Merge pull request #118 from StatProofBook/master

JoramSoch · web-flow · commit a7453caec7fd · 2023-11-24T13:17:39.000+01:00
update to master
diff --git a/D/iass.md b/D/iass.md
@@ -33,7 +33,7 @@ $$ \label{eq:anova}
 y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + \varepsilon_{ijk}, \; \varepsilon_{ijk} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; .
 $$
 
-Then, the interaction sum of squares is defined as the [explained sum of squares] (ESS) for each interaction, i.e. as the sum of squared deviations of the average for each cell from the average across all observations, controlling for the [treatment sums of squares](/D/trss) of the corresponding factors:
+Then, the interaction sum of squares is defined as the [explained sum of squares](/D/ess) (ESS) for each interaction, i.e. as the sum of squared deviations of the average for each cell from the average across all observations, controlling for the [treatment sums of squares](/D/trss) of the corresponding factors:
 
 $$ \label{eq:iass}
 \begin{split}
@@ -42,4 +42,4 @@ $$ \label{eq:iass}
 \end{split}
 $$
 
-Here, $\bar{y} _{i j \bullet}$ is the mean for the $(i,j)$-th cell (out of $a \times b$ cells), computed from $n_{ij}$ values $y_{ijk}$, $\bar{y} _{i \bullet \bullet}$ and $\bar{y} _{\bullet j \bullet}$ are the level means for the two factors and and $\bar{y} _{\bullet \bullet \bullet}$ is the mean across all values $y_{ijk}$.
+Here, $\bar{y}\_{i j \bullet}$ is the mean for the $(i,j)$-th cell (out of $a \times b$ cells), computed from $n\_{ij}$ values $y\_{ijk}$, $\bar{y}\_{i \bullet \bullet}$ and $\bar{y}\_{\bullet j \bullet}$ are the level means for the two factors and and $\bar{y}\_{\bullet \bullet \bullet}$ is the mean across all values $y\_{ijk}$.
diff --git a/D/trss.md b/D/trss.md
@@ -33,10 +33,10 @@ $$ \label{eq:anova}
 y_{ij} = \mu + \delta_i + \varepsilon_{ij}, \; \varepsilon_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; .
 $$
 
-Then, the treatment sum of squares is defined as the [explained sum of squares] (ESS) for each main effect, i.e. as the sum of squared deviations of the average for each level of the factor, from the average across all observations:
+Then, the treatment sum of squares is defined as the [explained sum of squares](/D/ess) (ESS) for each main effect, i.e. as the sum of squared deviations of the average for each level of the factor, from the average across all observations:
 
 $$ \label{eq:trss}
 \mathrm{SS}_\mathrm{treat} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (\bar{y}_i - \bar{y})^2 \; .
 $$
 
-Here, $\bar{y}_i$ is the mean for the $i$-th level of the factor (out of $k$ levels), computed from $n_i$ values $y_{ij}$, and $\bar{y}$ is the mean across all values $y_{ij}$.
+Here, $\bar{y}\_i$ is the mean for the $i$-th level of the factor (out of $k$ levels), computed from $n_i$ values $y\_{ij}$, and $\bar{y}$ is the mean across all values $y\_{ij}$.
diff --git a/I/PbA.md b/I/PbA.md
@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (393 proofs)
+### JoramSoch (394 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -142,6 +142,7 @@ title: "Proof by Author"
 - [Kullback-Leibler divergence for the binomial distribution](/P/bin-kl)
 - [Kullback-Leibler divergence for the continuous uniform distribution](/P/cuni-kl)
 - [Kullback-Leibler divergence for the Dirichlet distribution](/P/dir-kl)
+- [Kullback-Leibler divergence for the discrete uniform distribution](/P/duni-kl)
 - [Kullback-Leibler divergence for the gamma distribution](/P/gam-kl)
 - [Kullback-Leibler divergence for the matrix-normal distribution](/P/matn-kl)
 - [Kullback-Leibler divergence for the multivariate normal distribution](/P/mvn-kl)
@@ -433,8 +434,8 @@ title: "Proof by Author"
 - [Encompassing prior method for computing Bayes factors](/P/bf-ep)
 - [Mean of the ex-Gaussian distribution](/P/exg-mean)
 - [Mean of the Wald distribution](/P/wald-mean)
-- [Method of moments for ex-Gaussian distributed data](/P/exg-mome)
-- [Method of moments for Wald distributed data](/P/wald-mome)
+- [Method of moments for ex-Gaussian-distributed data](/P/exg-mome)
+- [Method of moments for Wald-distributed data](/P/wald-mome)
 - [Moment-generating function of the ex-Gaussian distribution](/P/exg-mgf)
 - [Moment-generating function of the exponential distribution](/P/exp-mgf)
 - [Moment-generating function of the Wald distribution](/P/wald-mgf)
diff --git a/I/PbN.md b/I/PbN.md
@@ -428,5 +428,6 @@ title: "Proof by Number"
 | P420 | bin-kl | [Kullback-Leibler divergence for the binomial distribution](/P/bin-kl) | JoramSoch | 2023-10-20 |
 | P421 | wald-skew | [Skewness of the Wald distribution](/P/wald-skew) | tomfaulkenberry | 2023-10-24 |
 | P422 | cuni-kl | [Kullback-Leibler divergence for the continuous uniform distribution](/P/cuni-kl) | JoramSoch | 2023-10-27 |
-| P423 | wald-mome | [Method of moments for Wald distributed data](/P/wald-mome) | tomfaulkenberry | 2023-10-30 |
-| P424 | exg-mome | [Method of moments for ex-Gaussian distributed data](/P/exg-mome) | tomfaulkenberry | 2023-10-30 |
+| P423 | wald-mome | [Method of moments for Wald-distributed data](/P/wald-mome) | tomfaulkenberry | 2023-10-30 |
+| P424 | exg-mome | [Method of moments for ex-Gaussian-distributed data](/P/exg-mome) | tomfaulkenberry | 2023-10-30 |
+| P425 | duni-kl | [Kullback-Leibler divergence for the discrete uniform distribution](/P/duni-kl) | JoramSoch | 2023-11-17 |
diff --git a/I/PbT.md b/I/PbT.md
@@ -169,6 +169,7 @@ title: "Proof by Topic"
 - [Kullback-Leibler divergence for the binomial distribution](/P/bin-kl)
 - [Kullback-Leibler divergence for the continuous uniform distribution](/P/cuni-kl)
 - [Kullback-Leibler divergence for the Dirichlet distribution](/P/dir-kl)
+- [Kullback-Leibler divergence for the discrete uniform distribution](/P/duni-kl)
 - [Kullback-Leibler divergence for the gamma distribution](/P/gam-kl)
 - [Kullback-Leibler divergence for the matrix-normal distribution](/P/matn-kl)
 - [Kullback-Leibler divergence for the multivariate normal distribution](/P/mvn-kl)
@@ -253,8 +254,8 @@ title: "Proof by Topic"
 - [Median of the normal distribution](/P/norm-med)
 - [Method of moments for beta-binomial data](/P/betabin-mome)
 - [Method of moments for beta-distributed data](/P/beta-mome)
-- [Method of moments for ex-Gaussian distributed data](/P/exg-mome)
-- [Method of moments for Wald distributed data](/P/wald-mome)
+- [Method of moments for ex-Gaussian-distributed data](/P/exg-mome)
+- [Method of moments for Wald-distributed data](/P/wald-mome)
 - [Mode of the continuous uniform distribution](/P/cuni-med)
 - [Mode of the exponential distribution](/P/exp-mode)
 - [Mode of the log-normal distribution](/P/lognorm-mode)
diff --git a/I/PwS.md b/I/PwS.md
@@ -47,6 +47,7 @@ title: "Proofs without Source"
 - [Joint likelihood is the product of likelihood function and prior density](/P/jl-lfnprior)
 - [Kullback-Leibler divergence for the Bernoulli distribution](/P/bern-kl)
 - [Kullback-Leibler divergence for the continuous uniform distribution](/P/cuni-kl)
+- [Kullback-Leibler divergence for the discrete uniform distribution](/P/duni-kl)
 - [Kullback-Leibler divergence for the matrix-normal distribution](/P/matn-kl)
 - [Kullback-Leibler divergence for the normal distribution](/P/norm-kl)
 - [Linear combination of independent normal random variables](/P/norm-lincomb)
@@ -85,8 +86,8 @@ title: "Proofs without Source"
 - [Median of the exponential distribution](/P/exp-med)
 - [Median of the log-normal distribution](/P/lognorm-med)
 - [Median of the normal distribution](/P/norm-med)
-- [Method of moments for ex-Gaussian distributed data](/P/exg-mome)
-- [Method of moments for Wald distributed data](/P/wald-mome)
+- [Method of moments for ex-Gaussian-distributed data](/P/exg-mome)
+- [Method of moments for Wald-distributed data](/P/wald-mome)
 - [Mode of the continuous uniform distribution](/P/cuni-med)
 - [Mode of the exponential distribution](/P/exp-mode)
 - [Mode of the normal distribution](/P/norm-mode)
diff --git a/P/anova1-fols.md b/P/anova1-fols.md
@@ -40,13 +40,13 @@ F = \frac{\frac{1}{k-1} \sum_{i=1}^{k} n_i \hat{\delta}_i^2}{\frac{1}{n-k} \sum_
 $$
 
 
-**Theorem:** The [F-statistic for the main effect in one-way ANOVA](/P/anova1-f) is given in terms of the [sample means](/D/mean-samp) as
+**Proof:** The [F-statistic for the main effect in one-way ANOVA](/P/anova1-f) is given in terms of the [sample means](/D/mean-samp) as
 
 $$ \label{eq:anova1-f}
 F = \frac{\frac{1}{k-1} \sum_{i=1}^{k} n_i (\bar{y}_i - \bar{y})^2}{\frac{1}{n-k} \sum_{i=1}^{k} \sum_{j=1}^{n_i} (y_{ij} - \bar{y}_i)^2}
 $$
 
-where $\bar{y} _i$ is the average of all values $y_{ij}$ from category $i$ and $\bar{y}$ is the grand mean of all values $y_{ij}$ from all categories $i = 1, \ldots, k$.
+where $\bar{y}\_i$ is the average of all values $y\_{ij}$ from category $i$ and $\bar{y}$ is the grand mean of all values $y\_{ij}$ from all categories $i = 1, \ldots, k$.
 
 1) The [ordinary least squares estimates for one-way ANOVA](/P/anova1-ols) are
 
diff --git a/P/anova2-pss.md b/P/anova2-pss.md
@@ -45,7 +45,7 @@ $$ \label{eq:anova2-pss}
 \mathrm{SS}_\mathrm{tot} = \mathrm{SS}_{A} + \mathrm{SS}_{B} + \mathrm{SS}_{A \times B} + \mathrm{SS}_\mathrm{res}
 $$
 
-where $\mathrm{SS} _\mathrm{tot}$ is the [total sum of squares](/D/tss), $\mathrm{SS} _{A}$, $\mathrm{SS} _{B}$ and $\mathrm{SS} _{A \times B}$ are [treatment](/D/trss) and [interaction sum of squares](/D/iass) (summing into the [explained sum of squares](/D/ess)) and $\mathrm{SS} _\mathrm{res}$ is the [residual sum of squares](/D/rss).
+where $\mathrm{SS}\_\mathrm{tot}$ is the [total sum of squares](/D/tss), $\mathrm{SS}\_{A}$, $\mathrm{SS}\_{B}$ and $\mathrm{SS}\_{A \times B}$ are [treatment](/D/trss) and [interaction sum of squares](/D/iass) (summing into the [explained sum of squares](/D/ess)) and $\mathrm{SS}\_\mathrm{res}$ is the [residual sum of squares](/D/rss).
 
 
 **Proof:** The [total sum of squares](/D/tss) for [two-way ANOVA](/D/anova2) is given by
@@ -54,7 +54,7 @@ $$ \label{eq:anova2-tss}
 \mathrm{SS}_\mathrm{tot} = \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{\bullet \bullet \bullet})^2
 $$
 
-where $\bar{y} _{\bullet \bullet \bullet}$ is the mean across all values $y_{ijk}$. This can be rewritten as
+where $\bar{y}\_{\bullet \bullet \bullet}$ is the mean across all values $y\_{ijk}$. This can be rewritten as
 
 $$ \label{eq:anova2-pss-s1}
 \begin{split}
diff --git a/P/bin-ent.md b/P/bin-ent.md
@@ -87,4 +87,4 @@ $$ \label{eq:bin-ent-s4}
 \mathrm{H}(X) = n \cdot \mathrm{H}_\mathrm{bern}(p) - \mathrm{E}_\mathrm{lbc}(n,p) \; .
 $$
 
-Note that, because $0 \leq \mathrm{H}_\mathrm{bern}(p) \leq 1$, we have $0 \leq n \cdot \mathrm{H}_\mathrm{bern}(p) \leq n$, and because the [entropy is non-negative](/P/ent-nonneg), it must hold that $n \geq \mathrm{E}_\mathrm{lbc}(n,p) \geq 0$.
+Note that, because $0 \leq \mathrm{H}\_\mathrm{bern}(p) \leq 1$, we have $0 \leq n \cdot \mathrm{H}\_\mathrm{bern}(p) \leq n$, and because the [entropy is non-negative](/P/ent-nonneg), it must hold that $n \geq \mathrm{E}\_\mathrm{lbc}(n,p) \geq 0$.
diff --git a/P/duni-kl.md b/P/duni-kl.md
@@ -43,7 +43,7 @@ $$ \label{eq:KL-disc}
 \mathrm{KL}[P\,||\,Q] = \sum_{x \in \mathcal{X}} p(x) \, \ln \frac{p(x)}{q(x)} \; .
 $$
 
-This means that the KL divergence of $P$ from $Q$ is only defined, if for all $x \in \mathcal{X}$, $q(x) = 0$ implies $p(x) = 0$. Thus, $\mathrm{KL}[P\,||\,Q]$ only exists, if $a_2 \leq a_1$ and $b_1 \leq b_2$, i.e. if $P$ only places non-zero probability where $Q$ also places non-zero probability, such that $q(x)$ is not zero for any $x \in \mathcal{X}$ where $p(x)$ is positive.
+This means that the KL divergence of $P$ from $Q$ is only defined, if for all $x \in \mathcal{X}$, $q(x) = 0$ implies $p(x) = 0$. Thus, $\mathrm{KL}[P\,\vert\vert\,Q]$ only exists, if $a_2 \leq a_1$ and $b_1 \leq b_2$, i.e. if $P$ only places non-zero probability where $Q$ also places non-zero probability, such that $q(x)$ is not zero for any $x \in \mathcal{X}$ where $p(x)$ is positive.
 
 If this requirement is fulfilled, we can write
 
diff --git a/P/exg-mome.md b/P/exg-mome.md
@@ -7,7 +7,7 @@ affiliation: "Tarleton State University"
 e_mail: "faulkenberry@tarleton.edu"
 date: 2023-10-30 12:00:00
 
-title: "Method of moments for ex-Gaussian distributed data"
+title: "Method of moments for ex-Gaussian-distributed data"
 chapter: "Probability Distributions"
 section: "Univariate continuous distributions"
 topic: "ex-Gaussian distribution"
diff --git a/P/ug-ttest2.md b/P/ug-ttest2.md
@@ -94,7 +94,7 @@ $$ \label{eq:mean-samp-dist}
 \end{split}
 $$
 
-Again employing the linear combination theorem and applying the null hypothesis from \eqref{eq:ttest2-h0}, the distribution of $Z = ( ( \bar{y}_1 - \bar{y}_2 ) - \mu_{\Delta} ) / ( \sigma \sqrt{1/n_1+1/n_2} )$ becomes [standard normal](/D/snorm)
+Again employing the linear combination theorem and applying the null hypothesis from \eqref{eq:ttest2-h0}, the distribution of $Z = ((\bar{y}\_1-\bar{y}\_2) - \mu\_{\Delta}) / (\sigma \sqrt{1/n\_1+1/n\_2})$ becomes [standard normal](/D/snorm)
 
 $$ \label{eq:Z-dist}
 Z = \frac{(\bar{y}_1-\bar{y}_2)-\mu_\Delta}{\sigma \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \sim \mathcal{N}\left( \frac{(\mu_1-\mu_2)-\mu_\Delta}{\sigma \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}, \left(\frac{1}{\sigma \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\right)^2 \left( \frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2} \right) \right) \overset{H_0}{=} \mathcal{N}\left( 0, 1 \right) \; .
diff --git a/P/wald-mome.md b/P/wald-mome.md
@@ -7,7 +7,7 @@ affiliation: "Tarleton State University"
 e_mail: "faulkenberry@tarleton.edu"
 date: 2023-10-30 12:00:00
 
-title: "Method of moments for Wald distributed data"
+title: "Method of moments for Wald-distributed data"
 chapter: "Probability Distributions"
 section: "Univariate continuous distributions"
 topic: "Wald distribution"
diff --git a/_layouts/definition.html b/_layouts/definition.html
@@ -23,11 +23,11 @@ <h1 class="post-title">Definition: {{ page.title }}</h1>
 	  <b>Sources:</b> <br><ul>
 	  {% for item in page.sources %}
 	    <li>
-	      {{item.authors}} ({{item.year}}): "{{item.title}}"
-		  {% if item.in %}; in: <i>{{item.in}}</i> {% endif %}
-		  {% if item.pages %}, {{item.pages}} {% endif %}
-		  {% if item.url %}; URL: <a href="{{item.url}}">{{item.url}}</a> {% endif %}
-		  {% if item.doi %}; DOI: <a href="https://doi.org/{{item.doi}}">{{item.doi}}</a> {% endif %}.
+	      {{item.authors}} ({{item.year}}): "{{item.title}}"<!--
+		  -->{% if item.in %}; in: <i>{{item.in}}</i>{% endif %}<!--
+		  -->{% if item.pages %}, {{item.pages}}{% endif %}<!--
+		  -->{% if item.url %}; URL: <a href="{{item.url}}">{{item.url}}</a>{% endif %}<!--
+		  -->{% if item.doi %}; DOI: <a href="https://doi.org/{{item.doi}}">{{item.doi}}</a>{% endif %}.
 	    </li>
 	  {% endfor %} </ul><br>
 	{% else %}
diff --git a/_layouts/proof.html b/_layouts/proof.html
@@ -23,11 +23,11 @@ <h1 class="post-title">Proof: {{ page.title }}</h1>
 	  <b>Sources:</b> <br><ul>
 	  {% for item in page.sources %}
 	    <li>
-	      {{item.authors}} ({{item.year}}): "{{item.title}}"
-		  {% if item.in %}; in: <i>{{item.in}}</i> {% endif %}
-		  {% if item.pages %}, {{item.pages}} {% endif %}
-		  {% if item.url %}; URL: <a href="{{item.url}}">{{item.url}}</a> {% endif %}
-		  {% if item.doi %}; DOI: <a href="https://doi.org/{{item.doi}}">{{item.doi}}</a> {% endif %}.
+	      {{item.authors}} ({{item.year}}): "{{item.title}}"<!--
+		  -->{% if item.in %}; in: <i>{{item.in}}</i>{% endif %}<!--
+		  -->{% if item.pages %}, {{item.pages}}{% endif %}<!--
+		  -->{% if item.url %}; URL: <a href="{{item.url}}">{{item.url}}</a>{% endif %}<!--
+		  -->{% if item.doi %}; DOI: <a href="https://doi.org/{{item.doi}}">{{item.doi}}</a>{% endif %}.
 	    </li>
 	  {% endfor %} </ul><br>
 	{% else %}
