Chapter 3 Beta-Binomial Bayesian Model Notes

library(bayesrules)
library(tidyverse)

The chapter is set up with an example of polling results. We are put into the scenario where we are managig the campaing for a candidate. We know that on average her support based on recent polls is around 45%. In the next few sections we’ll work through our Bayesian framework and incorporate a new tool the Beta-Binomial model. This model will take develop a continuous prior, as opposed to the discrete one’s we’ve been working with so far.

The Beta prior

Probability Density Function

Let \pi be a continuous random variable with probability density function (pdf) f(\pi). Then f(\pi) has the following properties:

1. f(\pi) \geq 0
2. \int_{\pi}f(\pi)d\pi = 1 (this is analogous to \sum in the case of pmfs)
3. P(a < \pi < b) = \int_a^bf(\pi)d\pi when a\leq b

Tip

a quick note on (1) above. Note that it does not place a restriction on f(\pi) being less than 1. This means that we can’t interpret values of f as probabilities, we can however use to interpret plausability of two different events, the greater the value of f the more plausible. To calculate probabilities using f we must determine the area under the curve it defines, as shown in (3).