The Poisson likelihood is often used to model count data since Poisson random variables are integer-valued, starting at 0.

Example scenario where could we appropriately model with a Poisson likelihood?

Predicting the number of goals scored in a hockey match.

Each of the following gamma distributions is being considered as a prior for a Poisson mean \(\lambda\). All have the same mean of 4. Which one expresses the most confidence in this prior mean? Equivalently, which has the greatest effective prior sample size?

\(\Gamma(1,1/4)\)

\(\Gamma(2,1/2)\)

\(\Gamma(5,5/4)\)

\(\Gamma(20,5)\) Correct - Of the four choices, this prior has the smallest standard deviation (0.89). It also has the largest effective sample size (5).

Consider the chocolate chip cookie example from the lesson. As in the lesson, we use the Poisson likelihood to model the number of chips per cookie, and a conjugate gamma prior on \(\lambda\), the expected number of chips per cookie. Suppose our prior expection is \(\lambda=8\).

  1. The conjugate prior with mean 8 and effective sample size of 2 is \(\Gamma(a,2)\). Find the value of \(a\). Answer: mean = \(\alpha/\beta = 8,\space a/2 = 8,\space a=16\)

  2. The conjugate prior with mean 8 and standard deviation of 1 is \(\Gamma(a,8)\). Find the value of \(a\). Answer: stddev = \(\sqrt\alpha/\beta = 1,\space \sqrt a/8 = 1,\space \sqrt a = 8, \space a=64\)

  3. Suppose we are not very confident in our prior guess of 8, so we want to use an effective sample size of 1/100 cookies. Then the conjugate prior is \(\Gamma(a,0.01)\). Find the value of \(a\). Answer: mean = \(\alpha/\beta = 8,\space a/0.01 = 8,\space a=0.08\)

  4. Suppose we decide on a prior \(\Gamma(8,1)\), which has a prior mean of 8 and sample size of 1 cookie. We collect data, sampling 5 cookies and counting the chips in each. We find 9, 12, 10, 15 and 13 chips. What is the posterior distribution for \(\lambda\)? Answer: posterior is \(\Gamma(\alpha+\Sigma y_i, \beta+n) = \Gamma(59+8, 1+5) = \Gamma(67,6)\)

chips_count <- list(c(9,12,10,15,13))
sapply(chips_count,sum)
## [1] 59
  1. What do the plots of the prior density (dotted line) and posterior density (solid line) look like?
lambda=seq(from=0,to=20,by=.01)
plot(lambda,dgamma(lambda,67,6),type="l",ylab='f(lambda | y)')
lines(lambda,dgamma(lambda,8,1),type="l",lty=2)