Proof by Topic

A

• Accuracy and complexity for Bayesian linear regression
• Accuracy and complexity for Bayesian linear regression with known covariance
• Accuracy and complexity for the univariate Gaussian
• Accuracy and complexity for the univariate Gaussian with known variance
• Addition law of probability
• Addition of the differential entropy upon multiplication with a constant
• Addition of the differential entropy upon multiplication with invertible matrix
• Additivity of the Kullback-Leibler divergence for independent distributions
• Additivity of the variance for independent random variables
• Akaike information criterion for multiple linear regression
• Application of Cochran’s theorem to two-way analysis of variance
• Approximation of log family evidences based on log model evidences

B

• Bayes’ rule
• Bayes’ theorem
• Bayesian information criterion for multiple linear regression
• Bayesian model averaging in terms of log model evidences
• Best linear unbiased estimator for the inverse general linear model
• Binomial test
• Brier scoring rule is strictly proper scoring rule

C

• Characteristic function of a function of a random variable
• Chi-squared distribution is a special case of gamma distribution
• Combined posterior distribution for Bayesian linear regression when analyzing conditionally independent data sets
• Combined posterior distributions in terms of individual posterior distributions obtained from conditionally independent data
• Concavity of the Shannon entropy
• Conditional distributions of the multivariate normal distribution
• Conditional distributions of the normal-gamma distribution
• Conjugate prior distribution for Bayesian linear regression
• Conjugate prior distribution for Bayesian linear regression with known covariance
• Conjugate prior distribution for binomial observations
• Conjugate prior distribution for multinomial observations
• Conjugate prior distribution for multivariate Bayesian linear regression
• Conjugate prior distribution for Poisson-distributed data
• Conjugate prior distribution for the Poisson distribution with exposure values
• Conjugate prior distribution for the univariate Gaussian
• Conjugate prior distribution for the univariate Gaussian with known variance
• Construction of confidence intervals using Wilks’ theorem
• Construction of unbiased estimator for variance
• Construction of unbiased estimator for variance in multiple linear regression
• Continuous uniform distribution maximizes differential entropy for fixed range
• Convexity of the cross-entropy
• Convexity of the Kullback-Leibler divergence
• Corrected Akaike information criterion converges to uncorrected Akaike information criterion when infinite data are available
• Corrected Akaike information criterion for multiple linear regression
• Corrected Akaike information criterion in terms of maximum log-likelihood
• Correlation always falls between -1 and +1
• Correlation coefficient in terms of standard scores
• Covariance and variance of the normal-gamma distribution
• Covariance matrices of the matrix-normal distribution
• Covariance matrix of the categorical distribution
• Covariance matrix of the multinomial distribution
• Covariance matrix of the multivariate normal distribution
• Covariance matrix of the sum of two random vectors
• Covariance of independent random variables
• Cross-validated log Bayes factor for the univariate Gaussian with known variance
• Cross-validated log model evidence for the univariate Gaussian with known variance
• Cumulative distribution function in terms of probability density function of a continuous random variable
• Cumulative distribution function in terms of probability mass function of a discrete random variable
• Cumulative distribution function of a strictly decreasing function of a random variable
• Cumulative distribution function of a strictly increasing function of a random variable
• Cumulative distribution function of a sum of independent random variables
• Cumulative distribution function of the beta distribution
• Cumulative distribution function of the beta-binomial distribution
• Cumulative distribution function of the continuous uniform distribution
• Cumulative distribution function of the discrete uniform distribution
• Cumulative distribution function of the exponential distribution
• Cumulative distribution function of the gamma distribution
• Cumulative distribution function of the log-normal distribution
• Cumulative distribution function of the normal distribution

D

• Derivation of Bayesian model averaging
• Derivation of R² and adjusted R²
• Derivation of the Bayesian information criterion
• Derivation of the family evidence
• Derivation of the log Bayes factor
• Derivation of the log family evidence
• Derivation of the log model evidence
• Derivation of the model evidence
• Derivation of the posterior model probability
• Deviance for multiple linear regression
• Deviance information criterion for multiple linear regression
• Differential entropy can be negative
• Differential entropy for the matrix-normal distribution
• Differential entropy of the continuous uniform distribution
• Differential entropy of the gamma distribution
• Differential entropy of the multivariate normal distribution
• Differential entropy of the normal distribution
• Differential entropy of the normal-gamma distribution
• Discrete uniform distribution maximizes entropy for finite support
• Distribution of parameter estimates for simple linear regression
• Distribution of residual sum of squares in multiple linear regression with weighted least squares
• Distribution of the inverse general linear model
• Distribution of the transformed general linear model
• Distributional transformation using cumulative distribution function
• Distributions of estimated parameters, fitted signal and residuals in multiple linear regression upon ordinary least squares
• Distributions of estimated parameters, fitted signal and residuals in multiple linear regression upon weighted least squares

E

• Effects of mean-centering on parameter estimates for simple linear regression
• Encompassing prior method for computing Bayes factors
• Entropy of the Bernoulli distribution
• Entropy of the binomial distribution
• Entropy of the categorical distribution
• Entropy of the discrete uniform distribution
• Entropy of the multinomial distribution
• Equivalence of log-likelihood ratio and mutual information for the general linear model
• Equivalence of log-likelihood ratios for regular and inverse general linear model
• Equivalence of matrix-normal distribution and multivariate normal distribution
• Equivalence of operations for model evidence and log model evidence
• Equivalence of parameter estimates from the transformed general linear model
• Exceedance probabilities for the Dirichlet distribution
• Exceedance probability for a random variable in terms of cumulative distribution function
• Existence of a corresponding forward model
• Expectation of a quadratic form
• Expectation of parameter estimates for simple linear regression
• Expectation of the cross-validated log Bayes factor for the univariate Gaussian with known variance
• Expectation of the log Bayes factor for the univariate Gaussian with known variance
• Expected value of a non-negative random variable
• Expected value of the trace of a matrix
• Expected value of x times ln(x) for a gamma distribution
• Exponential distribution is a special case of gamma distribution
• Expression of R² in terms of residual variances
• Expression of the cumulative distribution function of the normal distribution without the error function
• Expression of the noise precision posterior for Bayesian linear regression using prediction and parameter errors
• Expression of the probability mass function of the beta-binomial distribution using only the gamma function
• Extreme points of the probability density function of the normal distribution

F

• F-statistic in terms of ordinary least squares estimates in one-way analysis of variance
• F-statistics in terms of ordinary least squares estimates in two-way analysis of variance
• F-test for equality of variances in two independent samples
• F-test for grand mean in two-way analysis of variance
• F-test for interaction in two-way analysis of variance
• F-test for main effect in one-way analysis of variance
• F-test for main effect in two-way analysis of variance
• F-test for multiple linear regression using contrast-based inference
• First central moment is zero
• First raw moment is mean
• Full width at half maximum for the normal distribution

G

• Gamma distribution is a special case of Wishart distribution
• Gaussian integral
• Gibbs’ inequality

I

• Independence of estimated parameters and residuals in multiple linear regression
• Independence of products of multivariate normal random vector
• Independent random variables are uncorrelated
• Inflection points of the probability density function of the normal distribution
• Invariance of the covariance matrix under addition of constant vector
• Invariance of the differential entropy under addition of a constant
• Invariance of the Kullback-Leibler divergence under parameter transformation
• Invariance of the variance under addition of a constant
• Inverse transformation method using cumulative distribution function

J

• Joint likelihood is the product of likelihood function and prior density

K

• Kullback-Leibler divergence for the Bernoulli distribution
• Kullback-Leibler divergence for the binomial distribution
• Kullback-Leibler divergence for the continuous uniform distribution
• Kullback-Leibler divergence for the Dirichlet distribution
• Kullback-Leibler divergence for the discrete uniform distribution
• Kullback-Leibler divergence for the gamma distribution
• Kullback-Leibler divergence for the matrix-normal distribution
• Kullback-Leibler divergence for the multivariate normal distribution
• Kullback-Leibler divergence for the normal distribution
• Kullback-Leibler divergence for the normal-gamma distribution
• Kullback-Leibler divergence for the Wishart distribution

L

• Law of the unconscious statistician
• Law of total covariance
• Law of total expectation
• Law of total probability
• Law of total variance
• Linear combination of bivariate normal random variables
• Linear combination of independent normal random variables
• Linear transformation theorem for the matrix-normal distribution
• Linear transformation theorem for the moment-generating function
• Linear transformation theorem for the multivariate normal distribution
• Linearity of the expected value
• Log Bayes factor for Bayesian linear regression
• Log Bayes factor for binomial observations
• Log Bayes factor for multinomial observations
• Log Bayes factor for the univariate Gaussian with known variance
• Log Bayes factor in terms of log model evidences
• Log family evidences in terms of log model evidences
• Log model evidence for Bayesian linear regression
• Log model evidence for Bayesian linear regression with known covariance
• Log model evidence for binomial observations
• Log model evidence for multinomial observations
• Log model evidence for multivariate Bayesian linear regression
• Log model evidence for Poisson-distributed data
• Log model evidence for the Poisson distribution with exposure values
• Log model evidence for the univariate Gaussian
• Log model evidence for the univariate Gaussian with known variance
• Log model evidence in terms of prior and posterior distribution
• Log sum inequality
• Log-likelihood ratio for multiple linear regression
• Log-likelihood ratio for the general linear model
• Log-odds and probability in logistic regression
• Logarithmic expectation of the gamma distribution

M

• Marginal distribution of a conditional binomial distribution
• Marginal distributions for the matrix-normal distribution
• Marginal distributions of the multivariate normal distribution
• Marginal distributions of the normal-gamma distribution
• Marginal likelihood is a definite integral of the joint likelihood
• Marginally normal does not imply jointly normal
• Maximum likelihood estimation can result in biased estimates
• Maximum likelihood estimation for binomial observations
• Maximum likelihood estimation for Dirichlet-distributed data
• Maximum likelihood estimation for multinomial observations
• Maximum likelihood estimation for multiple linear regression
• Maximum likelihood estimation for Poisson-distributed data
• Maximum likelihood estimation for simple linear regression
• Maximum likelihood estimation for simple linear regression
• Maximum likelihood estimation for the general linear model
• Maximum likelihood estimation for the Poisson distribution with exposure values
• Maximum likelihood estimation for the univariate Gaussian
• Maximum likelihood estimation for the univariate Gaussian with known variance
• Maximum likelihood estimator of variance in multiple linear regression is biased
• Maximum likelihood estimator of variance is biased
• Maximum log-likelihood for binomial observations
• Maximum log-likelihood for multinomial observations
• Maximum log-likelihood for multiple linear regression
• Maximum log-likelihood for the general linear model
• Maximum-a-posteriori estimation for Bayesian linear regression
• Maximum-a-posteriori estimation for binomial observations
• Maximum-a-posteriori estimation for multinomial observations
• Mean of the Bernoulli distribution
• Mean of the beta distribution
• Mean of the binomial distribution
• Mean of the categorical distribution
• Mean of the continuous uniform distribution
• Mean of the ex-Gaussian distribution
• Mean of the exponential distribution
• Mean of the gamma distribution
• Mean of the log-normal distribution
• Mean of the matrix-normal distribution
• Mean of the multinomial distribution
• Mean of the multivariate normal distribution
• Mean of the normal distribution
• Mean of the normal-gamma distribution
• Mean of the normal-Wishart distribution
• Mean of the Poisson distribution
• Mean of the Wald distribution
• Median of the continuous uniform distribution
• Median of the exponential distribution
• Median of the log-normal distribution
• Median of the normal distribution
• Method of moments for beta-binomial data
• Method of moments for beta-distributed data
• Method of moments for ex-Gaussian-distributed data
• Method of moments for Wald-distributed data
• Mode of the continuous uniform distribution
• Mode of the exponential distribution
• Mode of the log-normal distribution
• Mode of the normal distribution
• Moment in terms of moment-generating function
• Moment-generating function of a function of a random variable
• Moment-generating function of a sum of independent random variables
• Moment-generating function of linear combination of independent random variables
• Moment-generating function of the beta distribution
• Moment-generating function of the ex-Gaussian distribution
• Moment-generating function of the exponential distribution
• Moment-generating function of the gamma distribution
• Moment-generating function of the multivariate normal distribution
• Moment-generating function of the normal distribution
• Moment-generating function of the Wald distribution
• Moments of the chi-squared distribution
• Monotonicity of probability
• Monotonicity of probability
• Monotonicity of the expected value
• Multinomial test
• Multiple linear regression is a special case of the general linear model
• Multivariate normal distribution is a special case of matrix-normal distribution
• Mutual information of dependent and independent variables in the general linear model
• Mutual information of the bivariate normal distribution
• Mutual information of the multivariate normal distribution

N

• Necessary and sufficient condition for independence of multivariate normal random variables
• Non-invariance of the differential entropy under change of variables
• (Non-)Multiplicativity of the expected value
• Non-negativity of the expected value
• Non-negativity of the Kullback-Leibler divergence
• Non-negativity of the Kullback-Leibler divergence
• Non-negativity of the Shannon entropy
• Non-negativity of the variance
• Non-symmetry of the Kullback-Leibler divergence
• Normal distribution is a special case of multivariate normal distribution
• Normal distribution maximizes differential entropy for fixed variance
• Normal-gamma distribution is a special case of normal-Wishart distribution
• Normally distributed and uncorrelated does not imply independent

O

• Omnibus F-test for multiple regressors in multiple linear regression
• One-sample t-test for independent observations
• One-sample z-test for independent observations
• Ordinary least squares for multiple linear regression
• Ordinary least squares for multiple linear regression
• Ordinary least squares for multiple linear regression
• Ordinary least squares for multiple linear regression with two regressors
• Ordinary least squares for one-way analysis of variance
• Ordinary least squares for simple linear regression
• Ordinary least squares for simple linear regression
• Ordinary least squares for the general linear model
• Ordinary least squares for two-way analysis of variance

P

• Paired t-test for dependent observations
• Paired z-test for dependent observations
• Parameter estimates for simple linear regression are uncorrelated after mean-centering
• Parameters of the corresponding forward model
• Partition of a covariance matrix into expected values
• Partition of covariance into expected values
• Partition of skewness into expected values
• Partition of sums of squares for multiple linear regression
• Partition of sums of squares for simple linear regression
• Partition of sums of squares in one-way analysis of variance
• Partition of sums of squares in two-way analysis of variance
• Partition of the log model evidence into accuracy and complexity
• Partition of the mean squared error into bias and variance
• Partition of variance into expected values
• Positive semi-definiteness of the covariance matrix
• Posterior credibility region against the omnibus null hypothesis for Bayesian linear regression
• Posterior density is proportional to joint likelihood
• Posterior distribution for Bayesian linear regression
• Posterior distribution for Bayesian linear regression with known covariance
• Posterior distribution for binomial observations
• Posterior distribution for multinomial observations
• Posterior distribution for multivariate Bayesian linear regression
• Posterior distribution for Poisson-distributed data
• Posterior distribution for the Poisson distribution with exposure values
• Posterior distribution for the univariate Gaussian
• Posterior distribution for the univariate Gaussian with known variance
• Posterior model probabilities in terms of Bayes factors
• Posterior model probabilities in terms of log model evidences
• Posterior model probability in terms of log Bayes factor
• Posterior predictive distribution is a marginal distribution of the joint likelihood
• Posterior probability of the alternative hypothesis for Bayesian linear regression
• Posterior probability of the alternative model for binomial observations
• Posterior probability of the alternative model for multinomial observations
• Probability and log-odds in logistic regression
• Probability density function is first derivative of cumulative distribution function
• Probability density function of a linear function of a continuous random vector
• Probability density function of a strictly decreasing function of a continuous random variable
• Probability density function of a strictly increasing function of a continuous random variable
• Probability density function of a sum of independent continuous random variables
• Probability density function of an invertible function of a continuous random vector
• Probability density function of the beta distribution
• Probability density function of the bivariate normal distribution
• Probability density function of the bivariate normal distribution in terms of correlation coefficient
• Probability density function of the chi-squared distribution
• Probability density function of the continuous uniform distribution
• Probability density function of the Dirichlet distribution
• Probability density function of the ex-Gaussian distribution
• Probability density function of the exponential distribution
• Probability density function of the F-distribution
• Probability density function of the gamma distribution
• Probability density function of the log-normal distribution
• Probability density function of the matrix-normal distribution
• Probability density function of the multivariate normal distribution
• Probability density function of the multivariate t-distribution
• Probability density function of the normal distribution
• Probability density function of the normal-gamma distribution
• Probability density function of the normal-Wishart distribution
• Probability density function of the t-distribution
• Probability density function of the Wald distribution
• Probability integral transform using cumulative distribution function
• Probability mass function of a strictly decreasing function of a discrete random variable
• Probability mass function of a strictly increasing function of a discrete random variable
• Probability mass function of a sum of independent discrete random variables
• Probability mass function of an invertible function of a random vector
• Probability mass function of the Bernoulli distribution
• Probability mass function of the beta-binomial distribution
• Probability mass function of the binomial distribution
• Probability mass function of the categorical distribution
• Probability mass function of the discrete uniform distribution
• Probability mass function of the multinomial distribution
• Probability mass function of the Poisson distribution
• Probability of exhaustive events
• Probability of exhaustive events
• Probability of normal random variable being within standard deviations from its mean
• Probability of the complement
• Probability of the empty set
• Probability of the empty set
• Probability under mutual exclusivity
• Probability under statistical independence
• Probability-generating function is expectation of function of random variable
• Probability-generating function of the binomial distribution
• Projection matrix and residual-forming matrix are idempotent
• Projection matrix and residual-forming matrix are symmetric
• Projection of a data point to the regression line
• Proof Template

Q

• Quantile function is inverse of strictly monotonically increasing cumulative distribution function
• Quantile function of the continuous uniform distribution
• Quantile function of the discrete uniform distribution
• Quantile function of the exponential distribution
• Quantile function of the gamma distribution
• Quantile function of the log-normal distribution
• Quantile function of the normal distribution

R

• Range of probability
• Range of the variance of the Bernoulli distribution
• Range of the variance of the binomial distribution
• Relation of continuous Kullback-Leibler divergence to differential entropy
• Relation of continuous mutual information to joint and conditional differential entropy
• Relation of continuous mutual information to marginal and conditional differential entropy
• Relation of continuous mutual information to marginal and joint differential entropy
• Relation of discrete Kullback-Leibler divergence to Shannon entropy
• Relation of mutual information to joint and conditional entropy
• Relation of mutual information to marginal and conditional entropy
• Relation of mutual information to marginal and joint entropy
• Relationship between chi-squared distribution and beta distribution
• Relationship between coefficient of determination and correlation coefficient in simple linear regression
• Relationship between correlation coefficient and slope estimate in simple linear regression
• Relationship between covariance and correlation
• Relationship between covariance matrix and correlation matrix
• Relationship between F-statistic and maximum log-likelihood
• Relationship between F-statistic and R²
• Relationship between gamma distribution and standard gamma distribution
• Relationship between gamma distribution and standard gamma distribution
• Relationship between multivariate normal distribution and chi-squared distribution
• Relationship between multivariate t-distribution and F-distribution
• Relationship between non-standardized t-distribution and t-distribution
• Relationship between normal distribution and chi-squared distribution
• Relationship between normal distribution and standard normal distribution
• Relationship between normal distribution and standard normal distribution
• Relationship between normal distribution and standard normal distribution
• Relationship between normal distribution and t-distribution
• Relationship between precision matrix and correlation matrix
• Relationship between R² and maximum log-likelihood
• Relationship between residual variance and sample variance in simple linear regression
• Relationship between second raw moment, variance and mean
• Relationship between signal-to-noise ratio and maximum log-likelihood
• Relationship between signal-to-noise ratio and R²
• Reparametrization for one-way analysis of variance

S

• Sampling from the matrix-normal distribution
• Sampling from the normal-gamma distribution
• Savage-Dickey density ratio for computing Bayes factors
• Scaling of a random variable following the gamma distribution
• Scaling of the covariance matrix upon multiplication with constant matrix
• Scaling of the variance upon multiplication with a constant
• Second central moment is variance
• Self-covariance equals variance
• Self-independence of random event
• Simple linear regression is a special case of multiple linear regression
• Skewness of the ex-Gaussian distribution
• Skewness of the exponential distribution
• Skewness of the Wald distribution
• Specific t-test for single regressor in multiple linear regression
• Square of expectation of product is less than or equal to product of expectation of squares
• Statistical significance test for the coefficient of determinantion based on an omnibus F-test
• Statistical test for comparing simple linear regression models with and without slope parameter
• Statistical test for intercept parameter in simple linear regression model
• Statistical test for slope parameter in simple linear regression model
• Sums of squares for simple linear regression
• Symmetry of the covariance
• Symmetry of the covariance matrix

T

• t-distribution is a special case of multivariate t-distribution
• t-test for multiple linear regression using contrast-based inference
• The expected value minimizes the mean squared error
• The geometric mean of independent log-normal random variables is a log-normal random variable
• The log probability scoring rule is a strictly proper scoring rule
• The median minimizes the mean absolute error
• The p-value follows a uniform distribution under the null hypothesis
• The product of independent log-normal random variables is a log-normal random variable
• The regression line goes through the center of mass point
• The residuals and the covariate are uncorrelated in simple linear regression
• The sum of residuals is zero in simple linear regression
• Transformation matrices for ordinary least squares
• Transformation matrices for simple linear regression
• Transitivity of Bayes Factors
• Transposition of a matrix-normal random variable
• Two-sample t-test for independent observations
• Two-sample z-test for independent observations

V

• Value of the probability-generating function for argument one
• Value of the probability-generating function for argument zero
• Variance of constant is zero
• Variance of parameter estimates for simple linear regression
• Variance of the Bernoulli distribution
• Variance of the beta distribution
• Variance of the binomial distribution
• Variance of the continuous uniform distribution
• Variance of the ex-Gaussian distribution
• Variance of the exponential distribution
• Variance of the gamma distribution
• Variance of the linear combination of two random variables
• Variance of the log-normal distribution
• Variance of the normal distribution
• Variance of the Poisson distribution
• Variance of the sum of two random variables
• Variance of the Wald distribution

W

• Weak law of large numbers
• Weighted least squares for multiple linear regression
• Weighted least squares for multiple linear regression
• Weighted least squares for simple linear regression
• Weighted least squares for simple linear regression
• Weighted least squares for the general linear model