added proof "bin-map"

Author: JoramSoch

P/bin-map.md

@@ -0,0 +1,72 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2023-12-01 14:36:54
+
+title: "Maximum-a-posteriori estimation for binomial observations"
+chapter: "Statistical Models"
+section: "Count data"
+topic: "Binomial observations"
+theorem: "Maximum-a-posteriori estimation"
+
+sources:
+
+proof_id: "P427"
+shortcut: "bin-map"
+username: "JoramSoch"
+---
+
+
+**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](/D/bin):
+
+$$ \label{eq:Bin}
+y \sim \mathrm{Bin}(n,p) \; .
+$$
+
+Moreover, assume a [beta prior distribution](/P/bin-prior) over the model parameter $p$:
+
+$$ \label{eq:Bin-prior}
+\mathrm{p}(p) = \mathrm{Bet}(p; \alpha_0, \beta_0) \; .
+$$
+
+Then, the [maximum-a-posteriori estimate](/D/map) of $p$ is
+
+$$ \label{eq:Bin-MLE}
+\hat{p}_\mathrm{MAP} = \frac{\alpha_0+y-1}{\alpha_0+\beta_0+n-2} \; .
+$$
+
+
+**Proof:** Given the [prior distribution](/D/prior) in \eqref{eq:Bin-prior}, the [posterior distribution](/D/post) for [binomial observations](/D/bin-data) [is also a beta distribution](/P/bin-post)
+
+$$ \label{eq:Bin-post}
+\mathrm{p}(p|y) = \mathrm{Bet}(p; \alpha_n, \beta_n)
+$$
+
+where the [posterior hyperparameters](/D/post) are equal to
+
+$$ \label{eq:Bin-post-par}
+\begin{split}
+\alpha_n &= \alpha_0 + y \\
+\beta_n &= \beta_0 + (n-y) \; .
+\end{split}
+$$
+
+The [mode of the beta distribution](/P/beta-mode) is given by:
+
+$$ \label{eq:beta-mode}
+X \sim \mathrm{Bet}(\alpha, \beta) \quad \Rightarrow \quad \mathrm{mode}(X) = \frac{\alpha-1}{\alpha+\beta-2} \; .
+$$
+
+Applying \eqref{eq:beta-mode} to \eqref{eq:Bin-post} with \eqref{eq:Bin-post-par}, the [maximum-a-posteriori estimate](/D/map) of $p$ follows as:
+
+$$ \label{eq:Bin-MAP}
+\begin{split}
+\hat{p}_\mathrm{MAP} &= \frac{\alpha_n-1}{\alpha_n+\beta_n-2} \\
+&\overset{\eqref{eq:Bin-post-par}}{=} \frac{\alpha_0+y-1}{\alpha_0+y+\beta_0+(n-y)-2} \\
+&= \frac{\alpha_0+y-1}{\alpha_0+\beta_0+n-2} \; .
+\end{split}
+$$