Bayesian Data Analysis 2

In the last post (Bayesian Data Analysis 1), I ran a Bayesian data analysis using a simple, first-principles approach. Armed with only the fact that a Poisson distribution is appropriate for modeling airplane accidents, Bayes Rule, and R, we got the correct answer to the problem through non-parametric simulation.

Before we get into precision and the other topics in the list of issues to explore, let’s tie up one loose end. I could have gotten the right answer for the wrong reason, so let’s look at the posterior distribution of $\theta$ as compared to the provided answer to make sure we’re close. In the answer, Gelman looks at the parametrical form for $\theta$ and considers conjugate priors and analytically determines that the correct parametric posterior distribution is a Gamma function with $\alpha=238$ and $\beta=10$ . (More on this in the next posting.) So I simulated out 20,000 samples from that distribution:

rgam <- rgamma (20000, 238, 10)

And then we can compare our posterior distribution with the official, parametric distribution:

Looks reasonably close (though biased by about 0.25) and it didn’t take a lot of fancy machinery to pull off. So why not use this method as our default? Why talk of things like conjugate priors and hyper parameters? And did we just get lucky with numeric precision because we only had 10 accidents and hence only 10 applications of Bayes Rule? Let’s cover that in the next posting, and finish this one with a graph of our answer (not $\theta$ but the predictive distribution of accidents) with the 95% CI :