Index: The Book of Statistical ProofsStatistical Models ▷ Categorical data ▷ Binomial observations ▷ Posterior distribution

Theorem: Let $y$ be the number of successes resulting from $n$ independent trials with unknown success probability $p$, such that $y$ follows a binomial distribution:

\[\label{eq:Bin} y \sim \mathrm{Bin}(n,p) \; .\]

Moreover, assume a beta prior distribution over the model parameter $p$:

\[\label{eq:Bin-prior} \mathrm{p}(p) = \mathrm{Bet}(p; \alpha_0, \beta_0) \; .\]

Then, the posterior distribution is also a beta distribution

\[\label{eq:Bin-post} \mathrm{p}(p|y) = \mathrm{Bet}(p; \alpha_n, \beta_n) \; .\]

and the posterior hyperparameters are given by

\[\label{eq:Bin-post-par} \begin{split} \alpha_n &= \alpha_0 + y \\ \beta_n &= \beta_0 + (n-y) \; . \end{split}\]

Proof: With the probability mass function of the binomial distribution, the likelihood function implied by \eqref{eq:Bin} is given by

\[\label{eq:Bin-LF} \mathrm{p}(y|p) = {n \choose y} \, p^y \, (1-p)^{n-y} \; .\]

Combining the likelihood function \eqref{eq:Bin-LF} with the prior distribution \eqref{eq:Bin-prior}, the joint likelihood of the model is given by

\[\label{eq:Bin-JL} \begin{split} \mathrm{p}(y,p) &= \mathrm{p}(y|p) \, \mathrm{p}(p) \\ &= {n \choose y} \, p^y \, (1-p)^{n-y} \cdot frac{1}{B(\alpha_0,\beta_0)} \, p^{\alpha_0-1} \, (1-p)^{\beta_0-1} \\ &= \frac{1}{B(\alpha_0,\beta_0)} {n \choose y} \, p^{\alpha_0+y-1} \, (1-p)^{\beta_0+(n-y)-1} \; . \end{split}\]

Note that the posterior distribution is proportional to the joint likelihood:

\[\label{eq:Bin-post-s1} \mathrm{p}(p|y) \propto \mathrm{p}(y,p) \; .\]

Setting $\alpha_n = \alpha_0 + y$ and $\beta_n = \beta_0 + (n-y)$, the posterior distribution is therefore proportional to

\[\label{eq:Bin-post-s2} \mathrm{p}(p|y) \propto p^{\alpha_n-1} \, (1-p)^{\beta_n-1}\]

which, when normalized to one, results in the probability density function of the beta distribution:

\[\label{eq:Bin-post-qed} \mathrm{p}(p|y) = \frac{1}{B(\alpha_n,\beta_n)} \, p^{\alpha_n-1} \, (1-p)^{\beta_n-1} = \mathrm{Bet}(p; \alpha_n, \beta_n) \; .\]

Metadata: ID: P30 | shortcut: bin-post | author: JoramSoch | date: 2020-01-24, 00:20.