A <conjugate prior> is a family of <prior distributions> whose members remain in that family after updating by the <likelihood function>. Here multiplying a shape-rate <gamma distribution> density by the <Poisson process> likelihood changes its power of $\lambda$ and its exponential rate, leaving a <gamma distribution>. The parameters change with the observations; <conjugate prior> does not mean that the <Bayesian posterior> equals the <prior distribution>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 35 / 1 / b / Solution

 Solution (source code)