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

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.