Merge pull request #33 from StatProofBook/master

update to master

Author: JoramSoch

D/encm.md

@@ -5,13 +5,13 @@ mathjax: true
 author: "Thomas J. Faulkenberry"
 affiliation: "Tarleton State University"
 e_mail: "faulkenberry@tarleton.edu"
-date: 2020-08-26 12:00:00
+date: 2020-09-02 12:00:00
 
 title: "Encompassing model"
 chapter: "Model Selection"
 section: "Bayesian model selection"
 topic: "Bayes factor"
-definition: "Definition"
+definition: "Encompassing model"
 
 sources:
   - authors: "Klugkist, I., Kato, B., and Hoijtink, H."

D/wald.md

@@ -0,0 +1,43 @@
+---
+layout: definition
+mathjax: true
+
+author: "Thomas J. Faulkenberry"
+affiliation: "Tarleton State University"
+e_mail: "faulkenberry@tarleton.edu"
+date: 2020-09-04 12:00:00
+
+title: "Wald distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Wald distribution"
+definition: "Definition"
+
+sources:
+  - authors: "Anders, R., Alario, F.-X., and van Maanen, L."
+    year: 2016
+    title: "The Shifted Wald Distribution for Response Time Data Analysis"
+    in: "Psychological Methods"
+    pages: "vol. 21, no. 3, pp. 309-327"
+    url: "https://dx.doi.org/10.1037/met0000066"
+    doi: "10.1037/met0000066"
+
+def_id: "D95"
+shortcut: "wald"
+username: "tomfaulkenberry"
+---
+
+
+**Definition**: Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a Wald distribution with drift rate $\gamma$ and threshold $\alpha$
+
+$$ \label{eq:wald}
+X \sim \mathrm{Wald}(\gamma, \alpha) \; ,
+$$
+
+if and only if its [probability density function](/D/pdf) is given by
+
+$$ \label{eq:wald-pdf}
+\mathrm{Wald}(x; \gamma, \alpha) = \frac{\alpha}{\sqrt{2\pi x^3}}\exp\left(-\frac{(\alpha-\gamma x)^2}{2x}\right)
+$$
+
+where $\gamma > 0$, $\alpha > 0$, and the density is zero if $x \leq 0$.

I/Definition_by_Author.md

@@ -4,7 +4,7 @@ title: "Definition by Author"
 ---
 
 
-### JoramSoch (91 definitions)
+### JoramSoch (92 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Bayesian information criterion](/D/bic)
@@ -92,6 +92,7 @@ title: "Definition by Author"
 - [Residual-forming matrix](/D/rfmat)
 - [Shannon entropy](/D/ent)
 - [Signal-to-noise ratio](/D/snr)
+- [Standard deviation](/D/std)
 - [Standard gamma distribution](/D/sgam)
 - [Statistical independence](/D/ind)
 - [Total sum of squares](/D/tss)
@@ -102,7 +103,8 @@ title: "Definition by Author"
 
 - [Definition Template](/D/-temp-)
 
-### tomfaulkenberry (2 definitions)
+### tomfaulkenberry (3 definitions)
 
 - [Bayes factor](/D/bf)
 - [Encompassing model](/D/encm)
+- [Wald distribution](/D/wald)

I/Definition_by_Number.md

@@ -99,4 +99,6 @@ title: "Definition by Number"
 | D90 | mom | [Moment](/D/mom) | JoramSoch | 2020-08-19 |
 | D91 | fwhm | [Full width at half maximum](/D/fwhm) | JoramSoch | 2020-08-19 |
 | D92 | bf | [Bayes factor](/D/bf) | tomfaulkenberry | 2020-08-26 |
-| D93 | encm | [Encompassing model](/D/encm) | tomfaulkenberry | 2020-08-26 |
+| D93 | encm | [Encompassing model](/D/encm) | tomfaulkenberry | 2020-09-02 |
+| D94 | std | [Standard deviation](/D/std) | JoramSoch | 2020-09-03 |
+| D95 | wald | [Wald distribution](/D/wald) | tomfaulkenberry | 2020-09-04 |

I/Definition_by_Topic.md

@@ -140,6 +140,7 @@ title: "Definition by Topic"
 
 - [Shannon entropy](/D/ent)
 - [Signal-to-noise ratio](/D/snr)
+- [Standard deviation](/D/std)
 - [Standard gamma distribution](/D/sgam)
 - [Statistical independence](/D/ind)
 
@@ -153,4 +154,5 @@ title: "Definition by Topic"
 
 ### W
 
+- [Wald distribution](/D/wald)
 - [Wishart distribution](/D/wish)

I/Proof_by_Author.md

@@ -4,7 +4,7 @@ title: "Proof by Author"
 ---
 
 
-### JoramSoch (155 proofs)
+### JoramSoch (159 proofs)
 
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Additivity of the Kullback-Leibler divergence for independent distributions](/P/kl-add)
@@ -18,11 +18,13 @@ title: "Proof by Author"
 - [Conjugate prior distribution for Bayesian linear regression](/P/blr-prior)
 - [Conjugate prior distribution for binomial observations](/P/bin-prior)
 - [Conjugate prior distribution for multinomial observations](/P/mult-prior)
+- [Conjugate prior distribution for multivariate Bayesian linear regression](/P/mblr-prior)
 - [Conjugate prior distribution for the Poisson distribution with exposure values](/P/poissexp-prior)
 - [Construction of confidence intervals using Wilks' theorem](/P/ci-wilks)
 - [Construction of unbiased estimator for variance](/P/resvar-unb)
 - [Convexity of the Kullback-Leibler divergence](/P/kl-conv)
 - [Convexity of the cross-entropy](/P/entcross-conv)
+- [Covariance of independent random variables](/P/cov-ind)
 - [Cumulative distribution function of the continuous uniform distribution](/P/cuni-cdf)
 - [Cumulative distribution function of the discrete uniform distribution](/P/duni-cdf)
 - [Cumulative distribution function of the exponential distribution](/P/exp-cdf)
@@ -59,6 +61,7 @@ title: "Proof by Author"
 - [Log model evidence for Bayesian linear regression](/P/blr-lme)
 - [Log model evidence for binomial observations](/P/bin-lme)
 - [Log model evidence for multinomial observations](/P/mult-lme)
+- [Log model evidence for multivariate Bayesian linear regression](/P/mblr-lme)
 - [Log model evidence for the Poisson distribution with exposure values](/P/poissexp-lme)
 - [Log-odds and probability in logistic regression](/P/logreg-lonp)
 - [Logarithmic expectation of the gamma distribution](/P/gam-logmean)
@@ -108,6 +111,7 @@ title: "Proof by Author"
 - [Posterior distribution for Bayesian linear regression](/P/blr-post)
 - [Posterior distribution for binomial observations](/P/bin-post)
 - [Posterior distribution for multinomial observations](/P/mult-post)
+- [Posterior distribution for multivariate Bayesian linear regression](/P/mblr-post)
 - [Posterior distribution for the Poisson distribution with exposure values](/P/poissexp-post)
 - [Posterior model probabilities in terms of Bayes factors](/P/pmp-bf)
 - [Posterior model probabilities in terms of log model evidences](/P/pmp-lme)
@@ -166,7 +170,9 @@ title: "Proof by Author"
 
 - [Proof Template](/P/-temp-)
 
-### tomfaulkenberry (2 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

@@ -163,4 +163,10 @@ title: "Proof by Number"
 | P154 | mgf-ltt | [Linear transformation theorem for the moment-generating function](/P/mgf-ltt) | JoramSoch | 2020-08-19 |
 | P155 | mgf-lincomb | [Moment-generating function of linear combination of independent random variables](/P/mgf-lincomb) | JoramSoch | 2020-08-19 |
 | P156 | bf-sddr | [Savage-Dickey Density Ratio for computing Bayes Factors](/P/bf-sddr) | tomfaulkenberry | 2020-08-26 |
-| P157 | bf-ep | [Encompassing Prior Method for computing Bayes Factors](/P/bf-ep) | tomfaulkenberry | 2020-08-26 |
+| P157 | bf-ep | [Encompassing Prior Method for computing Bayes Factors](/P/bf-ep) | tomfaulkenberry | 2020-09-02 |
+| P158 | cov-ind | [Covariance of independent random variables](/P/cov-ind) | JoramSoch | 2020-09-03 |
+| P159 | mblr-prior | [Conjugate prior distribution for multivariate Bayesian linear regression](/P/mblr-prior) | JoramSoch | 2020-09-03 |
+| 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

@@ -27,11 +27,13 @@ title: "Proof by Topic"
 - [Conjugate prior distribution for Bayesian linear regression](/P/blr-prior)
 - [Conjugate prior distribution for binomial observations](/P/bin-prior)
 - [Conjugate prior distribution for multinomial observations](/P/mult-prior)
+- [Conjugate prior distribution for multivariate Bayesian linear regression](/P/mblr-prior)
 - [Conjugate prior distribution for the Poisson distribution with exposure values](/P/poissexp-prior)
 - [Construction of confidence intervals using Wilks' theorem](/P/ci-wilks)
 - [Construction of unbiased estimator for variance](/P/resvar-unb)
 - [Convexity of the Kullback-Leibler divergence](/P/kl-conv)
 - [Convexity of the cross-entropy](/P/entcross-conv)
+- [Covariance of independent random variables](/P/cov-ind)
 - [Cumulative distribution function of the continuous uniform distribution](/P/cuni-cdf)
 - [Cumulative distribution function of the discrete uniform distribution](/P/duni-cdf)
 - [Cumulative distribution function of the exponential distribution](/P/exp-cdf)
@@ -90,6 +92,7 @@ title: "Proof by Topic"
 - [Log model evidence for Bayesian linear regression](/P/blr-lme)
 - [Log model evidence for binomial observations](/P/bin-lme)
 - [Log model evidence for multinomial observations](/P/mult-lme)
+- [Log model evidence for multivariate Bayesian linear regression](/P/mblr-lme)
 - [Log model evidence for the Poisson distribution with exposure values](/P/poissexp-lme)
 - [Log-odds and probability in logistic regression](/P/logreg-lonp)
 - [Logarithmic expectation of the gamma distribution](/P/gam-logmean)
@@ -151,13 +154,15 @@ title: "Proof by Topic"
 - [Posterior distribution for Bayesian linear regression](/P/blr-post)
 - [Posterior distribution for binomial observations](/P/bin-post)
 - [Posterior distribution for multinomial observations](/P/mult-post)
+- [Posterior distribution for multivariate Bayesian linear regression](/P/mblr-post)
 - [Posterior distribution for the Poisson distribution with exposure values](/P/poissexp-post)
 - [Posterior model probabilities in terms of Bayes factors](/P/pmp-bf)
 - [Posterior model probabilities in terms of log model evidences](/P/pmp-lme)
 - [Posterior model probability in terms of log Bayes factor](/P/pmp-lbf)
 - [Posterior probability of the alternative hypothesis for Bayesian linear regression](/P/blr-pp)
 - [Probability and log-odds in logistic regression](/P/logreg-pnlo)
 - [Probability density function of the Dirichlet distribution](/P/dir-pdf)
+- [Probability density function of the Wald distribution](/P/wald-pdf)
 - [Probability density function of the beta distribution](/P/beta-pdf)
 - [Probability density function of the continuous uniform distribution](/P/cuni-pdf)
 - [Probability density function of the exponential distribution](/P/exp-pdf)
@@ -208,6 +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

@@ -17,6 +17,7 @@ title: "Proofs without Source"
 - [Joint likelihood is the product of likelihood function and prior density](/P/jl-lfnprior)
 - [Linear transformation theorem for the matrix-normal distribution](/P/matn-ltt)
 - [Log model evidence for multinomial observations](/P/mult-lme)
+- [Log model evidence for multivariate Bayesian linear regression](/P/mblr-lme)
 - [Log model evidence for the Poisson distribution with exposure values](/P/poissexp-lme)
 - [Marginal distributions of the multivariate normal distribution](/P/mvn-marg)
 - [Marginal distributions of the normal-gamma distribution](/P/ng-marg)
@@ -36,6 +37,7 @@ title: "Proofs without Source"
 - [Ordinary least squares for the general linear model](/P/glm-ols)
 - [Posterior density is proportional to joint likelihood](/P/post-jl)
 - [Probability density function of the Dirichlet distribution](/P/dir-pdf)
+- [Probability density function of the Wald distribution](/P/wald-pdf)
 - [Probability density function of the beta distribution](/P/beta-pdf)
 - [Probability density function of the continuous uniform distribution](/P/cuni-pdf)
 - [Probability density function of the exponential distribution](/P/exp-pdf)
@@ -58,6 +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

@@ -5,7 +5,7 @@ title: "Table of Contents"
 
 
 **[Proofs](/P/-temp-)** are printed in **bold** – *[Definitions](/D/-temp-)* are set in *italics* <br>
-Proofs by **[Number](/I/Proof_by_Number)** and **[Topic](/I/Proof_by_Topic)** – Definitions by *[Number](/I/Definition_by_Number)* and *[Topic](/I/Definition_by_Topic)* <br>
+**Proofs**: [by Number](/I/Proof_by_Number), [by Topic](/I/Proof_by_Topic) – *Definitions*:  [by Number](/I/Definition_by_Number), [by Topic](/I/Definition_by_Topic) <br>
 <u>Specials:</u> [Probability Distributions](/S/Probability_Distributions), [Statistical Models](/S/Statistical_Models), [Model Selection Criteria](/S/Model_Selection) <br>
 
 
@@ -89,7 +89,6 @@ Proofs by **[Number](/I/Proof_by_Number)** and **[Topic](/I/Proof_by_Topic)** 
    1.10. Further moments <br>
    &emsp;&ensp; 1.10.1. *[Moment](/D/mom)* <br>
    &emsp;&ensp; 1.10.2. **[Moment in terms of moment-generating function](/P/mom-mgf)** <br>
-  
 
 2. Information theory
 
@@ -256,6 +255,10 @@ Proofs by **[Number](/I/Proof_by_Number)** and **[Topic](/I/Proof_by_Topic)** 
    3.5. Beta distribution <br>
    &emsp;&ensp; 3.5.1. *[Definition](/D/beta)* <br>
    &emsp;&ensp; 3.5.2. **[Probability density function](/P/beta-pdf)** <br>
+   
+   3.6. Wald distribution <br>
+   &emsp;&ensp; 3.6.1. *[Definition](/D/wald)* <br>
+   &emsp;&ensp; 3.6.2. **[Probability density function](/P/wald-pdf)** <br>
 
 4. Multivariate continuous distributions
 
@@ -421,9 +424,10 @@ Proofs by **[Number](/I/Proof_by_Number)** and **[Topic](/I/Proof_by_Topic)** 
 
    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>