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Beta Binomial Conjugacy

Prior h(σ)Beta(α,β)h(\sigma) \sim \text{Beta}(\alpha, \beta)

Likelihood g(xθ)Binomial(n,x,θ)g(x|\theta) \sim \text{Binomial}(n, x, \theta)

Posterior

f(θx)=g(xθ)h(θ)θg(xθ)h(θ)dθf(\theta | x) = {{g(x | \theta) h(\theta)} \over {\int\limits_{\theta} g(x|\theta) h(\theta) d \theta}}

f(θx)Beta(α+x, β+nx)\therefore f(\theta | x) \sim \text{Beta}(\alpha + x,~\beta + n - x)