Bayesian server error rate with a Gamma prior

Question

You model the number of server errors per hour as a Poisson process with unknown average rate $\lambda$ (errors per hour). Before collecting data your belief about $\lambda$ is a $\text{Gamma}(\alpha=3, \beta=2)$ prior, where $\alpha$ is the shape and $\beta$ is the rate. You then watch the server for 5 hours and record a total of 14 errors.

Using Bayesian updating, what is the posterior mean estimate of the error rate $\lambda$ ?

Accepted Answer

Intuition first. For counts of events per time, the Poisson is the natural model and the Gamma is its conjugate partner, meaning the answer stays a Gamma. A Gamma prior acts like pretend past experience: think of \beta as a number of pretend hours already watched, carrying \alpha pretend events. Updating is bookkeeping: add the real events to the shape, add the real hours to the rate. The rule. 	ext{Gamma}(\alpha,\beta)\ +\ (	ext{total events } 	extstyle\sum xi 	ext{ over } n 	ext{ hours})\ ightarrow\ 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + n\Big). In words: new shape = old shape…

Bayesian server error rate with a Gamma prior

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