Gamma-Poisson Conjugacy

Question

Busses arrive at a certain bus stop according to a poisson process with rate $\lambda$ buses per hour, where $\lambda$ is unknown. The uncertainty about $\lambda$ is quantified using the prior $\lambda \sim Gamma(r_{0}, b_{0})$, where $r_{0}$ amd $b_{0}$ are known, positive constants with $r_{0}$ an integer.

Why is $\lambda$ distributed as a Gamma distribution, when a Gamma distribution represents the sum of inter-arrival times of the busses?

Answer 1

The true rate of the Poisson process is unknown, so in statistical practice, we can either:

1. Treat $\lambda$ as a fixed unknown parameter (as a frequentist would), collect a sample of arrival times, and use this sample to calculate a statistic $\hat \lambda$ with the assumption that it estimates the true value; or,
2. Treat $\lambda$ as a random variable (as a Bayesian would), impose a prior probability distribution on its value, collect a sample of arrival times, and use this sample to update their belief of the probability distribution of $\lambda$, obtaining a posterior distribution given the observed data.

The second scenario is what is being done in your question. It is not to say that the prior distribution is "accurate" or "faithful" to the state of reality, but rather, that this choice of prior exhibits certain nice properties that lends itself well to the calculation of the posterior distribution. That it is gamma distributed isn't a direct consequence of the fact that the sum of the interarrival times of the buses is also gamma distributed, or that they "ought" to be identically distributed.