Goal of PDF Projection

The goal is to approximate some unknown distribution, denoted by $ p({\bf x}\vert H)$ or just by $ p({\bf x})$, by converting $ g({\bf z})$ into a PDF of $ {\bf x}$. We call this PDF projection since the PDF $ g({\bf z})$ is projected onto the input data space. So, given that we know (or have estimated) $ g({\bf z})$, what does this tell us about $ p({\bf x})$? Let $ {\cal G}(T,g)$ be the set of PDFs which generate $ g({\bf z})$ through $ T({\bf x})$. More precisely, if $ p({\bf x}) \in {\cal G}(T,g),$ then samples drawn from PDF $ p({\bf x})$, and passed through $ T({\bf x})$ must have PDF $ g({\bf z})$. We denote PDFs that are members of $ {\cal G}(T,g)$ by $ G({\bf x};H_0,T,g)$. Thus, PDF projection is a matter of finding a member of $ {\cal G}(T,g)$. The reference hypothesis appears in the argument list because each member distribution of $ {\cal G}(T,g)$ is uniquely determined by $ H_0$ [5]. The entirety of all PDFs in $ {\cal G}(T,g)$ are found by selection of $ H_0$.

Baggenstoss 2017-05-19