Merge pull request #121 from StatProofBook/master

update to mastwer

Author: JoramSoch

D/map.md

@@ -33,7 +33,7 @@ $$ \label{eq:post}
 \theta|y \sim \mathcal{D}(\phi) \; .
 $$
 
-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 $\theta$:
+Then, the value of $\theta$ at which the [posterior density](/D/post) attains its maximum is called the "maximum-a-posteriori estimate", "MAP estimate" or "posterior mode" of $\theta$:
 
 $$ \label{eq:prior-pdf}
 \hat{\theta}_\mathrm{MAP} = \operatorname*{arg\,max}_\theta \mathcal{D}(\theta; \phi) \; .

D/mle.md

@@ -21,7 +21,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let there be a [generative model](/D/gm) $m$ describing measured data $y$ using model parameters $\theta$. Then, the parameter values maximizing the [likelihood function](/D/lf) or [log-likelihood function](/D/llf) are called maximum likelihood estimates of $\theta$:
+**Definition:** Let there be a [generative model](/D/gm) $m$ describing measured data $y$ using model parameters $\theta$. Then, the parameter values maximizing the [likelihood function](/D/lf) or [log-likelihood function](/D/llf) are called "maximum likelihood estimates" of $\theta$:
 
 $$ \label{eq:mle}
 \hat{\theta} = \operatorname*{arg\,max}_\theta \mathcal{L}_m(\theta) = \operatorname*{arg\,max}_\theta \mathrm{LL}_m(\theta) \; .

I/PbA.md

@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (396 proofs)
+### JoramSoch (397 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -199,6 +199,7 @@ title: "Proof by Author"
 - [Maximum log-likelihood for multinomial observations](/P/mult-mll)
 - [Maximum log-likelihood for multiple linear regression](/P/mlr-mll)
 - [Maximum-a-posteriori estimation for binomial observations](/P/bin-map)
+- [Maximum-a-posteriori estimation for multinomial observations](/P/mult-map)
 - [Mean of the Bernoulli distribution](/P/bern-mean)
 - [Mean of the beta distribution](/P/beta-mean)
 - [Mean of the binomial distribution](/P/bin-mean)

I/PbN.md

@@ -433,3 +433,4 @@ title: "Proof by Number"
 | P425 | duni-kl | [Kullback-Leibler divergence for the discrete uniform distribution](/P/duni-kl) | JoramSoch | 2023-11-17 |
 | P426 | gam-scal | [Scaling of a random variable following the gamma distribution](/P/gam-scal) | JoramSoch | 2023-11-24 |
 | P427 | bin-map | [Maximum-a-posteriori estimation for binomial observations](/P/bin-map) | JoramSoch | 2023-12-01 |
+| P428 | mult-map | [Maximum-a-posteriori estimation for multinomial observations](/P/mult-map) | JoramSoch | 2023-12-08 |

I/PbT.md

@@ -232,6 +232,7 @@ title: "Proof by Topic"
 - [Maximum log-likelihood for multinomial observations](/P/mult-mll)
 - [Maximum log-likelihood for multiple linear regression](/P/mlr-mll)
 - [Maximum-a-posteriori estimation for binomial observations](/P/bin-map)
+- [Maximum-a-posteriori estimation for multinomial observations](/P/mult-map)
 - [Mean of the Bernoulli distribution](/P/bern-mean)
 - [Mean of the beta distribution](/P/beta-mean)
 - [Mean of the binomial distribution](/P/bin-mean)

I/PwS.md

@@ -76,6 +76,7 @@ title: "Proofs without Source"
 - [Maximum log-likelihood for binomial observations](/P/bin-mll)
 - [Maximum log-likelihood for multinomial observations](/P/mult-mll)
 - [Maximum-a-posteriori estimation for binomial observations](/P/bin-map)
+- [Maximum-a-posteriori estimation for multinomial observations](/P/mult-map)
 - [Mean of the categorical distribution](/P/cat-mean)
 - [Mean of the continuous uniform distribution](/P/cuni-mean)
 - [Mean of the ex-Gaussian distribution](/P/exg-mean)

P/bin-map.md

@@ -35,7 +35,7 @@ $$
 
 Then, the [maximum-a-posteriori estimate](/D/map) of $p$ is
 
-$$ \label{eq:Bin-MLE}
+$$ \label{eq:Bin-MAP}
 \hat{p}_\mathrm{MAP} = \frac{\alpha_0+y-1}{\alpha_0+\beta_0+n-2} \; .
 $$
 
@@ -57,13 +57,13 @@ $$
 
 The [mode of the beta distribution](/P/beta-mode) is given by:
 
-$$ \label{eq:beta-mode}
+$$ \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:
+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:
 
-$$ \label{eq:Bin-MAP}
+$$ \label{eq:Bin-MAP-qed}
 \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} \\

P/mult-map.md

@@ -71,10 +71,11 @@ $$
 Since $y_1 + \ldots + y_k = n$ [by definition](/D/mult-data), this becomes
 
 $$ \label{eq:Mult-MAP-s2}
-\hat{p}_{i,\mathrm{MAP}} = \frac{\alpha_{0i} + y_i - 1}{\sum_j \alpha_{0j} + n - k} \end{equation}
+\hat{p}_{i,\mathrm{MAP}} = \frac{\alpha_{0i} + y_i - 1}{\sum_j \alpha_{0j} + n - k}
+$$
 
 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}
+$$ \label{eq:Mult-MAP-qed}
 \hat{p}_\mathrm{MAP} = \frac{\alpha_0+y-1}{\sum_{j=1}^k \alpha_{0j} + n - k} \; .
 $$

_includes/footer.html

@@ -21,7 +21,7 @@ <h2 class="footer-heading">{{ site.title | escape }}</h2>
 			{%- endif -%}
 		  </li>
           <li>
-		    <a href="https://dx.doi.org/10.5281/zenodo.4305950"><img src="https://zenodo.org/badge/204500928.svg" alt="DOI"></a>
+		    <a href="https://doi.org/10.5281/zenodo.4305949"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.4305949" alt="DOI"></a>
 		  </li>
         </ul>
       </div>