added proof "mult-map"

Author: JoramSoch

P/mult-map.md

@@ -0,0 +1,80 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2023-12-08 15:14:47
+
+title: "Maximum-a-posteriori estimation for multinomial observations"
+chapter: "Statistical Models"
+section: "Count data"
+topic: "Multinomial observations"
+theorem: "Maximum-a-posteriori estimation"
+
+sources:
+
+proof_id: "P428"
+shortcut: "mult-map"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $y = [y_1, \ldots, y_k]$ be the number of observations in $k$ categories resulting from $n$ independent trials with unknown category probabilities $p = [p_1, \ldots, p_k]$, such that $y$ follows a [multinomial distribution](/D/mult):
+
+$$ \label{eq:Mult}
+y \sim \mathrm{Mult}(n,p) \; .
+$$
+
+Moreover, assume a [Dirichlet prior distribution](/P/mult-prior) over the model parameter $p$:
+
+$$ \label{eq:Mult-prior}
+\mathrm{p}(p) = \mathrm{Dir}(p; \alpha_0) \; .
+$$
+
+Then, the [maximum-a-posteriori estimates](/D/map) of $p$ are
+
+$$ \label{eq:Mult-MAP}
+\hat{p}_\mathrm{MAP} = \frac{\alpha_0+y-1}{\sum_{j=1}^k \alpha_{0j} + n - k} \; .
+$$
+
+
+**Proof:** Given the [prior distribution](/D/prior) in \eqref{eq:Mult-prior}, the [posterior distribution](/D/post) for [multinomial observations](/D/mult-data) [is also a Dirichlet distribution](/P/mult-post)
+
+$$ \label{eq:Mult-post}
+\mathrm{p}(p|y) = \mathrm{Dir}(p; \alpha_n)
+$$
+
+where the [posterior hyperparameters](/D/post) are equal to
+
+$$ \label{eq:Mult-post-par}
+\alpha_{nj} = \alpha_{0j} + y_j, \; j = 1,\ldots,k \; .
+$$
+
+The [mode of the Dirichlet distribution](/P/dir-mode) is given by:
+
+$$ \label{eq:Dir-mode}
+X \sim \mathrm{Dir}(\alpha) \quad \Rightarrow \quad \mathrm{mode}(X_i) = \frac{\alpha_i-1}{\sum_j \alpha_j - k} \; .
+$$
+
+Applying \eqref{eq:Dir-mode} to \eqref{eq:Mult-post} with \eqref{eq:Mult-post-par}, the [maximum-a-posteriori estimates](/D/map) of $p$ follow as
+
+$$ \label{eq:Mult-MAP-s1}
+\begin{split}
+\hat{p}_{i,\mathrm{MAP}} &= \frac{\alpha_{ni} - 1}{\sum_j \alpha_{nj} - k} \\
+&\overset{\eqref{eq:Mult-post-par}}{=} \frac{\alpha_{0i} + y_i - 1}{\sum_j (\alpha_{0j} + y_j) - k} \\
+&= \frac{\alpha_{0i} + y_i - 1}{\sum_j \alpha_{0j} + \sum_j y_j - k} \; .
+\end{split}
+$$
+
+Since $y_1 + \ldots + y_k = n$ [by definition](/D/mult-data), this becomes
+
+$$ \label{eq:Mult-MAP-s2}
+\hat{p}_{i,\mathrm{MAP}} = \frac{\alpha_{0i} + y_i - 1}{\sum_j \alpha_{0j} + n - k} \end{equation}
+
+which, using the $1 \times k$ [vectors](/D/mult-data) $y$, $p$ and $\alpha_0$, can be written as:
+
+\begin{equation} \label{eq:Mult-MAP-qed}
+\hat{p}_\mathrm{MAP} = \frac{\alpha_0+y-1}{\sum_{j=1}^k \alpha_{0j} + n - k} \; .
+$$