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.

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(↵

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, …

This article presents an overview of the gamma and beta functions and their relation to a variety of integrals. We will touch on several other techniques along the way, as well as allude to some …

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).

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).

B(a; b) = xa 1(1 0 x)b 1 dx: Claim: The gamma and beta functions are related as ( a)( b) B(a; b) = : ( a + b)