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updated special pages
The special pages were renamed to shortcuts and updated with the recently added proofs/definitions.
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‎404.html‎

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<div class="list">
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<ul>
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<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>
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<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>
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<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>
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<li>Alternatively, you may also use the <a href="/search">search</a> function of the website.</li>
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</ul>
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</div>

‎I/Table_of_Contents.md‎

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**[Proofs](/P/-temp-)** are printed in **bold** – *[Definitions](/D/-temp-)* are set in *italics* <br>
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**Proofs**: [by Number](/I/PbN), [by Topic](/I/PbT) – *Definitions*: [by Number](/I/DbN), [by Topic](/I/DbT) <br>
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<u>Specials:</u> [Probability Distributions](/S/Probability_Distributions), [Statistical Models](/S/Statistical_Models), [Model Selection Criteria](/S/Model_Selection) <br>
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<u>Specials:</u> [Probability Distributions](/S/ChII), [Statistical Models](/S/ChIII), [Model Selection Criteria](/S/ChIV) <br>
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‎I/ToC.md‎

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**[Proofs](/P/-temp-)** are printed in **bold** – *[Definitions](/D/-temp-)* are set in *italics* <br>
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**Proofs**: [by Number](/I/PbN), [by Topic](/I/PbT) – *Definitions*: [by Number](/I/DbN), [by Topic](/I/DbT) <br>
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<u>Specials:</u> [Probability Distributions](/S/Probability_Distributions), [Statistical Models](/S/Statistical_Models), [Model Selection Criteria](/S/Model_Selection) <br>
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<u>Specials:</u> [Probability Distributions](/S/ChII), [Statistical Models](/S/ChIII), [Model Selection Criteria](/S/ChIV) <br>
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‎S/ChII.md‎

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---
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layout: page
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title: "Special: Probability Distributions"
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---
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**Proofs** and *Definitions* on [Probability Distributions](/I/ToC#Probability%20Distributions) in the StatProofBook, as of 2021-11-05.
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| Distribution | Def | PDF | CDF | QF | MGF | Mean | Med | Mode | Var | Ent | KL | Marg | Cond | Other |
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|:------------ |:--- |:--- |:--- |:-- |:--- |:---- |:--- |:---- |:--- |:--- |:-- |:---- |:---- |:----- |
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| **Univariate<br>discrete<br>distributions** | | | | | | | | | | | | | | |
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| Discrete uniform distribution | *[duni](/D/duni)* | **[duni-pmf](/P/duni-pmf)** | **[duni-cdf](/P/duni-cdf)** | **[duni-qf](/P/duni-qf)** | | | | | | | | | | |
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| Bernoulli distribution | *[bern](/D/bern)* | **[bern-pmf](/P/bern-pmf)** | | | | **[bern-mean](/P/bern-mean)** | | | | | | | | |
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| Binomial distribution | *[bin](/D/bin)* | **[bin-pmf](/P/bin-pmf)** | | | | **[bin-mean](/P/bin-mean)** | | | | | | | | |
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| Poisson distribution | *[poiss](/D/poiss)* | **[poiss-pmf](/P/poiss-pmf)** | | | | **[poiss-mean](/P/poiss-mean)** | | | **[poiss-var](/P/poiss-var)** | | | | | |
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| **Multivariate<br>discrete<br>distributions** | | | | | | | | | | | | | | |
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| Categorical distribution | *[cat](/D/cat)* | **[cat-pmf](/P/cat-pmf)** | | | | **[cat-mean](/P/cat-mean)** | | | | | | | | |
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| Multinomial distribution | *[mult](/D/mult)* | **[mult-pmf](/P/mult-pmf)** | | | | **[mult-mean](/P/mult-mean)** | | | | | | | | |
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| **Univariate<br>continuous<br>distributions** | | | | | | | | | | | | | | |
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| Continuous uniform distribution | *[cuni](/D/cuni)* | **[cuni-pdf](/P/cuni-pdf)** | **[cuni-cdf](/P/cuni-cdf)** | **[cuni-qf](/P/cuni-qf)** | | **[cuni-mean](/P/cuni-mean)** | **[cuni-med](/P/cuni-med)** | **[cuni-mode](/P/cuni-mode)** | | | | | | |
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| Normal distribution | *[norm](/D/norm)*<br>*[snorm](/D/snorm)* | **[norm-pdf](/P/norm-pdf)** | **[norm-cdf](/P/norm-cdf)**<br>**[norm-cdfwerf](/P/norm-cdfwerf)** | **[norm-qf](/P/norm-qf)** | **[norm-mgf](/P/norm-mgf)** | **[norm-mean](/P/norm-mean)** | **[norm-med](/P/norm-med)** | **[norm-mode](/P/norm-mode)** | **[norm-var](/P/norm-var)** | **[norm-dent](/P/norm-dent)**<br>**[norm-maxent](/P/norm-maxent)** | **[norm-kl](/P/norm-kl)** | | | **[norm-snorm](/P/norm-snorm)**<br>**[norm-chi2(/P/norm-chi2)**<br>**[norm-t](/P/norm-t)**<br>**[norm-lincomb](/P/norm-lincomb)**<br>**[norm-gi](/P/norm-gi)**<br>**[norm-fwhm](/P/norm-fwhm)**<br>**[norm-extr](/P/norm-extr)**<br>**[norm-infl](/P/norm-infl)** |
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| t-distribution | *[t](/D/t)*<br>*[nst](/D/nst)* | **[t-pdf](/P/t-pdf)** | | | | | | | | | | | | **[nst-t](/P/nst-t)** |
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| Gamma distribution | *[gam](/D/gam)*<br>*[sgam](/D/sgam)* | **[gam-pdf](/P/gam-pdf)** | **[gam-cdf](/P/gam-cdf)** | **[gam-qf](/P/gam-qf)** | | **[gam-mean](/P/gam-mean)**<br>**[gam-logmean](/P/gam-logmean)**<br>**[gam-xlogx](/P/gam-xlogx)** | | | **[gam-var](/P/gam-var)** | **[gam-dent](/P/gam-dent)** | **[gam-kl](/P/gam-kl)** | | | **[gam-sgam](/P/gam-sgam)** |
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| Exponential distribution | *[exp](/D/exp)* | **[exp-pdf](/P/exp-pdf)** | **[exp-cdf](/P/exp-cdf)** | **[exp-qf](/P/exp-qf)** | | **[exp-mean](/P/exp-mean)** | **[exp-med](/P/exp-med)** | **[exp-mode](/P/exp-mode)** | | | | | | **[exp-gam](/P/exp-gam)** |
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| Chi-squared distribution | *[chi2](/D/chi2)* | **[chi2-pdf](/P/chi2-pdf)** | | | | | | | | | | | | **[chi2-gam](/P/chi2-gam)**<br>**[chi2-mom](/P/chi2-mom)** |
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| F-distribution | *[f](/D/f)* | **[f-pdf](/P/f-pdf)** | | | | | | | | | | | | |
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| Beta distribution | *[beta](/D/beta)* | **[beta-pdf](/P/beta-pdf)** | **[beta-cdf](/P/beta-cdf)** | | **[beta-mgf](/P/beta-mgf)** | **[beta-mean](/P/beta-mean)** | | | **[beta-var](/P/beta-var)** | | | | | |
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| Wald distribution | *[wald](/D/wald)* | **[wald-pdf](/P/wald-pdf)** | | | **[wald-mgf](/P/wald-mgf)** | **[wald-mean](/P/wald-mean)** | | | **[wald-var](/P/wald-var)** | | | | | |
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| **Multivariate<br>continuous<br>distributions** | | | | | | | | | | | | | | |
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| Multivariate normal distribution | *[mvn](/D/mvn)* | **[mvn-pdf](/P/mvn-pdf)** | | | | | | | | **[mvn-dent](/P/mvn-dent)** | **[mvn-kl](/P/mvn-kl)** | **[mvn-marg](/P/mvn-marg)** | **[mvn-cond](/P/mvn-cond)** | **[mvn-ltt](/P/mvn-ltt)**<br>**[mvn-ind](/P/mvn-ind)** |
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| Multivariate t-distribution | *[mvt](/D/mvt)* | | | | | | | | | | | | | **[mvt-f](/P/mvt-f)** |
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| Normal-gamma distribution | *[ng](/D/ng)* | **[ng-pdf](/P/ng-pdf)** | | | | **[ng-mean](/P/ng-mean)** | | | | **[ng-dent](/P/ng-dent)** | **[ng-kl](/P/ng-kl)** | **[ng-marg](/P/ng-marg)** | **[ng-cond](/P/ng-cond)** | |
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| Dirichlet distribution | *[dir](/D/dir)* | **[dir-pdf](/P/dir-pdf)** | | | | | | | | | | | | **[dir-ep](/P/dir-ep)** |
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| **Matrix-variate<br>continuous<br>distributions** | | | | | | | | | | | | | | |
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| Matrix-normal distribution | *[matn](/D/matn)* | **[matn-pdf](/P/matn-pdf)** | | | | | | | | | | | | **[matn-mvn](/P/matn-mvn)**<br>**[matn-ltt](/P/matn-ltt)**<br>**[matn-trans](/P/matn-trans)** |
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| Wishart distribution | *[wish](/D/wish)* | | | | | | | | | | | | | |

‎S/ChIII.md‎

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title: "Special: Statistical Models"
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**Proofs** and *Definitions* on [Statistical Models](/I/ToC#Statistical%20Models) in the StatProofBook, as of 2021-11-05.
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| Model | Definition | Estimation | Maximum<br>likelihood<br>estimation | Conjugate<br>prior | Posterior<br>distribution | Log<br>model<br>evidence | Other |
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|:----- |:---------- |:---------- |:----------------------------------- |:------------------ |:------------------------- |:------------------------ |:----- |
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| **Univariate<br>normal data** | | | | | | | |
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| Univariate Gaussian | *[ug](/D/ug)* | | **[ug-mle](/P/ug-mle)** | **[ug-prior](/P/ug-prior)** | **[ug-post](/P/ug-post)** | **[ug-lme](/P/ug-lme)**<br>**[ug-anc](/P/ug-anc)** | **[ug-ttest1](/P/ug-ttest1)**<br>**[ug-ttest2](/P/ug-ttest2)**<br>**[ug-ttestp](/P/ug-ttestp)** |
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| Univariate Gaussian with known variance | *[ugkv](/D/ugkv)* | | **[ugkv-mle](/P/ugkv-mle)** | **[ugkv-prior](/P/ugkv-prior)** | **[ugkv-post](/P/ugkv-post)** | **[ugkv-lme](/P/ugkv-lme)**<br>**[ugkv-anc](/P/ugkv-anc)**<br>**[ugkv-cvlme](/P/ugkv-cvlme)** | **[ugkv-ztest1](/P/ugkv-ztest1)**<br>**[ugkv-ztest2](/P/ugkv-ztest2)**<br>**[ugkv-ztestp](/P/ugkv-ztestp)**<br>**[ugkv-lbf](/P/ugkv-lbf)**<br>**[ugkv-lbfmean](/P/ugkv-lbfmean)**<br>**[ugkv-cvlbf](/P/ugkv-cvlbf)**<br>**[ugkv-cvlbfmean](/P/ugkv-cvlbfmean)** |
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| Simple linear regression | *[slr](/D/slr)* | **[slr-ols](/P/slr-ols)**<br>**[slr-olsmean](/P/slr-olsmean)**<br>**[slr-olsvar](/P/slr-olsvar)** | | | | | **[slr-meancent](/P/slr-meancent)**<br>**[slr-comp](/P/slr-comp)**<br>**[slr-ressum](/P/slr-ressum)**<br>**[slr-rescorr](/P/slr-rescorr)**<br>**[slr-resvar](/P/slr-resvar)**<br>**[slr-corr](/P/slr-corr)**<br>**[slr-rsq](/P/slr-rsq)** |
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| Multiple linear regression | *[mlr](/D/mlr)* | **[mlr-ols](/P/mlr-ols)**<br>**[mlr-wls](/P/mlr-wls)** | **[mlr-mle](/P/mlr-mle)** | | | | **[mlr-pss](/P/mlr-pss)**<br>**[mlr-mat](/P/mlr-mat)**<br>**[mlr-idem](/P/mlr-idem)** |
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| Bayesian linear regression | | | | **[blr-prior](/P/blr-prior)** | **[blr-post](/P/blr-post)** | **[blr-lme](/P/blr-lme)** | **[blr-pp](/P/blr-pp)**<br>**[blr-pcr](/P/blr-pcr)** |
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| **Multivariate<br>normal data** | | | | | | | |
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| General linear model | *[glm](/D/glm)* | **[glm-ols](/P/glm-ols)**<br>**[glm-wls](/P/glm-wls)** | **[glm-mle](/P/glm-mle)** | | | | |
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| Transformed general linear model | *[tglm](/D/tglm)* | | | | | | **[tglm-dist](/P/tglm-dist)**<br>**[tglm-para](/P/tglm-para)** |
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| Inverse general linear model | *[iglm](/D/iglm)*<br>*[cfm](/D/cfm)* | | | | | | **[iglm-dist](/P/iglm-dist)**<br>**[iglm-blue](/P/iglm-blue)**<br>**[cfm-para](/P/cfm-para)**<br>**[cfm-exist](/P/cfm-exist)** |
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| Multivariate Bayesian linear regression | | | | **[mblr-prior](/P/mblr-prior)** | **[mblr-post](/P/mblr-post)** | **[mblr-lme](/P/mblr-lme)** | |
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| **Poisson data** | | | | | | | |
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| Poisson-distributed data | *[poiss](/D/poiss-data)* | | **[poiss-mle](/P/poiss-mle)** | **[poiss-prior](/P/poiss-prior)** | **[poiss-post](/P/poiss-post)** | **[poiss-lme](/P/poiss-lme)** | |
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| Poisson distribution with exposure values | *[poissexp](/D/poissexp)* | | **[poissexp-mle](/P/poissexp-mle)** | **[poissexp-prior](/P/poissexp-prior)** | **[poissexp-post](/P/poissexp-post)** | **[poissexp-lme](/P/poissexp-lme)** | |
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| **Probability data** | | | | | | | |
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| Beta-distributed data | *[beta](/D/beta-data)* | **[beta-mom](/P/beta-mom)** | | | | | |
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| Dirichlet-distributed data | *[dir](/D/dir-data)* | | **[dir-mle](/P/dir-mle)** | | | | |
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| **Categorical data** | | | | | | | |
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| Binomial observations | *[bin](/D/bin-data)* | | | **[bin-prior](/P/bin-prior)** | **[bin-post](/P/bin-post)** | **[bin-lme](/P/bin-lme)** | |
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| Multinomial observations | *[mult](/D/mult-data)* | | | **[mult-prior](/P/mult-prior)** | **[mult-post](/P/mult-post)** | **[mult-lme](/P/mult-lme)** | |
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| Logistic regression | *[logreg](/D/logreg)* | | | | | | **[logreg-pnlo](/P/logreg-pnlo)**<br>**[logreg-lonp](/P/logreg-lonp)** |

‎S/ChIV.md‎

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title: "Special: Model Selection Criteria"
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**Proofs** and *Definitions* on [Model Selection Criteria](/I/ToC#Model%20Selection) in the StatProofBook, as of 2021-11-05.
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| Criterion | Definition | Derivation | Relationships | Other |
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|:--------- |:---------- |:---------- |:------------- |:----- |
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| **Goodness-of-<br>fit measures** | | | | |
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| Residual variance | *[resvar](/D/resvar)* | | | **[resvar-bias](/P/resvar-bias)**<br>**[resvar-unb](/P/resvar-unb)** |
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| R-squared | *[rsq](/D/rsq)* | **[rsq-der](/P/rsq-der)** | **[rsq-mll](/P/rsq-mll)** | |
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| Signal-to-noise ratio | *[snr](/D/snr)* | | **[snr-rsq](/P/snr-rsq)** | |
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| **Classical infor-<br>mation criteria** | | | | |
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| Akaike information criterion | *[aic](/D/aic)* | | | |
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| Bayesian information criterion | *[bic](/D/bic)* | **[bic-der](/P/bic-der)** | | |
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| Deviance information criterion | *[dic](/D/dic)* | | | |
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| **Bayesian<br>model selection** | | | | |
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| Log model evidence | *[lme](/D/lme)*<br>*[uplme](/D/uplme)*<br>*[cvlme](/D/cvlme)*<br>*[eblme](/D/eblme)*<br>*[vblme](/D/vblme)* | **[lme-der](/P/lme-der)** | | **[lme-anc](/P/lme-anc)** |
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| Log family evidence | *[lfe](/D/lfe)* | **[lfe-der](/P/lfe-der)** | **[lfe-lme](/P/lfe-lme)** | |
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| Log Bayes factor | *[lbf](/D/lbf)* | **[lbf-der](/P/lbf-der)** | **[lbf-lme](/P/lbf-lme)** | |
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| Bayes factor | *[bf](/D/bf)* | | | **[bf-trans](/P/bf-trans)**<br>**[bf-sddr](/P/bf-sddr)**<br>**[bf-ep](/P/bf-ep)** |
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| Posterior model probability | *[pmp](/D/pmp)* | **[pmp-der](/P/pmp-der)** | **[pmp-bf](/P/pmp-bf)**<br>**[pmp-lbf](/P/pmp-lbf)**<br>**[pmp-lme](/P/pmp-lme)** | |
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| Bayesian model averaging | *[bma](/D/bma)* | **[bma-der](/P/bma-der)** | **[bma-lme](/P/bma-lme)** | |

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