The Elliptical Quartic Exponential Distribution An Annular Distribution Obtained via Maximum Entropy Christopher K I Williams

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The Elliptical Quartic Exponential Distribution: An

Annular Distribution Obtained via Maximum Entropy

Christopher K I Williams

School of Informatics, University of Edinburgh, UK

October 11, 2022

Abstract

This paper describes the Elliptical Quartic Exponential distribution in RD, obtained via

a maximum entropy construction by imposing second and fourth moment constraints. I

discuss relationships to related work, analytical expressions for the normalization constant

and the entropy, and the conditional and marginal distributions.

The maximum entropy construction allows the speciﬁcation of a probability distribution in

terms of constraints, see e.g., Cover and Thomas (1991, ch. 12). Consider a radially symmetric

zero-mean distribution in RDwith r2=xTx. Constraints are imposed on the variance E[r2] =

c2and on E[r4] = c4; the latter implies a constraint on the “variance of the variance”. The

intuition behind the construction here is that if this “variance of the variance” is small then the

distribution should be similar to an annulus at some radius R. We deﬁne an annular distribution

to be one where the distribution as a function of ris unimodal with the mode away from zero.

The maximum entropy construction gives

p(x) = 1

ZD(λ1, λ2)exp λ1xTx−λ2(xTx)2,(1)

where ZD(λ1, λ2)is the normalization constant in RD. We require λ2>0so that the distri-

bution is normalizable. λ1>0produces an annular distribution, while for λ1≤0the density

decays monotonically from r= 0.

Consider the exponential term in eq. 1 as a function of rfor λ1>0. By differentiation we

have at the maximum that 2λ1r−4λ2r3= 0. Let the value of rat which the maximum is

reached be denoted by R. Hence λ2=λ1/(2R2). By setting λ1=α/R2for α > 0, we have

p(x) = 1

ZD(α, R)exp αxTx

R2−(xTx)2

2R4.(2)

As αincreases the thickness of the ring decreases. A plot of p(x)in 2D with R= 1 and α= 8

is shown in Figure 1.

arXiv:2210.04221v1 [stat.ME] 9 Oct 2022

Figure 1: A plot of the annular distribution in 2D for R= 1 and α= 8.

The distribution can clearly be shifted to a non-zero mean µand xTxcan transformed

to xTΣ−1xfor some SPD matrix Σ. I term the distribution in eq. 1 under this transformation

the Elliptical Quartic Exponential distribution, by analogy with the Elliptical Gamma distribution

discussed below; both have elliptical contours.

Below I discuss related work, analytical expressions for the normalization constant and

the entropy, and the conditional and marginal distributions of the Elliptical Quartic Exponential

distribution.

1.1 Related work

Fisher (1922) discussed univariate probability densities having the form p(x)∝exp(−Qk(x)),

where Qk(x) = Pk

q=1 αqxq, with keven and αk>0. Matz (1978) discusses the case

with k= 4 known as the quartic exponential distribution, and the special case of p(x)∝

exp(−βx2−γx4)with γ > 0and βunrestricted in sign; this is termed the symmetric quartic

exponential distribution. The quartic exponential distribution can be obtained via maximum

entropy considerations given the ﬁrst four moments, see e.g., Zellner and Highﬁeld (1988).

In the multivariate case, Urzúa (1997) considers p(x)∝exp(−Q(x)). If Q(x)is a poly-

nomial of degree kin Ddimensions, it can be written as Q(x) = Pk

q=1 Q(q)(x), where each

Q(q)(x)is a homogeneous polynomial of degree q, i.e.

Q(q)(x) = Xα(q)

j1...jD

i=1

xji

i,(3)

with the summation taken over all non-negative integer D-tuples (j1, . . . , jD)such that j1+

. . . +jD=q. This is known as the multivariate quartic exponential distribution. The maximum

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TheEllipticalQuarticExponentialDistribution:AnAnnularDistributionObtainedviaMaximumEntropyChristopherKIWilliamsSchoolofInformatics,UniversityofEdinburgh,UKOctober11,2022AbstractThispaperdescribestheEllipticalQuarticExponentialdistributioninRD,obtainedviaamaximumentropyconstructionbyimposingsecondand...

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The Elliptical Quartic Exponential Distribution An Annular Distribution Obtained via Maximum Entropy Christopher K I Williams

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