Statement of the PDF Projection Theorem

Theorem 1   (PDF Projection theorem), see [3,5]. Let $ {\bf z}=T({\bf x})$ be a mapping from $ {\cal X} \subset {\cal R}^N$ to $ {\cal Z} \subset {\cal R}^D$, $ D<N$. Let $ g({\bf z})$ be an arbitrary feature PDF with support $ {\cal Z}$. Let $ p({\bf x}\vert H_0)$ be a reference distribution with support $ {\cal X}$. Let $ p({\bf z}\vert H_0;T)$, the distribution imposed on $ {\cal Z}$ when $ {\bf z}=T({\bf x})$ and $ {\bf x}\sim p({\bf x}\vert H_0)$. Let $ p({\bf z}\vert H_0;T)$ be non-zero and have finite value everywhere on $ {\cal Z}$. Then, the function (2.2) is a PDF (integrates to 1 over $ {\cal X}$ ), and is a member of $ {\cal G}(T,g)$.

Proof: These assertions are proved in Section 16.4. See also [3] or [5], Theorem 2.

Baggenstoss 2017-05-19