Reference Hypothesis

In order to proceed with PDF projection, we need to have a reference hypothesis, which we have explained, provides a way of specifying the distribution of the unobservable dimensions of $ {\bf x}$. The reference hypothesis can be any statistical hypothesis under which the distribution of both $ {\bf x}$ and $ {\bf z}$ are known, and denoted by the pair $ p({\bf x}\vert H_0)$, $ p({\bf z}\vert H_0)$.

Note that $ H_0$ is a purely mathematical concept, and does not need to represent any type of ``noise-only" condition or realistic data or such. The reader, at this point, should have a bit of healthy skepticism. It should ring an alarm bell, that PDF projection is based on defining this seemingly arbitrary reference hypothesis. These concerns will be addressed two ways. First, in Section 2.2.2, we will discuss how $ H_0$ and $ T({\bf x})$ can be selected jointly to approximate the condition of sufficiency, and thereby approximate as close as possbile the distribution of $ {\bf x}$ under some arbitrary hypothesis $ H$. Later, in Chapter 3 we will discuss how $ H_0$ can be selected according to the principle of maxumum entropy (MaxEnt), and why this is a good thing to do. For now, let's just assume that $ H_0$ is given. Normally, $ p({\bf z}\vert H_0)$ must be derived analytically. Only in rare situations can $ p({\bf z}\vert H_0)$ can be estimated. We discuss type of problem in Section 2.2.3.

Baggenstoss 2017-05-19