The Projected PDF

The formula for PDF projection is

$$G({\bf x};H_0,T,g) = \frac{p({\bf x}\vert H_0)}{p({\bf z}\vert H_0)} g({\bf z}).$$ (2.2)

Note that $G({\bf x};H_0,T,g)$ is a function of ${\bf x}$ only because, although ${\bf z}$ appears on the right hand side, ${\bf z}$ is uniquely determined by ${\bf x}$ through (2.1). We show in the appendix (Section 17.4) that indeed $G({\bf x};H_0,T,g)$ is a PDF (integrates to 1), and is a member of ${\cal G}(T,g)$. The term

$$J({\bf x};H_0,T) = \frac{p({\bf x}\vert H_0)}{p({\bf z}\vert H_0)},$$ (2.3)

is called the “J-function" because of the analog with the Jacobian in the change of variables theorem[6]. The projected PDF is therefore decomposed into two terms,

$$G({\bf x};H_0,T,g)=J({\bf x};H_0,T)\; g({\bf z}).$$

The first depends only on the feature transformation and reference hypothesis and can be exactly determined, so does not need to be learned. The second term is the estimated or given feature distribution.