Palindromic Bernoulli distributions

TL;DR

This paper introduces palindromic Bernoulli distributions, a subclass where joint probabilities are symmetric under reversal, and characterizes their properties and parameterizations across different models.

Contribution

It establishes the equivalence between the palindromic property and zero odd-order interactions, providing explicit transformations and MLE formulas.

Findings

Palindromic Bernoulli distributions have symmetric joint probabilities.

Zero odd-order interaction parameters characterize these distributions.

Explicit parametric transformations and maximum likelihood estimates are derived.

Abstract

We introduce and study a subclass of joint Bernoulli distributions which has the palindromic property. For such distributions the vector of joint probabilities is unchanged when the order of the elements is reversed. We prove for binary variables that the palindromic property is equivalent to zero constraints on all odd-order interaction parameters, be it in parameterizations which are log-linear, linear or multivariate logistic. In particular, we derive the one-to-one parametric transformations for these three types of model specifications and give simple closed forms of maximum likelihood estimates. Some special cases and a case study are described.