BDA3 Chapter 3 Exercise 7

Here’s my solution to exercise 7, chapter 3, of Gelman’s Bayesian Data Analysis (BDA), 3rd edition. There are solutions to some of the exercises on the book’s webpage.

Suppose we observe b bikes and v other vehicles passing a section of road within an hour. We can model the counts as Poisson distributed

b∣θb∼Poisson(θb)v∣θv∼Poisson(θv)

or as binomial distributed

b∣n,p∼Binomial(n,p)

where n is the number of trials and p is the probability of observing a bike. Let

p:=θbθb+θv.

We are supposed to show that this definition of p gives the two models the same likelihood, but I’m stuck. At best I can show that the expectations are different

E(b∣θb)=θbE(b∣n,p)=np=nθbθb+θv

which suggests the conditioning should be done differently.