Statement of the PDF Projection Theorem

Theorem 1   (PDF Projection theorem), see [4,3]. 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 17.4. See also [4] or [3], Theorem 2.