A Fresh Look at Bivariate Binomial Distributions

TL;DR

This paper introduces a new categorical formulation of dependent bivariate binomial distributions, exploring their properties, behavior under various operations, and how to fit mixtures using Expectation Maximisation.

Contribution

It provides a novel categorical framework for dependent bivariate binomial distributions and analyzes their properties and mixture modeling techniques.

Findings

Categorical formulation simplifies understanding of bivariate binomials.

Derived properties include mean, variance, covariance, and behavior under convolution.

Demonstrated EM algorithm for mixture recognition in data.

Abstract

Binomial distributions capture the probabilities of `heads' outcomes when a (biased) coin is tossed multiple times. The coin may be identified with a distribution on the two-element set {0,1}, where the 1 outcome corresponds to `head'. One can also toss two separate coins, with different biases, in parallel and record the outcomes. This paper investigates a slightly different `bivariate' binomial distribution, where the two coins are dependent (also called: entangled, or entwined): the two-coin is a distribution on the product {0,1} x {0,1}. This bivariate binomial exists in the literature, with complicated formulations. Here we use the language of category theory to give a new succint formulation. This paper investigates, also in categorically inspired form, basic properties of these bivariate distributions, including their mean, variance and covariance, and their behaviour under convolution and under updating, in Laplace's rule of succession. Furthermore, it is shown how Expectation Maximisation works for these bivariate binomials, so that mixtures of bivariate binomials can be recognised in data. This paper concentrates on the bivariate case, but the binomial distributions may be generalised to the multivariate case, with multiple dimensions, in a straightforward manner.