Help-desk call rate with a Gamma prior

Question

You model the number of help-desk calls per day as a Poisson process with unknown average rate $\lambda$ (calls per day). Your prior is $\text{Gamma}(\alpha=4, \beta=1)$ , with shape $\alpha$ and rate $\beta$ . Over 6 days you record a total of 50 calls.

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

Accepted Answer

Intuition first. Counts per day fit a Poisson, and its conjugate partner is the Gamma, so the answer stays a Gamma. Read \beta as a number of pretend days already observed and \alpha as the pretend events they held. Updating is bookkeeping: add real events to the shape, add real days to the rate. The rule. 	ext{Gamma}(\alpha,\beta)\ +\ (	ext{total calls } 	extstyle\sum xi 	ext{ over } n 	ext{ days})\ ightarrow\ 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + n\Big). Plug in. Here \alpha=4, \beta=1, total calls \sum xi = 50, days n = 6: 	ext{posterior} = 	ext{Gamma}(4+50,\ 1+6) = 	ext{…

Help-desk call rate with a Gamma prior

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