A pair of sufficient statistics for a beta

Question

You have independent observations X1, \dots, Xn in (0, 1) from a Beta distribution with density proportional to x^{\alpha - 1}(1 - x)^{\beta - 1}, where both shapes \alpha and \beta are unknown. Find a pair of statistics that are jointly sufficient for (\alpha, \beta).

Accepted Answer

The plan. Factor the joint density into a parameter-piece times a data-only piece, and read off which data summaries the parameter-piece requires. Multiply the densities. Up to the normalizing constant (which depends only on \alpha, \beta), f(x; \alpha, \beta) \propto \prod{i=1}^n xi^{\alpha - 1}(1 - xi)^{\beta - 1} = \Big(\prodi xi\Big)^{\alpha - 1}\Big(\prodi (1 - xi)\Big)^{\beta - 1}. Turn each product into a sum of logs: f(x; \alpha, \beta) \propto \exp\!\Big((\alpha - 1)\sumi \log xi + (\beta - 1)\sumi \log(1 - xi)\Big). The parameter-piece touches the data through exactly two totals. So…

A pair of sufficient statistics for a beta

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