I’m interested in learning about the pattern of text message use for my college son. I pay the monthly cell phone bill and I want to be pretty sure that he won’t exceed his monthly allowance of 5000 messages.
We’ll put this problem in the context of a normal distribution inference problem. Suppose
We’ll talk about this problem in three steps.
- First I talk about some prior beliefs about
that we’ll model by independent conjugate priors.
- I’ll discuss the use of Gibbs sampling to simulate from the posterior distribution.
- Last, we’ll use the output of the Gibbs sampler to get a prediction interval for the sum of text messages in the next 17 days.
- First I assume that my prior beliefs about the mean
and standard deviation of the population of text messages are independent. This seems reasonable, especially since it is easier to think about each parameter separately.
- I’ll use conjugate priors to model beliefs about each parameter. I believe my son makes, on average, 40 messages per day but I could easily be off by 15. So I let
- It is harder to think about my beliefs about the standard deviation
of the text message population. After some thought, I decide that my prior mean and standard deviation of are 5 and 2, respectively. We’ll see shortly that it is convenient to model the precision by a distribution. It turns out that is a reasonable match to my prior information about .