Let’s imagine that we’re measuring the approval rating of an unpopular politician. Suppose we sample ten polls and get the values
How can we construct a posterior distribution for our belief in the politician’s mean approval rating?
Let’s assume that the polls are independent and identically distributed random variables, X_1, …, X_n. The central limit theorem tells us that the sample mean will asymptotically approach a normal distribution with variance σ²/n
where μ and σ² are the mean and variance of X_i.
Motivated by this asymptotic limit, let’s approximate the likelihood of observed data y with
Using the objective prior
(more on this later) and integrating out σ² gives us a t distribution for the posterior, π(µ|y)
where
Let’s look at the posterior distribution for the data in Table 1.