The formula for PDF projection is
is a function of only because,
although appears on the right hand side,
is uniquely determined by through
We show in the appendix (Section 16.4)
is a PDF (integrates to 1), and
is a member of
is called the ``J-function" because of the
analog with the Jacobian in the change of variables theorem.
The projected PDF is therefore decomposed into two terms,
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.