Proofs without Source

Missing source (190 proofs)

• Accuracy and complexity for the univariate Gaussian
• Accuracy and complexity for the univariate Gaussian with known variance
• Bayesian information criterion for multiple linear regression
• Bernoulli distribution is a special case of categorical distribution
• Beta distribution is a special case of Dirichlet distribution
• Binomial distribution is a special case of multinomial distribution
• 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
• Conditional distributions of the normal-gamma distribution
• Construction of unbiased estimator for variance in multiple linear regression
• Covariance and variance of the normal-gamma distribution
• Covariance matrix of the categorical distribution
• Cross-covariance matrices of the matrix-normal distribution
• Cross-validated log Bayes factor for binomial observations
• Cross-validated log Bayes factor for multinomial observations
• Cross-validated log Bayes factor for the univariate Gaussian
• Cross-validated log Bayes factor for the univariate Gaussian with known variance
• Cross-validated log model evidence for binomial observations
• Cross-validated log model evidence for multinomial observations
• Cross-validated log model evidence for the univariate Gaussian
• 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 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 multinomial distribution
• Derivation of Bayesian model averaging
• 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 for the matrix-normal distribution
• Differential entropy of the continuous uniform distribution
• Differential entropy of the normal-gamma distribution
• Effects of mean-centering on parameter estimates for simple linear regression
• Entropy of the binomial distribution
• Entropy of the categorical distribution
• Entropy of the discrete uniform distribution
• Entropy of the multinomial distribution
• Entropy of the Poisson distribution
• Expectation of a bilinear form
• Expectation of a quadratic form for the multivariate normal distribution
• 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
• Exponential distribution is a special case of gamma distribution
• Expression of R² in terms of residual variances
• Expression of the noise precision posterior for Bayesian linear regression using prediction and parameter errors
• 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
• Gamma distribution is a special case of Wishart distribution
• Joint likelihood is the product of likelihood function and prior density
• Kullback-Leibler divergence for the Bernoulli distribution
• Kullback-Leibler divergence for the continuous uniform distribution
• Kullback-Leibler divergence for the discrete uniform distribution
• Kullback-Leibler divergence for the matrix-normal distribution
• Kullback-Leibler divergence for the normal distribution
• Linear combination of independent normal random variables
• Linear transformation theorem for the matrix-normal distribution
• 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 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 with known variance
• Log model evidence in terms of prior and posterior distribution
• Log-likelihood ratio for the general linear model
• Marginal distributions for the matrix-normal distribution
• Marginal distributions of the multinomial 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
• Maximum likelihood estimation can result in biased estimates
• Maximum likelihood estimation for bivariate normally 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 Poisson distribution with exposure values
• Maximum log-likelihood for binomial observations
• Maximum log-likelihood for multinomial observations
• 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, mode and median of the coefficient of determination under the null hypothesis
• Mean of the categorical distribution
• Mean of the continuous uniform distribution
• Mean of the ex-Gaussian distribution
• Mean of the multinomial distribution
• Mean of the normal-gamma distribution
• Mean of the normal-Wishart 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 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 gamma distribution
• Mode of the multivariate normal distribution
• Mode of the normal 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
• Mutual information of dependent and independent variables in the general linear model
• Necessary and sufficient condition for independence of multivariate normal random variables
• Normal-gamma distribution is a special case of normal-Wishart distribution
• 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
• Parameter estimates for simple linear regression are uncorrelated after mean-centering
• Partition of a cross-covariance matrix into expected values
• Posterior density is proportional to joint likelihood
• Posterior predictive distribution is a marginal distribution of the joint likelihood
• Posterior probability of the alternative model for binomial observations
• Posterior probability of the alternative model for multinomial observations
• Prior predictive distribution is a marginal distribution of the joint likelihood
• Probability density function of the beta distribution
• Probability density function of the bivariate normal 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 gamma 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 Wald distribution
• Probability mass function of the Bernoulli 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
• Quantile function of the continuous uniform distribution
• Quantile function of the discrete uniform distribution
• Quantile function of the exponential distribution
• Range of the variance of the binomial distribution
• Relationship between covariance and correlation
• Relationship between F-statistic and maximum log-likelihood
• Relationship between gamma distribution and standard gamma distribution
• Relationship between gamma distribution and standard gamma distribution
• Relationship between non-standardized t-distribution and t-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 precision matrix and correlation matrix
• Relationship between R² and maximum log-likelihood
• Relationship between signal-to-noise ratio and maximum log-likelihood
• Relationship between signal-to-noise ratio and R²
• Scaling of a random variable following the gamma distribution
• 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
• 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
• The first raw moment equals the mean
• The first standardized moment equals zero
• The second raw moment equals squared mean plus squared standard deviation
• The second standardized moment equals one
• The third standardized moment equals the skewness
• Transformation matrices for simple linear regression
• Transitivity of Bayes Factors
• Transposition of a matrix-normal random variable
• Variance of the coefficient of determination under the null hypothesis
• Variance of the continuous uniform distribution
• Variance of the ex-Gaussian distribution
• Variance of the Wald distribution
• 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