbs mode xmode ; (iii) the log-Gibbs mean xmean (in general there is no closed-form formula for the logSubbotin mean); (iv) the skewness ratio Rmode of Eq. (6); and (v) the skewness ratio Rmean of Eq. (9).

4

Gamma-Pareto model

The measurement of socioeconomic inequality is of major importance in economics and the social sciences [75]-[78], as well as in econophysics and sociophysics [79]-[82]. The gauges that are used in order to measure and quantify socioeconomic inequality are termed inequality indices [83]-[86]. In this section we introduce a gamma-Pareto model that provides an inequality explanation to the “Average is Over” phenomenon. The gamma-Pareto model is based on a pair of inequality indices: the poverty index and the riches index [87]-[88]. An inequality index I of a given human society takes values in the unit interval, 0 I 1, and is required to satisfy several basic properties [89]-[90]. 12

The lower bound I = 0 characterizes the scenario of perfect equality: a purely communist society in which wealth is equally distributed among all the society members. The greater the value of the inequality index I –the less egalitarian the human society, and the larger the gap between the poor and the rich. The most popular inequality indices are the Gini index [91]-[96] and the Pietra index [97]-[102], for which the upper bound I = 1 characterizes the scenario of perfect inequality: a society in which 100% of the wealth is held by 0% of the population. In the context of the log-Gibbs model we already encountered a couple of inequality indices: (i) the skewness ratio Rmode of Eq. (6), for which the upper bound I = 1 characterizes the zero-mode scenario, xmode = 0; and (i) the skewness ratio Rmean of Eq. (9), for which the upper bound I = 1 characterizes the scenario of a zero mode-to-mean ratio, xmode =xmean = 0.

4.1

Poverty index

Consider a given human society, and let X denote the wealth of a population member that is sampled at random. It is implicitly assumed that the random variable X is non-negative valued, and that it possesses a positive mean. In terms of the random variable X the society’s poverty index is given by [87]-[88]: IP overty = 1

exp fE [ln (X)]g . E [X]

(10)

The poverty index of Eq. (10) emanates from entropy and from relativeentropy [87], it is related to the Atkinson index [88],[103], and it displays the following sensitivity to extreme poverty: if a positive fraction of the population has zero wealth then the inequality upper bound IP overty = 1 is met. We note that the aforementioned Gini and Pietra indices do not display such sensitivity to extreme poverty. Evidently, the poverty index is based on the …rst moment E [X] and on the logarithmic moment E [ln (X)] of the random variable X. The …rst moment E [X] measures the mean linear distance of the random variable X from the origin 0, i.e.: the mean number of walking steps from the level 0 to the level X. The logarithmic moment E [ln (X)] measures the mean exponential distance of the random variable X from the unit value 1, i.e.: the mean number of doubling steps from the level 1 to the level X. Given these two moments, E [X] and E [ln (X)], the max-entropy principle implies the following gamma density function [105]: 1 x x 1 ' (x) = exp (11) s ( ) s s (0 x < 1), where s is a positive scale parameter, and where is a positive exponent. These two parameters are tuned to meet the two moments, E [X] and E [ln (X)] (see the Appendix for details). The gamma density ' (x) of Eq. (11) admits the three shape scenarios described in section 2; moreover, these shape scenarios are determined by the exponent as speci…ed in section 2. In the exponent range > 1 the gamma 13

mode is given by xmode = s ( 1); and in the exponent range 1 the gamma mode vanishes, xmode = 0. The gamma mean is given by xmean = s . Hence, for the gamma density ' (x) the skewness ratio of Eq. (9) is given by 8 if > 1, < 1= Rmean = (12) : 1 if 1.

4.2

Riches index

Consider a given human society, and employ two di¤erent sampling mechanisms. The ‘population mechanism’is as above: it samples at random a single population member, and it denotes by X the wealth of the randomly sampled member; also, as above, it is implicitly assumed that the random variable X is non-negative valued, and that it has a positive mean. The ‘wealth mechanism’samples at random a single dollar from the society’s overall wealth, and ~ the wealth of the population member to whom the randomlyit denotes by X ~ the society’s sampled dollar belongs. In terms of the random variables X and X riches index is given by [87]-[88]: IRiches = 1

E [X] . ~ expfE[ln(X)]g

(13)

The riches index of Eq. (13) emanates from entropy and from relativeentropy [87], it is related to the Theil index [88],[104], and it displays the following sensitivity to extreme riches: if a zero fraction of the population possesses a positive fraction of the society’s overall wealth then the inequality upper bound IRiches = 1 is met. We note that the aforementioned Gini and Pietra indices do not display such sensitivity to extreme riches. Evidently, the riches index is based on the …rst moment E [X] of the random ~ of the random variable variable X, and on the logarithmic moment E[ln(X)] ~ The …rst moment E [X] emanates from the ‘population mechanism’, and it X. measures the mean linear distance of the random variable X from the origin 0, i.e.: the mean number of walking steps from the level 0 to the level X. The ~ emanates from the ‘wealth mechanism’, and it logarithmic moment E[ln(X)] ~ from the measures the mean exponential distance of the random variable X unit value 1, i.e.: the mean number of doubling steps from the level 1 to the ~ As the two moments correspond to di¤erent sampling mechanisms we level X. address them separately. Given the …rst moment E [X], the max-entropy principle implies the following exponential density function [105] in the context of the ‘population mechanism’: 1 x ' (x) = exp (14) s s (0 x < 1), where s = E [X]. The exponential density ' (x) of Eq. (14) is a special case of the gamma density ' (x) of Eq. (11) – obtained by setting the gamma exponent to be one, = 1. 14

~ the max-entropy principle implies Given the logarithmic moment E[ln(X)], a power-law density function ' ~ (x). As power-laws are not integrable over the non-negative half-line (0 x < 1), bounds must be set. If we set s to be a positive upper bound then we obtain the following inverse-Pareto density [106] in the context of the ‘wealth mechanism’: ' ~ (x) =

s

x s

1

(15)

(0 x s), where is a positive exponent. If we set s to be a positive lower bound then we obtain the following Pareto density function [106] in the context of the ‘wealth mechanism’: ' ~ (x) =

s

x s

1

(16)

(s x < 1), where is a positive exponent. The bound s appearing in Eqs. (15) and (16) is a positive scale parameter. The exponents and appearing ~ (see in Eqs. (15) and (16) are tuned to meet the logarithmic moment E[ln(X)] the Appendix for details). The Pareto density ' ~ (x) of Eq. (16) is monotone decreasing and bounded, and it admits the two mean scenarios described in section 2; moreover, these mean scenarios are determined by the exponent as speci…ed in section 2. In the exponent range > 1 the Pareto mean is given by xmean = s = ( 1); in the exponent range 1 the Pareto mean diverges, xmean = 1. The Pareto mode is the lower bound, xmode = s. Hence, for the Pareto density ' ~ (x) of Eq. (16) the skewness ratio of Eq. (9) is given by 8 if > 1, < 1= Rmean = (17) : 1 if 1. The density ' ~ (x) of Eq. (16) was discovered by the Italian economist Vilfredo Pareto, in his seminal work on the distribution of wealth and income in human societies [107].

4.3

Discussion

So, the poverty index led us to the gamma density of Eq. (11), and the riches index led us to three densities: the exponential density of Eq. (14), the inversePareto density of Eq. (15), and the Pareto density of Eq. (16). Can we somehow ‘reconcile’these four densities? The answer, as we shall now argue, is a¢ rmative indeed. Firstly, the gamma density of Eq. (11) and the exponential density of Eq. (14) share the same sampling mechanism, and, as noted above, the exponential density is a special case of the gamma density. Secondly, the connection between the density of the ‘population mechanism’ ' (x) of the random variable X, and the corresponding density of the ‘wealth mechanism’ ' ~ (x) of the random 15

~ is given by [87]-[88]: ' variable X, ~ (x) = x' (x) =E [X] (0 x < 1). This connection implies that the density ' (x) is gamma / inverse-Pareto / Pareto if and only if the density ' ~ (x) is gamma / inverse-Pareto / Pareto, respectively. Thirdly, near zero the gamma density is asymptotically inverse-Pareto (i.e. as x ! 0). Due to the three aforementioned observations, ‘reconciling’ the four densities boils down to intertwining the gamma and the Pareto densities. As the Pareto density has a ‘built in’ lower bound, intertwining the two densities is straightforward: gamma up to a threshold level, and Pareto above the threshold level. Requiring the intertwined density to be smooth, i.e. to have a continuous and di¤erentiable ‘stitching’at the threshold level, yields the following gamma-Pareto density: 8 > > < exp ( + )

1 ' (x) = h cs > > :

x ( + )s

i

x s 1

h

x ( + )s

i

1

if

x s

if

x s

+ (18) >

+

(0 x < 1), where and are positive exponents (see the Appendix for details). The gamma-Pareto density ' (x) of Eq. (18) satis…es the setting of section 2. This density has the following socioeconomic interpretation: a dominant bulk governed by gamma statistics, and a rich-class tail governed by Pareto statistics. For exponents > 1 the bulk comprises of a poor class and of a middle class; for exponents < 1 the bulk is a vast poor class. The skewness ratio of Eq. (12) can be used for the ‘gamma side’ of gamma-Pareto density, and the skewness ratio of Eq. (17) can be used for the ‘Pareto side’of gamma-Pareto density. At the critical exponent value = 1 the gamma-Pareto density reduces to the following Exponential-Pareto density: 8 x if xs 1 + > exp (1 + ) s 1 < ' (x) = (19) h i 1 cs > x : if xs > 1 + (1+ )s (0 x < 1; see the Appendix for details). There is substantial empirical evidence for the gamma-Pareto statistics in general, and for the ExponentialPareto statistics in particular [79]-[81],[108]-[111] (see also references in [110]).

5

Beta models

One of the illustrative examples of the bell curve, which we noted in the introduction, was the distributions of test scores. On the one hand, test scores take values in bounded ranges; on the other hand, the bell curve is supported over the entire real line. Hence, in the context of test scores –and, in general, in the context of quantities that take values in bounded ranges – the bell curve is a 16

debatable approximation. In this section we address the case of quantities that take values in a bounded range: a segment of the real line with a left bound l and with a right bound r (where 1 < l < r < 1). This approach will lead us to a beta model, and thereafter to a beta-prime model.

5.1

Trial and error

Consider a quantity Z that is con…ned to the real bounded range: l Z r. The mean linear distance of the random variable Z from the left bound l is E [Z l] = E [Z] l, and the mean linear distance of the random variable Z from the right bound r is E [r Z] = r E [Z]. Hence, in e¤ect, both these mean linear distances are determined by the …rst moment E [X]. Given the …rst moment E [X], the max-entropy principle implies the following exponential density function: (z) = 1c exp ( z) (l z u), where is a real exponent that is tuned to meet the …rst moment E [X]. The shape of this exponential density is: monotone decreasing if < 0, ‡at if = 0, and monotone increasing if > 0. Similarly to the transition from the Gibbs density to the log-Gibbs density, we apply the exponential mapping X = exp (Z), and thus shift from the ‘linearworld’quantity Z to the ‘exponential-world’quantity X. The quantity X is also con…ned to a bounded range: exp (l) X exp (r). Moreover, the statistical distribution of the quantity X is governed by the following power-law density function: ' (x) = 1c x 1 (exp (l) x exp (r)), where is the real exponent. The shape of this power-law density is: monotone decreasing if < 1, ‡at if = 1, and monotone increasing if > 1. The two densities obtained in this subsection –the exponential density (z) and the power-law density ' (x) – admit simple shapes: either monotone or ‡at. Neither of these densities encompasses a unimodal bell shape, and hence neither of these densities can provide a phase transition a-lá the setting of section 2. Thus, an alternative approach is required, and such an approached will be presented indeed in the coming subsection.

5.2

Beta model

Consider a quantity X that is con…ned to the real bounded range: l X r. The linear distance of the random variable X from the left bound l is X l; the mean exponential length of this distance is given by the logarithmic moment E [ln (X l)]. The linear distance of the random variable X from the right bound r is r X; the mean exponential length of this distance is given by the logarithmic moment E [ln (r X)]. Namely, when setting of from the unit length 1, E [ln (X l)] is the mean number of doubling steps required to reach the length X l, and E [ln (r X)] is the mean number of doubling steps required to reach the length r X. With respect to the two logarithmic moments, E [ln (X l)] and E [ln (r X)], the max-entropy principle implies the following beta density function [105]: ' (x) =

1 (x c

l) 17

1

(r

x)

1

(20)

(l x r), where and are positive exponents that are tuned to meet the two logarithmic moments (see the Appendix for details). The beta density ' (x) of Eq. (20) is highly versatile, and it yields nine qualitatively di¤erent shapes – two of which are of particular importance. For exponents > 1 and > 1 the shape is unimodal : in the range l x < xmode the beta density ' (x) increases monotonically from ' (l) = 0 to its peak value, in the range xmode < x r the beta density ' (x) decreases monotonically from its peak value to ' (r) = 0, and xmode =

( (

1) r + ( 1) + (

1) l . 1)

(21)

This is a bell-shape scenario which admits the following sociopolitical interpretation: a dominant center, a thin left-wing tail, and a thin right-wing tail. In other words, this scenario describes a cohesive society with a ‘normal’distribution of political votes. For exponents < 1 and < 1 the shape is bipolar : in the range l x < xlow the beta density ' (x) decreases monotonically from ' (l) = 1 to its low value, in the range xlow < x r the beta density ' (x) increases monotonically from its low value to ' (r) = 1, and xlow given by the right-hand side of Eq. (21). This is an unbounded U-shape scenario which admits the following sociopolitical interpretation: a void center, a massive left-wing, and a massive right-wing. In other words, this scenario describes a polarized society with an ‘anomalous’distribution of political votes. The shape of the beta density ' (x) is symmetric when the exponents coincide, = , and is asymmetric and skewed otherwise. In the symmetric case the shape of the beta density ' (x) admits the three following shape scenarios. (I) = > 1 , unimodal : as described above, with xmode = xmedian = xmean = 21 (r + l). (II) = = 1 , ‡at: the density ' (x) is constant, and hence it manifests a uniform distribution over the bounded range l x r. (III) = < 1 , bipolar : as described above, with xlow = xmedian = xmean = 21 (r + l). The critical value = = 1 is a tipping point at which a dramatic phase transition takes place: the bell-shape turns into an unbounded U-shape, the center is whipped out, and the left and right wings explode.

5.3

Beta-prime model

In the previous subsection we used the left bound l and the right bound r, of the underlying bounded range, as ‘anchors’. Can a similar approach to the one taken in the previous subsection be applied also when the underlying range is the non-negative half-line? In such a case one ‘anchor’is the origin 0, the left bound of the negative half-line. Another ‘anchor’is an arbitrary positive scale s. The anchoring is performed as follows. Setting o¤ from a non-negative starting point p, and moving a linear distance X to the right, we arrive at the point p + X. The logarithmic moment E [ln (p + X)] measures the mean exponential distance of the point p + X from

18

the point 1, i.e.: the mean number of doubling steps from the point 1 to the point p + X. Thus, setting o¤ from the …rst anchor point, p = 0, yields the logarithmic moment E [ln (X)]; and, setting o¤ from the second anchor point, p = s, yields the logarithmic moment E [ln (s + X)]. With respect to the two logarithmic moments, E [ln (X)] and E [ln (s + X)], the max-entropy principle implies the following beta-prime density function [105]: 1 x 1 x 1+ (22) ' (x) = cs s s (0 x < 1), where and are positive exponents that are tuned to meet the two logarithmic moments (see the Appendix for details). The beta-prime density ' (x) of Eq. (22) satis…es the setting of section 2. In the exponent range > 1 the beta-prime mode is given by xmode = s( 1) = ( + 1); in the exponent range 1 the beta-prime mode vanishes, xmode = 0. In the exponent range > 1 the beta-prime mean is given by xmean = s = ( 1); in the exponent range 1 the beta-prime mean diverges, xmean = 1. Hence, for the beta-prime density ' (x) the skewness ratio of Eq. (9) is given by 8 1 1 if > 1 & > 1, < +1 Rmean = 1 (23) : 0 otherwise.

6

Revisiting the log-Gibbs model

In section 4 we constructed the gamma-Pareto model, and in section 5 we constructed the beta-prime model. Interestingly, both these models can be manifested via the log-Gibbs model of section 3. In this section we ‘close the circle’ by revisit the log-Gibbs model. Representing the gamma-Pareto density of Eq. (18) in terms of the log-Gibbs density of Eq. (3) yields the following underlying force: 8 if z ln ( + ) < exp (z) 1 F (z) = (24) : if z > ln ( + ) ( 1 < z < 1). Also, representing the beta-prime density of Eq. (20) in terms of the log-Gibbs density of Eq. (3) yields the following underlying sigmoid force: 1

+ 1 + exp (z)

F (z) =

(25)

( 1 < z < 1). The gamma-Pareto force of Eq. (24) and the beta-prime force of Eq. (25) admit a common shape: 1 F (z) is bounded and monotone increasing from the negative lower bound = 1 F ( 1) to the positive upper bound = 1 F (1). 19

Note that this is also the shape of the force that yields the setting of section 2 in the context of the log-Gibbs model. In section 3 we considered the underlying force to pass through the origin, F (0) = 0. Let us further consider the force to be di¤erentiable at the origin and set = 1 F 0 (0), where is a positive parameter. Also, assume that the aforementioned common shape holds. Then, the force admits the following piecewise-a¢ ne approximation [112]-[113]: 8 if z < > > > > > < 1 z if z (26) F (z) = > > > > > : if z >

( 1 < z < 1). In turn, the approximate force of Eq. (26) yields the following composite approximation of the log-Gibbs density of Eq. (3): i 8 h x > if xs < exp > s > exp 2 > > > > n o 1 < x 2 x exp if exp exp ' (x) = (27) 2 ln s s cx > > > > > h i > > : exp s if xs > exp 2 x (0

x < 1; see the Appendix for details). The piecewise-a¢ ne force of Eq. (26) is a continuous function, and it is determined by three positive parameters: its lower bound , its upper bound , and its slope at the origin . The composite log-Gibbs density of Eq. (27) is a continuous and smooth function, and it is determined by the three parameters f ; ; g and by the positive scale parameter s. The composite log-Gibbs model of Eq. (27) comprises of three parts: (i) an inverse-Pareto left part –the canonical power-law statistics for small values [114]-[115]; (ii) a lognormal middle part –the canonical ‘middle-class’statistics [51]-[58]; (iii) a Pareto right part – the canonical power-law statistics for large values [115]-[116]. In the case of a ‘normal’ statistical behavior the lognormal part dominates, and in the case of an ‘anomalous’statistical behavior the Pareto extremes dominate. The composite log-Gibbs model o¤ers a universal approximation to any smooth log-Gibbs density that exhibits the setting of section 2. Consequently, this composite model is widely observed empirically, as well as widely applied [79]-[81],[117]-[124]. As our ‘base camp’was the setting of section 2, and as the starting point of section 3 was lognormal statistics – the composite log-Gibbs model of Eq. (27) ‘closes the circle’indeed.

20

7

Conclusion

In this paper we presented three models for the “Average is Over”phenomenon, which was proclaimed by Thomas Friedman. The models share a common setting, and are founded on two common pillars: entropy maximization and exponential growth. The common setting is a probability density function, de…ned over the non-negative half-line, that is determined by two parameters. The parameter is an exponent that governs the density’s small values, and as it crosses the critical tipping point = 1 the density’s mode vanishes. The parameter is an exponent that governs the density’s large values, and as it crosses the critical tipping point = 1 the density’s mean diverges. When the mode and the mean are both positive then the density has a unimodal shape which manifests a ‘normal’statistical behavior: a tail of small values, a dominant bulk of ‘middle-class’values, and a tail of large values. When the mode is zero and the mean is in…nite then the density has an unbounded monotone shape which manifests an ‘anomalous’statistical behavior: an explosion of small values on the left hand, super large values on the right hand, and an in…nite rift between the small and the super large. In the transition from the ‘normal’behavior to the ‘anomalous’behavior the ‘middle-class’is whipped out and “Average is Over” indeed. The …rst model is Gibbsian, and the above setting emerges when shifting from the max-entropy Gibbs density to the corresponding log-Gibbs density. The second model is based on two measures of socioeconomic inequality, the poverty index and the riches index, and the above setting emerges by interlacing together four di¤erent max-entropy densities that are induces by the two inequality indices –thus yielding the gamma-Pareto density. The third model is based on measuring exponential distances from two ‘anchors’, the origin and an arbitrary scale, and the above setting emerges from the induced max-entropy density –the beta-prime density. On route to the beta-prime model we also derived, in the context of bounded ranges, a beta model. This model is based on measuring exponential distances from the ranges’ bounds, the left bound and the right bound, and the maxentropy density it induces is the beta density. The beta model is a boundedrange counterpart of the above setting, and it provides a rich assortment of shapes. In particular, the beta model facilitates a phase transition from a ‘normal’unimodal shape to an ‘anomalous’bipolar shape. We showed that the gamma-Pareto model and the beta-prime model can be represented in terms of the log-Gibbs model. Also, we saw that all smooth log-Gibbs models admit a universal composite approximation that comprises of three parts: an inverse-Pareto left part, a lognormal middle part, and a Pareto right part. This paper o¤ers researchers a cohesive framework for modeling the “Average is Over” phenomenon. On the one hand, the framework provides several general theoretical insights: geometric, probabilistic, socioeconomic, and statistical-physical. On the other hand the framework proposes several speci…c concrete models: log-Gibbs, gamma-Pareto, beta-prime, and the composite ap21

proximation. The theoretical insights illuminate the application of the concrete models.

8

Appendix

In the context of the log-Gibbs model of section 3 consider, as an additional example, the following potential: G (z) = ln [cosh (pz)] ( 1 < z < 1), where p is a positive parameter. This potential is symmetric, and its force is the sigmoid function 2 (28) F (z) = p tanh (pz) = p 1 1 + exp (2pz) 1=

( 1 < z < 1). The resulting Gibbs density is (z) = (2p=c ) [cosh (pz)] 2 ( 1 < z < 1), with c = (1=2 ) = (1= ); the temperature = 21 yields the logistic density [74]. In turn, the resulting log-Gibbs density is ' (x) =

2p h x c x s

p

+

p i 1=

s x

(0 x < 1). The corresponding exponents are sponding skewness ratio of Eq. (6) is given by: p p+

Rmode = 1

=

(29) = p= , and the corre-

1=2p

(30)

for sub-critical temperatures < p, and Rmode = 0 for critical and super-critical temperatures p. The log-Gibbs density of Eq. (29) is a generalization of the log-logistic density.

Details regarding probability density functions appearing along the paper: Gamma density of Eq. (11): the exponent is implicitly given by ln ( ) 0 ( ) E [ln (X)], and the scale parameter is given by s = ( ) = ln (E [X]) 1 E [X]. Inverse-Pareto density of Eq. (15): the exponent is given by Pareto density of Eq. (16): the exponent is given by

= 1= fln (s)

= 1= fE [ln (X)]

ln (s)g.

Gamma-Paretoh density of Eq. (18): the normalization i constant is given + ) R + by c = ( + ) 1 + exp( exp ( u) u 1 du . ( + ) 0

Exponential-Pareto density of Eq. (19): the normalization constant is given by c = 1 + exp (1 + ).

22

E [ln (X)]g.

Beta density of Eq. (20): the exponents are implicitly given by 0

( + ) ( + )

ln (r

= E [ln (X

l)]

ln (r

l) and

0

( ) ( )

0

( + ) ( + )

l), and the normalization constant is given by c =

0

( ) ( )

= E [ln (r

X)]

( ) ( ) ( + )

l)

(r

+

Beta-prime density of Eq. (22): the exponents are implicitly given by 0 0 0 0 ( ) ( ) ( ) ln (s) and (( ++ )) ( ) ( ) = E [ln (X)] ( ) = E [ln (s + X)] ln (s), and the normalization constant is given by c =

( ) ( ) ( + ) .

Composite log-Gibbs density q of Eq. constant is h (27): the normalization i 2 2 1 2 1 p p exp + + exp . 2 2

given by c =

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Figure Captions Figure 1: schematic illustrations of density shapes. (a) Bell-curve shape: a symmetric and unimodal density over the real line. (b) Skewed bell shape: an asymmetric and unimodal density over the positive half line. (c) Skewed monotone shape: a bounded density over the positive half line. (d) Skewed monotone shape: an ubounded density over the positive half line.

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Figure 1a

Figure 1b

Figure 1c

Figure 1d