# Bayesian Data Analysis 1

I’ve read quite a few presentations on Bayesian data analysis, but I always seem to fall into the crack between the first, one-step problem (the usual being how likely you are to have a disease), and more advanced problems that involve conjugate priors and quite a few other concepts. So, when I was recently reading Bayesian Data Analysis[1], I decided to tackle Exercise 13 from Chapter 2 using only Bayes rule updates and simulation. I think it’s been illuminating, so decided to write it up here, using my favorite tool R.

Exercise 13 involves annual airline accidents from 1976 to 1985, modeled as a Poisson$(\theta)$ distribution. The data is :

accidents <- c(24, 25, 31, 31, 22, 21, 26, 20, 16, 22)

A little playing around with graphs and R‘s dpois gives me the impression that $\theta$ is probably around 24, but I’ll make three priors (for $\theta$):

r <- seq (10, 45, 0.2) theta1 <- dnorm (r, 15, 5) theta1 <- theta1 / sum (theta1) theta2 <- dnorm (r, 35, 6) theta2 <- theta2 / sum (theta2) theta3 <- 20 - abs (r - mean (r)) theta3 <- theta3 / sum (theta3)

Where theta1 is probably low, theta2 is probably high, and theta3 is not even parametric. (More on this later.) I normalized them to create proper priors so that they graph well together, but that wasn’t necessary. Remember, I’m doing all of my calculations over the discretized range, $r$. Plotting the theta’s together:

So let’s run the numbers (accidents) through a repeated set of Bayes Rule updates to get a posterior distribution for $\theta$ based on the prior distribution, theta1: