Letter to the Editor

Abstract

Hutson and Vexler (2018) demonstrate an example of aliasing with the beta and normal distribution. This letter presents another illustration of aliasing using the beta and normal distributions via an infinite mixture model, inspired by the problem of modeling placebo response.

Hutson and Vexler (2018) present interesting results showing how the 4-parameter beta-normal distribution can become “aliased” with a normal distribution meaning that under particular parameter settings, the beta-normal becomes almost indistinguishable from a normal distribution. It is curious to note that strong aliasing can also occur using infinite mixtures (or convolutions) of beta and normal distributions via a latent regression model

$$y={\beta }_0+{\beta }_{1}x+ϵ,$$ (1)

where x ~ beta(p,q) is unobserved (or latent) and independent of ϵ ~ N(0,σ²). It is straightforward to show that the Kullback-Liebler distance between y and a normal distribution is minimized when the normal distribution has mean and variance equal to that of y. In the limiting case when p and q go to zero at the same rate, x degenerates to a Bernoulli and y becomes a 2-component normal mixture. Model (1) is useful in modeling placebo response to medical treatment (Tarpey and Petkova, 2010) and in the field of psychiatric nosology when interest lies in determining if distinct disease states exist (finite mixture model) or if disease severity varies over a spectrum (x continuous). The distribution of y in (1) can become indistinguishable from a normal distribution (i.e., aliased) even when the latent x component explains a substantial degree of variability in y and has a bimodal or strongly skewed pdf.