In this chapter we are going to have a very meta discussion about how we represent probabilities. Until now probabilities have just been numbers in the range 0 to 1.

he beta function. It is related to the gamma fu. 0 x 1: 1 ∫ (x) = ta 1(1 t)b 1dt; 0 x 1: B(a; b) 0 We will denote the beta distribution by Beta(a; b): It is often used for modeling random variables, …

Beta-Bernoulli model: posterior prediction (marginalization) ta D = {X1, 1}n, contains N1 ones and model M: Xi are generated i.i.d. from a Ber( ) distribution ) p( |D) Beta(↵

On a log-log scale, the pdf forms a straight line, of the form log p(x) = a log x + c for some constants a and c (power law, Zipf’s law).

We’ll start by introducing the beta distribution and using it as a conjugate prior with a binomial likelihood. After that we’ll look at other conjugate pairs.

The beta function (p; q) is the name used by Legen-dre and Whittaker and Watson(1990) for the beta integral (also called the Eulerian integral of the rst kind).

Relationship Between the Gamma and Beta Functions Recall that the gamma funciton is de ned, for