Goals per match with a Gamma prior

Question

You model the number of goals a team scores per match as a Poisson process with unknown average rate $\lambda$ (goals per match). Your prior is $\text{Gamma}(\alpha=2, \beta=2)$ , with shape $\alpha$ and rate $\beta$ . Across 4 matches the team scores a total of 6 goals.

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

Accepted Answer

Intuition first. Goal counts per match fit a Poisson, and the Gamma is its conjugate partner, so the answer stays a Gamma. Treat \beta as pretend matches already seen and \alpha as the pretend goals in them. Updating is bookkeeping: add real goals to the shape, add real matches to the rate. The rule. 	ext{Gamma}(\alpha,\beta)\ +\ (	ext{total goals } 	extstyle\sum xi 	ext{ over } n 	ext{ matches})\ ightarrow\ 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + n\Big). Plug in. Here \alpha=2, \beta=2, total goals \sum xi = 6, matches n = 4: 	ext{posterior} = 	ext{Gamma}(2+6,\ 2+4) = 	ext{Ga…

Goals per match with a Gamma prior

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