# Glossary¶

assumed density filtering

Assumed Density Filtering (ADF) is a concept from Bayesian inference related to Expectation Propagation (EP). Expectation propagation tries to iteratively approximate intractable multivariate probability distribution by distributions that factorize into several simpler parts. The approximation is done with respect to Kullback-Leibler divergence. In the case of assuming that the simpler distributions are Gaussians this comes down to moment matching (matching the mean and covariance of the intractable distribution). ADF basically does only one step of the EP iteration, finding the best approximation for each of the factors in a chosen order just once. This means that approximations of the factors found first can not depend on the approximations of factors found later (unlike for EP, where several iterations of recalculating the approximations can capture correlations across factors). In our setting of neural networks, we chose the factors and their order according to the layers of the network and for simplicity assume Gaussian distributions for each layer.

moment matching

Moment matching of distributions refers to the process of finding the parameters in a family of parametrized probability distributions that match the first few moments of a given distribution. The given distribution is typically not in the same family. This is for example done to approximate an intractable distribution by a simpler distribution from a family of tractable distributions. Approximating any distribution by a Gaussian distribution with respect to Kullback-Leibler divergence boils down to matching the first two moments (mean and covariance).

moments of a probability distribution

The moments of a probability distribution or a random variable distributed according to that distributions are the expected values of the powers of that random variable. Central moments are the expectation values of the powers of the random variables shifted by its mean. We usually consider only central moments even though we not always explicitly state it. For example the first moment is just the expectation value, the second moment is variance, the third moment is skewness, and so on. For multivariate random vectors we also refer to the mean vector and covariances matrix as the first and second moments.

probabilistic networks

Probabilistic networks refer to any type of neural network involving probabilistic quantities (random variables). Typically these arise in context of Bayesian inference (Bayesian neural networks). The randomness can be used to model uncertainties in the input data, or also the model parameters. Applications include uncertainty quantification for neural networks and explaining or interpreting network predictions.