Typos per page with a weak Gamma prior

Question

You model the number of typos per page in a manuscript as a Poisson process with unknown average rate $\lambda$ (typos per page). Your prior is $\text{Gamma}(\alpha=1, \beta=4)$ , with shape $\alpha$ and rate $\beta$ , a light belief centered at $0.25$ per page. A proofreader checks 10 pages and finds a total of 7 typos.

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

Accepted Answer

Intuition first. Typos per page fit a Poisson, and the Gamma is its conjugate partner, so the answer stays a Gamma. Read \beta = 4 as 4 pretend pages already checked, holding \alpha = 1 pretend typo, a soft opinion. Updating is bookkeeping: add real typos to the shape, add real pages to the rate. The rule. 	ext{Gamma}(\alpha,\beta)\ +\ (	ext{total typos } 	extstyle\sum xi 	ext{ over } n 	ext{ pages})\ ightarrow\ 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + n\Big). Plug in. Here \alpha=1, \beta=4, total typos \sum xi = 7, pages n = 10: 	ext{posterior} = 	ext{Gamma}(1+7,\ 4+10) = 	ex…

Typos per page with a weak Gamma prior

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