### Mathematical Notation

In what follows, the symbol is always used for the (high-dimensional) input data, and the symbol is always used for (lower-dimensional) features, the output of . We view both and as random variables (RV) 2.1. We will use a simplified notation for a RV and the associated probability density function (PDF). We use the same symbol to represent the RV and a sample of that RV. Distributions are identified by their argument, so and are understood to be the distributions of RVs and , respectively. When there is the possibility of confusion, we use a subscript, for example, is understood to be the evaluation of the density at sample . If there are more than one possible distributions of a given RV, we index them by the statistical hypothesis, such as , , etc. For a generic fixed feature PDF, we use .

Baggenstoss 2017-05-19