integrated new content

Proof "bf-trans" wwas added to the table of contents; index pages updated.

Author: JoramSoch

I/Proof_by_Author.md

@@ -170,8 +170,9 @@ title: "Proof by Author"
 
 - [Proof Template](/P/-temp-)
 
-### tomfaulkenberry (3 proofs)
+### tomfaulkenberry (4 proofs)
 
 - [Encompassing Prior Method for computing Bayes Factors](/P/bf-ep)
 - [Probability density function of the Wald distribution](/P/wald-pdf)
 - [Savage-Dickey Density Ratio for computing Bayes Factors](/P/bf-sddr)
+- [Transitivity of Bayes Factors](/P/bf-trans)

I/Proof_by_Number.md

@@ -169,3 +169,4 @@ title: "Proof by Number"
 | P160 | mblr-post | [Posterior distribution for multivariate Bayesian linear regression](/P/mblr-post) | JoramSoch | 2020-09-03 |
 | P161 | mblr-lme | [Log model evidence for multivariate Bayesian linear regression](/P/mblr-lme) | JoramSoch | 2020-09-03 |
 | P162 | wald-pdf | [Probability density function of the Wald distribution](/P/wald-pdf) | tomfaulkenberry | 2020-09-04 |
+| P163 | bf-trans | [Transitivity of Bayes Factors](/P/bf-trans) | tomfaulkenberry | 2020-09-07 |

I/Proof_by_Topic.md

@@ -213,6 +213,7 @@ title: "Proof by Topic"
 ### T
 
 - [Transformation matrices for ordinary least squares](/P/mlr-mat)
+- [Transitivity of Bayes Factors](/P/bf-trans)
 - [Transposition of a matrix-normal random variable](/P/matn-trans)
 
 ### V

I/Proofs_without_Source.md

@@ -60,6 +60,7 @@ title: "Proofs without Source"
 - [Relationship between covariance and correlation](/P/cov-corr)
 - [Relationship between precision matrix and correlation matrix](/P/precmat-corrmat)
 - [Relationship between signal-to-noise ratio and R²](/P/snr-rsq)
+- [Transitivity of Bayes Factors](/P/bf-trans)
 - [Transposition of a matrix-normal random variable](/P/matn-trans)
 - [Weighted least squares for multiple linear regression](/P/mlr-wls2)
 - [Weighted least squares for the general linear model](/P/glm-wls)

I/Table_of_Contents.md

@@ -424,9 +424,10 @@ title: "Table of Contents"
 
    3.3. Bayes factor <br>
    &emsp;&ensp; 3.3.1. *[Definition](/D/bf)* <br>
-   &emsp;&ensp; 3.3.2. *[Encompassing model](/D/encm)* <br>
+   &emsp;&ensp; 3.3.2. **[Transitivity](/P/bf-trans)** <br>
    &emsp;&ensp; 3.3.3. **[Computation using Savage-Dickey Density Ratio](/P/bf-sddr)** <br>
    &emsp;&ensp; 3.3.4. **[Computation using Encompassing Prior Method](/P/bf-ep)** <br>
+   &emsp;&ensp; 3.3.5. *[Encompassing model](/D/encm)* <br>
 
    3.4. Log Bayes factor <br>
    &emsp;&ensp; 3.4.1. *[Definition](/D/lbf)* <br>

P/bf-tran.md P/bf-trans.mdP/bf-tran.md renamed to P/bf-trans.md

@@ -11,19 +11,19 @@ title: "Transitivity of Bayes Factors"
 chapter: "Model Selection"
 section: "Bayesian model selection"
 topic: "Bayes factor"
-theorem: "Transitivity of Bayes Factors"
+theorem: "Transitivity"
 
 sources:
 
 proof_id: "P163"
-shortcut: "bf-tran"
+shortcut: "bf-trans"
 username: "tomfaulkenberry"
 ---
 
 
 **Theorem:** Consider three competing [models](/D/gm) $m_1$, $m_2$, and $m3$ for observed data $y$. Then the [Bayes factor](/D/bf) for $m_1$ over $m_3$ can be written as:
 
-$$ \label{eq:bf-tran}
+$$ \label{eq:bf-trans}
 \text{BF}_{13} = \text{BF}_{12}\cdot \text{BF}_{23}.
 $$
 
@@ -33,17 +33,15 @@ $$ \label{eq:bf}
 \text{BF}_{13}=\frac{p(y \mid m_1)}{p(y \mid m_3)}.
 $$
 
-We can equivlently write 
+We can equivalently write 
 
 $$
 \begin{split}
   \text{BF}_{13} &\overset{\eqref{eq:bf}}{=} \frac{p(y \mid m_1)}{p(y \mid m_3)}\\
   &= \frac{p(y \mid m_1)}{p(y \mid m_3)} \cdot \frac{p(y \mid m_2)}{p(y \mid m_2)}\\
-  &= \frac{p(y \mid m_1) \cdot p(y \mid m_2)}{p(y \mid m_3)\cdot p(y \mid m_2)}\\
-  &= \frac{p(y \mid m_1) \cdot p(y \mid m_2)}{p(y \mid m_2)\cdot p(y \mid m_3)}\\
   &=\frac{p(y \mid m_1)}{p(y \mid m_2)} \cdot \frac{p(y \mid m_2)}{p(y \mid m_3)}\\
   &\overset{\eqref{eq:bf}}{=}\text{BF}_{12} \cdot \text{BF}_{23},
 \end{split}
 $$
 
-which completes the proof of \eqref{eq:tran}.
+which completes the proof of \eqref{eq:bf-trans}.