Merge pull request #32 from StatProofBook/master

update to master

Author: JoramSoch

404.html

@@ -24,7 +24,7 @@ <h1>This page doesn't exist!</h1>
 
 <div class="list">
   <ul>
-    <li>If the URL reads <code class="language-plaintext highlighter-rouge">.../Proof/[shortcut].html</code>, try to call <code class="language-plaintext highlighter-rouge">.../P/[shortcut]</code>.</li>
+    <li>If the URL reads <code class="language-plaintext highlighter-rouge">.../Proofs/[shortcut].html</code>, try to call <code class="language-plaintext highlighter-rouge">.../P/[shortcut]</code>.</li>
 	<li>If the URL reads <code class="language-plaintext highlighter-rouge">.../P/[shortcut]</code>, create a <a href="/contribute/">new page</a> using the <a href="https://raw.githubusercontent.com/StatProofBook/StatProofBook.github.io/master/P/-temp-.md">proof template</a>.</li>
     <li>If the URL reads <code class="language-plaintext highlighter-rouge">.../D/[shortcut]</code>, create a <a href="/contribute/">new page</a> using the <a href="https://raw.githubusercontent.com/StatProofBook/StatProofBook.github.io/master/D/-temp-.md">definition template</a>.</li>
   </ul>

D/bf.md

@@ -0,0 +1,49 @@
+---
+layout: definition
+mathjax: true
+
+author: "Thomas J. Faulkenberry"
+affiliation: "Tarleton State University"
+e_mail: "faulkenberry@tarleton.edu"
+date: 2020-08-26 12:00:00
+
+title: "Bayes factor"
+chapter: "Model Selection"
+section: "Bayesian model selection"
+topic: "Bayes factor"
+definition: "Definition"
+
+sources:
+  - authors: "Kass, Robert E. and Raftery, Adrian E."
+    year: 1995
+    title: "Bayes Factors"
+    in: "Journal of the American Statistical Association"
+    pages: "vol. 90, no. 430, pp. 773-795"
+    url: "https://dx.doi.org/10.1080/01621459.1995.10476572"
+    doi: "10.1080/01621459.1995.10476572"
+
+def_id: "D92"
+shortcut: "bf"
+username: "tomfaulkenberry"
+---
+
+
+**Definition:** Consider two competing [generative models](/D/gm) $m_1$ and $m_2$ for observed data $y$. Then the Bayes factor in favor $m_1$ over $m_2$ is the ratio of [marginal likelihoods](/D/ml) of $m_1$ and $m_2$:
+
+$$ \label{eq:BF}
+\text{BF}_{12} = \frac{p(y\mid m_1)}{p(y\mid m_2)}.
+$$
+
+Note that by [Bayes' theorem](/P/bayes-th), the ratio of [posterior model probabilities](/D/pmp) (i.e., the posterior model odds) can be written as
+
+$$ \label{eq:odds}
+\frac{p(m_1 \mid y)}{p(m_2 \mid y)} = \frac{p(m_1)}{p(m_2)} \cdot \frac{p(y\mid m_1)}{p(y\mid m_2)},
+$$
+
+or equivalently by \eqref{eq:BF},
+
+$$ \label{eq:odds2}
+\frac{p(m_1 \mid y)}{p(m_2 \mid y)} = \frac{p(m_1)}{p(m_2)} \cdot \text{BF}_{12}.
+$$
+
+In other words, the Bayes factor can be viewed as the factor by which the prior model odds are updated (after observing data $y$) to posterior model odds – which is also expressed by [Bayes' rule](/P/bayes-rule).

D/emat.md

@@ -9,7 +9,7 @@ date: 2020-07-22 05:17:00
 
 title: "Estimation matrix"
 chapter: "Statistical Models"
-section: "Normal data"
+section: "Univariate normal data"
 topic: "Multiple linear regression"
 definition: "Estimation matrix"
 

D/encm.md

@@ -0,0 +1,31 @@
+---
+layout: definition
+mathjax: true
+
+author: "Thomas J. Faulkenberry"
+affiliation: "Tarleton State University"
+e_mail: "faulkenberry@tarleton.edu"
+date: 2020-08-26 12:00:00
+
+title: "Encompassing model"
+chapter: "Model Selection"
+section: "Bayesian model selection"
+topic: "Bayes factor"
+definition: "Definition"
+
+sources:
+  - authors: "Klugkist, I., Kato, B., and Hoijtink, H."
+    year: 2005
+    title: "Bayesian model selection using encompassing priors"
+    in: "Statistica Neerlandica"
+    pages: "vol. 59, no. 1, pp. 57-69"
+    url: "https://dx.doi.org/10.1111/j.1467-9574.2005.00279.x"
+    doi: "10.1111/j.1467-9574.2005.00279.x"
+
+def_id: "D93"
+shortcut: "encm"
+username: "tomfaulkenberry"
+---
+
+
+**Definition:** Consider a family $f$ of [generative models](/D/gm) $m$ on data $y$, where each $m \in f$ is defined by placing an inequality constraint on model parameter(s) $\theta$ (e.g., $m:\theta>0$). Then the encompassing model $m_e$ is constructed such that each $m$ is nested within $m_e$ and all inequality constraints on the parameter(s) $\theta$ are removed.

D/ent-cross.md

@@ -11,7 +11,7 @@ title: "Cross-entropy"
 chapter: "General Theorems"
 section: "Information theory"
 topic: "Shannon entropy"
-definition: "Definition"
+definition: "Cross-entropy"
 
 sources:
   - authors: "Wikipedia"

D/ess.md

@@ -9,7 +9,7 @@ date: 2020-03-21 21:57:00
 
 title: "Explained sum of squares"
 chapter: "Statistical Models"
-section: "Normal data"
+section: "Univariate normal data"
 topic: "Multiple linear regression"
 definition: "Explained sum of squares"
 

D/fpm.md

@@ -9,7 +9,7 @@ date: 2020-03-03 16:16:00
 
 title: "Full probability model"
 chapter: "General Theorems"
-section: "Bayesian inference"
+section: "Bayesian statistics"
 topic: "Probabilistic modeling"
 definition: "Full probability model"
 

D/fwhm.md

@@ -10,7 +10,7 @@ date: 2020-08-19 05:40:00
 title: "Full width at half maximum"
 chapter: "General Theorems"
 section: "Probability theory"
-topic: "Further moments"
+topic: "Measures of statistical dispersion"
 definition: "Full width at half maximum"
 
 sources:

D/glm.md

@@ -9,7 +9,7 @@ date: 2020-03-21 22:24:00
 
 title: "General linear model"
 chapter: "Statistical Models"
-section: "Normal data"
+section: "Multivariate normal data"
 topic: "General linear model"
 definition: "Definition"
 

D/gm.md

@@ -9,7 +9,7 @@ date: 2020-03-03 15:50:00
 
 title: "Generative model"
 chapter: "General Theorems"
-section: "Bayesian inference"
+section: "Bayesian statistics"
 topic: "Probabilistic modeling"
 definition: "Generative model"