Bayesian updating for a Poisson rate, the Gamma conjugate

Question

You model the number of trades per day as $X_i \sim \text{Poisson}(\lambda)$ with unknown rate $\lambda$ , and place a Gamma prior $\lambda \sim \text{Gamma}(\alpha, \beta)$ (shape $\alpha$ , rate $\beta$ ).

Derive the posterior after observing $x_1,\dots,x_n$ , give its mean, and interpret the prior's weight.

Accepted Answer

Likelihood. For n i.i.d. Poisson observations, p(x{1:n}\mid\lambda) = \prod{i=1}^n \frac{e^{-\lambda}\lambda^{xi}}{xi!} \ \propto\ \lambda^{\sumi xi}\, e^{-n\lambda}. Prior. The Gamma(\alpha,\beta) density is p(\lambda) \propto \lambda^{\alpha-1} e^{-\beta\lambda}. Posterior. Multiply and collect powers of \lambda: p(\lambda \mid x{1:n}) \ \propto\ \lambda^{\alpha - 1}e^{-\beta\lambda}\cdot \lambda^{\sumi xi}e^{-n\lambda} = \lambda^{\,\alpha + \sumi xi - 1}\, e^{-(\beta + n)\lambda}. This is the kernel of a Gamma, so \boxed{\ \lambda \mid x{1:n} \ \sim\ 	ext{Gamma}\!\Big(\alpha + 	extstyle\…

Bayesian updating for a Poisson rate, the Gamma conjugate

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