added definition "map"

Author: JoramSoch

D/map.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2023-12-01 14:32:38
+
+title: "Maximum-a-posteriori estimation"
+chapter: "General Theorems"
+section: "Bayesian statistics"
+topic: "Probabilistic modeling"
+definition: "Maximum-a-posteriori estimation"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2023
+    title: "Maximum a posteriori estimation"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2023-12-01"
+    url: "https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation#Description"
+
+def_id: "D191"
+shortcut: "map"
+username: "JoramSoch"
+---
+
+
+**Definition:** Consider a [posterior distribution](/D/post) of an unknown parameter $\theta$, given measured data $y$, parametrized by posterior hyperparameters $\phi$:
+
+$$ \label{eq:post}
+\theta|y \sim \mathcal{D}(\phi) \; .
+$$
+
+Then, the value of $\theta$ at which the [posterior density](/D/post) attains its maximum is called the "maximum-a-posteriori estimate" or "MAP estimate" of $\theta$:
+
+$$ \label{eq:prior-pdf}
+\hat{\theta}_\mathrm{MAP} = \operatorname*{arg\,max}_\theta \mathcal{D}(\theta; \phi) \; .
+$$