In one particular iteration of this course, we talked about reasons why grade distributions might be well suited to be described as a Beta distribution. Let’s say that we are given a set of student …

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, …

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.

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).

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

Plots of the Beta pdf for various values of α and β can help inform the prior specification (see R examples).

X evaluated at E[X]. We don’t have a formula for the CDF of a Beta distribution but all modern programming languages will have a Beta CDF function. In JavaScript we can execute: …