Website signup rate with a Gamma prior

Question

You model the number of website signups per hour as a Poisson process with unknown average rate $\lambda$ (signups per hour). Your prior is $\text{Gamma}(\alpha=10, \beta=5)$ , with shape $\alpha$ and rate $\beta$ , a fairly confident belief centered at 2 per hour. Over 8 hours you record a total of 30 signups.

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

Accepted Answer

Intuition first. Signups per hour fit a Poisson, and the Gamma is its conjugate partner, so the answer stays a Gamma. Here \beta = 5 acts like 5 pretend hours already watched, carrying \alpha = 10 pretend signups, a moderately strong opinion. Updating is bookkeeping: add real signups to the shape, add real hours to the rate. The rule. 	ext{Gamma}(\alpha,\beta)\ +\ (	ext{total signups } 	extstyle\sum xi 	ext{ over } n 	ext{ hours})\ ightarrow\ 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + n\Big). Plug in. Here \alpha=10, \beta=5, total signups \sum xi = 30, hours n = 8: 	ext{posterior}…

Website signup rate with a Gamma prior

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