ID Shortcut Theorem Author Date
P0 -temp- Proof Template StatProofBook 2019-09-27
P1 mvn-ltt Linear transformation theorem for the multivariate normal distribution JoramSoch 2019-08-27
P2 mlr-ols Ordinary least squares for multiple linear regression JoramSoch 2019-09-27
P3 lme-anc Partition of the log model evidence into accuracy and complexity JoramSoch 2019-09-27
P4 bayes-th Bayes’ theorem JoramSoch 2019-09-27
P5 mse-bnv Partition of the mean squared error into bias and variance JoramSoch 2019-11-27
P6 ng-kl Kullback-Leibler divergence for the normal-gamma distribution JoramSoch 2019-12-06
P7 glm-mle Maximum likelihood estimation for the general linear model JoramSoch 2019-12-06
P8 rsq-der Derivation of R² and adjusted R² JoramSoch 2019-12-06
P9 blr-prior Conjugate prior distribution for Bayesian linear regression JoramSoch 2020-01-03
P10 blr-post Posterior distribution for Bayesian linear regression JoramSoch 2020-01-03
P11 blr-lme Log model evidence for Bayesian linear regression JoramSoch 2020-01-03
P12 bayes-rule Bayes’ rule JoramSoch 2020-01-06
P13 lme-der Derivation of the log model evidence JoramSoch 2020-01-06
P14 rsq-mll Relationship between R² and maximum log-likelihood JoramSoch 2020-01-08
P15 norm-mean Mean of the normal distribution JoramSoch 2020-01-09
P16 norm-med Median of the normal distribution JoramSoch 2020-01-09
P17 norm-mode Mode of the normal distribution JoramSoch 2020-01-09
P18 norm-var Variance of the normal distribution JoramSoch 2020-01-09
P19 dmi-mce Relation of mutual information to marginal and conditional entropy JoramSoch 2020-01-13
P20 dmi-mje Relation of mutual information to marginal and joint entropy JoramSoch 2020-01-13
P21 dmi-jce Relation of mutual information to joint and conditional entropy JoramSoch 2020-01-13
P22 bern-mean Mean of the Bernoulli distribution JoramSoch 2020-01-16
P23 bin-mean Mean of the binomial distribution JoramSoch 2020-01-16
P24 cat-mean Mean of the categorical distribution JoramSoch 2020-01-16
P25 mult-mean Mean of the multinomial distribution JoramSoch 2020-01-16
P26 matn-mvn Equivalence of matrix-normal distribution and multivariate normal distribution JoramSoch 2020-01-20
P27 poiss-mle Maximum likelihood estimation for Poisson-distributed data JoramSoch 2020-01-20
P28 beta-mome Method of moments for beta-distributed data JoramSoch 2020-01-22
P29 bin-prior Conjugate prior distribution for binomial observations JoramSoch 2020-01-23
P30 bin-post Posterior distribution for binomial observations JoramSoch 2020-01-24
P31 bin-lme Log model evidence for binomial observations JoramSoch 2020-01-24
P32 bic-der Derivation of the Bayesian information criterion JoramSoch 2020-01-26
P33 norm-pdf Probability density function of the normal distribution JoramSoch 2020-01-27
P34 mvn-pdf Probability density function of the multivariate normal distribution JoramSoch 2020-01-27
P35 mvn-marg Marginal distributions of the multivariate normal distribution JoramSoch 2020-01-29
P36 ng-marg Marginal distributions of the normal-gamma distribution JoramSoch 2020-01-29
P37 cuni-pdf Probability density function of the continuous uniform distribution JoramSoch 2020-01-31
P38 cuni-cdf Cumulative distribution function of the continuous uniform distribution JoramSoch 2020-01-02
P39 cuni-qf Quantile function of the continuous uniform distribution JoramSoch 2020-01-02
P40 mlr-ols2 Ordinary least squares for multiple linear regression JoramSoch 2020-02-03
P41 poissexp-prior Conjugate prior distribution for the Poisson distribution with exposure values JoramSoch 2020-02-04
P42 poissexp-post Posterior distribution for the Poisson distribution with exposure values JoramSoch 2020-02-04
P43 poissexp-lme Log model evidence for the Poisson distribution with exposure values JoramSoch 2020-02-04
P44 ng-pdf Probability density function of the normal-gamma distribution JoramSoch 2020-02-07
P45 gam-pdf Probability density function of the gamma distribution JoramSoch 2020-02-08
P46 exp-pdf Probability density function of the exponential distribution JoramSoch 2020-02-08
P47 exp-mean Mean of the exponential distribution JoramSoch 2020-02-10
P48 exp-cdf Cumulative distribution function of the exponential distribution JoramSoch 2020-02-11
P49 exp-med Median of the exponential distribution JoramSoch 2020-02-11
P50 exp-qf Quantile function of the exponential distribution JoramSoch 2020-02-12
P51 exp-mode Mode of the exponential distribution JoramSoch 2020-02-12
P52 mean-nonneg Non-negativity of the expected value JoramSoch 2020-02-13
P53 mean-lin Linearity of the expected value JoramSoch 2020-02-13
P54 mean-mono Monotonicity of the expected value JoramSoch 2020-02-17
P55 mean-mult (Non-)Multiplicativity of the expected value JoramSoch 2020-02-17
P56 ci-wilks Construction of confidence intervals using Wilks’ theorem JoramSoch 2020-02-19
P57 ent-nonneg Non-negativity of the Shannon entropy JoramSoch 2020-02-19
P58 cmi-mcde Relation of continuous mutual information to marginal and conditional differential entropy JoramSoch 2020-02-21
P59 cmi-mjde Relation of continuous mutual information to marginal and joint differential entropy JoramSoch 2020-02-21
P60 cmi-jcde Relation of continuous mutual information to joint and conditional differential entropy JoramSoch 2020-02-21
P61 resvar-bias Maximum likelihood estimator of variance is biased JoramSoch 2020-02-24
P62 resvar-unb Construction of unbiased estimator for variance JoramSoch 2020-02-25
P63 snr-rsq Relationship between signal-to-noise ratio and R² JoramSoch 2020-02-26
P64 lbf-lme Log Bayes factor in terms of log model evidences JoramSoch 2020-02-27
P65 lfe-lme Log family evidences in terms of log model evidences JoramSoch 2020-02-27
P66 pmp-lme Posterior model probabilities in terms of log model evidences JoramSoch 2020-02-27
P67 bma-lme Bayesian model averaging in terms of log model evidences JoramSoch 2020-02-27
P68 dent-neg Differential entropy can be negative JoramSoch 2020-03-02
P69 exp-gam Exponential distribution is a special case of gamma distribution JoramSoch 2020-03-02
P70 matn-pdf Probability density function of the matrix-normal distribution JoramSoch 2020-03-02
P71 norm-mgf Moment-generating function of the normal distribution JoramSoch 2020-03-03
P72 logreg-lonp Log-odds and probability in logistic regression JoramSoch 2020-03-03
P73 pmp-lbf Posterior model probability in terms of log Bayes factor JoramSoch 2020-03-03
P74 pmp-bf Posterior model probabilities in terms of Bayes factors JoramSoch 2020-03-03
P75 mlr-mat Transformation matrices for ordinary least squares JoramSoch 2020-03-09
P76 mlr-pss Partition of sums of squares in ordinary least squares JoramSoch 2020-03-09
P77 mlr-wls Weighted least squares for multiple linear regression JoramSoch 2020-03-11
P78 mlr-mle Maximum likelihood estimation for multiple linear regression JoramSoch 2020-03-11
P79 mult-prior Conjugate prior distribution for multinomial observations JoramSoch 2020-03-11
P80 mult-post Posterior distribution for multinomial observations JoramSoch 2020-03-11
P81 mult-lme Log model evidence for multinomial observations JoramSoch 2020-03-11
P82 cuni-mean Mean of the continuous uniform distribution JoramSoch 2020-03-16
P83 cuni-med Median of the continuous uniform distribution JoramSoch 2020-03-16
P84 cuni-med Mode of the continuous uniform distribution JoramSoch 2020-03-16
P85 norm-cdf Cumulative distribution function of the normal distribution JoramSoch 2020-03-20
P86 norm-cdfwerf Expression of the cumulative distribution function of the normal distribution without the error function JoramSoch 2020-03-20
P87 norm-qf Quantile function of the normal distribution JoramSoch 2020-03-20
P88 mvn-cond Conditional distributions of the multivariate normal distribution JoramSoch 2020-03-20
P89 jl-lfnprior Joint likelihood is the product of likelihood function and prior density JoramSoch 2020-05-05
P90 post-jl Posterior density is proportional to joint likelihood JoramSoch 2020-05-05
P91 ml-jl Marginal likelihood is a definite integral of joint likelihood JoramSoch 2020-05-05
P92 mvn-kl Kullback-Leibler divergence for the multivariate normal distribution JoramSoch 2020-05-05
P93 gam-kl Kullback-Leibler divergence for the gamma distribution JoramSoch 2020-05-05
P94 beta-pdf Probability density function of the beta distribution JoramSoch 2020-05-05
P95 dir-pdf Probability density function of the Dirichlet distribution JoramSoch 2020-05-05
P96 bern-pmf Probability mass function of the Bernoulli distribution JoramSoch 2020-05-11
P97 bin-pmf Probability mass function of the binomial distribution JoramSoch 2020-05-11
P98 cat-pmf Probability mass function of the categorical distribution JoramSoch 2020-05-11
P99 mult-pmf Probability mass function of the multinomial distribution JoramSoch 2020-05-11
P100 mvn-dent Differential entropy of the multivariate normal distribution JoramSoch 2020-05-14
P101 norm-dent Differential entropy of the normal distribution JoramSoch 2020-05-14
P102 poiss-pmf Probability mass function of the Poisson distribution JoramSoch 2020-05-14
P103 mean-nnrvar Expected value of a non-negative random variable JoramSoch 2020-05-18
P104 var-mean Partition of variance into expected values JoramSoch 2020-05-19
P105 logreg-pnlo Probability and log-odds in logistic regression JoramSoch 2020-05-19
P106 glm-ols Ordinary least squares for the general linear model JoramSoch 2020-05-19
P107 glm-wls Weighted least squares for the general linear model JoramSoch 2020-05-19
P108 gam-mean Mean of the gamma distribution JoramSoch 2020-05-19
P109 gam-var Variance of the gamma distribution JoramSoch 2020-05-19
P110 gam-logmean Logarithmic expectation of the gamma distribution JoramSoch 2020-05-25
P111 norm-snorm Relationship between normal distribution and standard normal distribution JoramSoch 2020-05-26
P112 gam-sgam Relationship between gamma distribution and standard gamma distribution JoramSoch 2020-05-26
P113 kl-ent Relation of Kullback-Leibler divergence to entropy JoramSoch 2020-05-27
P114 kl-dent Relation of continuous Kullback-Leibler divergence to differential entropy JoramSoch 2020-05-27
P115 kl-inv Invariance of the Kullback-Leibler divergence under parameter transformation JoramSoch 2020-05-28
P116 kl-add Additivity of the Kullback-Leibler divergence for independent distributions JoramSoch 2020-05-31
P117 kl-nonneg Non-negativity of the Kullback-Leibler divergence JoramSoch 2020-05-31
P118 cov-mean Partition of covariance into expected values JoramSoch 2020-06-02
P119 cov-corr Relationship between covariance and correlation JoramSoch 2020-06-02
P120 covmat-mean Partition of a covariance matrix into expected values JoramSoch 2020-06-06
P121 covmat-corrmat Relationship between covariance matrix and correlation matrix JoramSoch 2020-06-06
P122 precmat-corrmat Relationship between precision matrix and correlation matrix JoramSoch 2020-06-06
P123 var-nonneg Non-negativity of the variance JoramSoch 2020-06-06
P124 var-const Variance of constant is zero JoramSoch 2020-06-27
P126 var-inv Invariance of the variance under addition of a constant JoramSoch 2020-07-07
P127 var-scal Scaling of the variance upon multiplication with a constant JoramSoch 2020-07-07
P128 var-sum Variance of the sum of two random variables JoramSoch 2020-07-07
P129 var-lincomb Variance of the linear combination of two random variables JoramSoch 2020-07-07
P130 var-add Additivity of the variance for independent random variables JoramSoch 2020-07-07
P131 mean-qf Expectation of a quadratic form JoramSoch 2020-07-13
P132 lfe-der Derivation of the log family evidence JoramSoch 2020-07-13
P133 blr-pp Posterior probability of the alternative hypothesis for Bayesian linear regression JoramSoch 2020-07-17
P134 blr-pcr Posterior credibility region against the omnibus null hypothesis for Bayesian linear regression JoramSoch 2020-07-17
P135 mlr-idem Projection matrix and residual-forming matrix are idempotent JoramSoch 2020-07-22
P136 mlr-wls2 Weighted least squares for multiple linear regression JoramSoch 2020-07-22
P137 lbf-der Derivation of the log Bayes factor JoramSoch 2020-07-22
P138 mean-lotus Law of the unconscious statistician JoramSoch 2020-07-22
P139 pmp-der Derivation of the posterior model probability JoramSoch 2020-07-28
P140 duni-pmf Probability mass function of the discrete uniform distribution JoramSoch 2020-07-28
P141 duni-cdf Cumulative distribution function of the discrete uniform distribution JoramSoch 2020-07-28
P142 duni-qf Quantile function of the discrete uniform distribution JoramSoch 2020-07-28
P143 bma-der Derivation of Bayesian model averaging JoramSoch 2020-08-03
P144 matn-trans Transposition of a matrix-normal random variable JoramSoch 2020-08-03
P145 matn-ltt Linear transformation theorem for the matrix-normal distribution JoramSoch 2020-08-03
P146 ng-cond Conditional distributions of the normal-gamma distribution JoramSoch 2020-08-05
P147 kl-nonsymm Non-symmetry of the Kullback-Leibler divergence JoramSoch 2020-08-11
P148 kl-conv Convexity of the Kullback-Leibler divergence JoramSoch 2020-08-11
P149 ent-conc Concavity of the Shannon entropy JoramSoch 2020-08-11
P150 entcross-conv Convexity of the cross-entropy JoramSoch 2020-08-11
P151 poiss-mean Mean of the Poisson distribution JoramSoch 2020-08-19
P152 norm-fwhm Full width at half maximum for the normal distribution JoramSoch 2020-08-19
P153 mom-mgf Moment in terms of moment-generating function JoramSoch 2020-08-19
P154 mgf-ltt Linear transformation theorem for the moment-generating function JoramSoch 2020-08-19
P155 mgf-lincomb Moment-generating function of linear combination of independent random variables JoramSoch 2020-08-19
P156 bf-sddr Savage-Dickey Density Ratio for computing Bayes Factors tomfaulkenberry 2020-08-26
P157 bf-ep Encompassing Prior Method for computing Bayes Factors tomfaulkenberry 2020-09-02
P158 cov-ind Covariance of independent random variables JoramSoch 2020-09-03
P159 mblr-prior Conjugate prior distribution for multivariate Bayesian linear regression JoramSoch 2020-09-03
P160 mblr-post Posterior distribution for multivariate Bayesian linear regression JoramSoch 2020-09-03
P161 mblr-lme Log model evidence for multivariate Bayesian linear regression JoramSoch 2020-09-03
P162 wald-pdf Probability density function of the Wald distribution tomfaulkenberry 2020-09-04
P163 bf-trans Transitivity of Bayes Factors tomfaulkenberry 2020-09-07
P164 gibbs-ineq Gibbs’ inequality JoramSoch 2020-09-09
P165 logsum-ineq Log sum inequality JoramSoch 2020-09-09
P166 kl-nonneg2 Non-negativity of the Kullback-Leibler divergence JoramSoch 2020-09-09
P167 momcent-1st First central moment is zero JoramSoch 2020-09-09
P168 wald-mgf Moment-generating function of the Wald distribution tomfaulkenberry 2020-09-13
P169 wald-mean Mean of the Wald distribution tomfaulkenberry 2020-09-13
P170 wald-var Variance of the Wald distribution tomfaulkenberry 2020-09-13
P171 momraw-1st First raw moment is mean JoramSoch 2020-10-08
P172 momraw-2nd Relationship between second raw moment, variance and mean JoramSoch 2020-10-08
P173 momcent-2nd Second central moment is variance JoramSoch 2020-10-08
P174 chi2-gam Chi-squared distribution is a special case of gamma distribution kjpetrykowski 2020-10-12
P175 chi2-mom Moments of the chi-squared distribution kjpetrykowski 2020-10-13
P176 norm-snorm2 Relationship between normal distribution and standard normal distribution JoramSoch 2020-10-15
P177 gam-sgam2 Relationship between gamma distribution and standard gamma distribution JoramSoch 2020-10-15
P178 gam-cdf Cumulative distribution function of the gamma distribution JoramSoch 2020-10-15
P179 gam-xlogx Expected value of x times ln(x) for a gamma distribution JoramSoch 2020-10-15
P180 norm-snorm3 Relationship between normal distribution and standard normal distribution JoramSoch 2020-10-22
P181 dir-ep Exceedance probabilities for the Dirichlet distribution JoramSoch 2020-10-22
P182 dir-mle Maximum likelihood estimation for Dirichlet-distributed data JoramSoch 2020-10-22
P183 cdf-sifct Cumulative distribution function of a strictly increasing function of a random variable JoramSoch 2020-10-29
P184 pmf-sifct Probability mass function of a strictly increasing function of a discrete random variable JoramSoch 2020-10-29
P185 pdf-sifct Probability density function of a strictly increasing function of a continuous random variable JoramSoch 2020-10-29
P186 cdf-sdfct Cumulative distribution function of a strictly decreasing function of a random variable JoramSoch 2020-11-06
P187 pmf-sdfct Probability mass function of a strictly decreasing function of a discrete random variable JoramSoch 2020-11-06
P188 pdf-sdfct Probability density function of a strictly decreasing function of a continuous random variable JoramSoch 2020-11-06
P189 cdf-pmf Cumulative distribution function in terms of probability mass function of a discrete random variable JoramSoch 2020-11-12
P190 cdf-pdf Cumulative distribution function in terms of probability density function of a continuous random variable JoramSoch 2020-11-12
P191 pdf-cdf Probability density function is first derivative of cumulative distribution function JoramSoch 2020-11-12
P192 qf-cdf Quantile function is inverse of strictly monotonically increasing cumulative distribution function JoramSoch 2020-11-12
P193 norm-kl Kullback-Leibler divergence for the normal distribution JoramSoch 2020-11-19
P194 gam-qf Quantile function of the gamma distribution JoramSoch 2020-11-19
P195 beta-cdf Cumulative distribution function of the beta distribution JoramSoch 2020-11-19
P196 norm-gi Gaussian integral JoramSoch 2020-11-25
P197 chi2-pdf Probability density function of the chi-squared distribution JoramSoch 2020-11-25
P198 beta-mgf Moment-generating function of the beta distribution JoramSoch 2020-11-25
P199 dent-inv Invariance of the differential entropy under addition of a constant JoramSoch 2020-12-02
P200 dent-add Addition of the differential entropy upon multiplication with a constant JoramSoch 2020-12-02
P201 ug-prior Conjugate prior distribution for the univariate Gaussian JoramSoch 2021-03-03
P202 ug-post Posterior distribution for the univariate Gaussian JoramSoch 2021-03-03
P203 ug-lme Log model evidence for the univariate Gaussian JoramSoch 2021-03-03
P204 ug-ttest1 One-sample t-test for independent observations JoramSoch 2021-03-12
P205 ug-ttest2 Two-sample t-test for independent observations JoramSoch 2021-03-12
P206 ug-ttestp Paired t-test for dependent observations JoramSoch 2021-03-12
P207 ugkv-mle Maximum likelihood estimation for the univariate Gaussian with known variance JoramSoch 2021-03-24
P208 ugkv-ztest1 One-sample z-test for independent observations JoramSoch 2021-03-24
P209 ugkv-ztest2 Two-sample z-test for independent observations JoramSoch 2021-03-24
P210 ugkv-ztestp Paired z-test for dependent observations JoramSoch 2021-03-24
P211 ugkv-prior Conjugate prior distribution for the univariate Gaussian with known variance JoramSoch 2021-03-24
P212 ugkv-post Posterior distribution for the univariate Gaussian with known variance JoramSoch 2021-03-24
P213 ugkv-lme Log model evidence for the univariate Gaussian with known variance JoramSoch 2021-03-24
P214 ugkv-anc Accuracy and complexity for the univariate Gaussian with known variance JoramSoch 2021-03-24
P215 ugkv-lbf Log Bayes factor for the univariate Gaussian with known variance JoramSoch 2021-03-24
P216 ugkv-lbfmean Expectation of the log Bayes factor for the univariate Gaussian with known variance JoramSoch 2021-03-24
P217 ugkv-cvlme Cross-validated log model evidence for the univariate Gaussian with known variance JoramSoch 2021-03-24
P218 ugkv-cvlbf Cross-validated log Bayes factor for the univariate Gaussian with known variance JoramSoch 2021-03-24
P219 ugkv-cvlbfmean Expectation of the cross-validated log Bayes factor for the univariate Gaussian with known variance JoramSoch 2021-03-24
P220 cdf-pit Probability integral transform using cumulative distribution function JoramSoch 2021-04-07
P221 cdf-itm Inverse transformation method using cumulative distribution function JoramSoch 2021-04-07
P222 cdf-dt Distributional transformation using cumulative distribution function JoramSoch 2021-04-07
P223 ug-mle Maximum likelihood estimation for the univariate Gaussian JoramSoch 2021-04-16
P224 poissexp-mle Maximum likelihood estimation for the Poisson distribution with exposure values JoramSoch 2021-04-16
P225 poiss-prior Conjugate prior distribution for Poisson-distributed data JoramSoch 2020-04-21
P226 poiss-post Posterior distribution for Poisson-distributed data JoramSoch 2020-04-21
P227 poiss-lme Log model evidence for Poisson-distributed data JoramSoch 2020-04-21
P228 beta-mean Mean of the beta distribution JoramSoch 2021-04-29
P229 beta-var Variance of the beta distribution JoramSoch 2021-04-29
P230 poiss-var Variance of the Poisson distribution JoramSoch 2021-04-29
P231 mvt-f Relationship between multivariate t-distribution and F-distribution JoramSoch 2021-05-04
P232 nst-t Relationship between non-standardized t-distribution and t-distribution JoramSoch 2021-05-11
P233 norm-chi2 Relationship between normal distribution and chi-squared distribution JoramSoch 2021-05-20
P234 norm-t Relationship between normal distribution and t-distribution JoramSoch 2021-05-27
P235 norm-lincomb Linear combination of independent normal random variables JoramSoch 2021-06-02
P236 mvn-ind Necessary and sufficient condition for independence of multivariate normal random variables JoramSoch 2021-06-02