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More generally, other distributions can be used than the Bernoulli distribution, such as the gaussian noise: = +, (,) where = and =, with and being the mean and variance of the -th output neuron. The reparameterization trick can be applied to all such cases, resulting in the variational dropout method.
The probability content of the multivariate normal in a quadratic domain defined by () = ′ + ′ + > (where is a matrix, is a vector, and is a scalar), which is relevant for Bayesian classification/decision theory using Gaussian discriminant analysis, is given by the generalized chi-squared distribution. [17]
The same result is used when approximating the posterior with Laplace's approximation, where the Fisher information appears as the covariance of the fitted Gaussian. [3] Statistical systems of a scientific nature (physical, biological, etc.) whose likelihood functions obey shift invariance have been shown to obey maximum Fisher information. [4]
This generalized inverse Wishart distribution has been applied to estimating the distributions of multivariate autoregressive processes. [11] A different type of generalization is the normal-inverse-Wishart distribution, essentially the product of a multivariate normal distribution with an inverse Wishart distribution.
A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference. [7] [23] Given any set of N points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian. For ...
An alternative parameterization of the logistic distribution can be derived by expressing the scale parameter, , in terms of the standard deviation, , using the substitution =, where = / = …. The alternative forms of the above functions are reasonably straightforward.
The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. To distinguish the two families, they are referred to below as "symmetric" and "asymmetric"; however ...
In probability theory and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic form of a multinormal variable (normal vector), or a linear combination of different normal variables and squares of normal variables.