Let $(\Omega, \mathcal{A}, P)$ be a measure space. If the probability measure admits a density $p$ it should hold by the product and summation rules that
$\displaystyle p(a|b,c,d) = \int p(a|b,e)p(e|c,d)\mathrm{d}e$
which I am not able to proof.
I can show that
$\displaystyle p(a|b)=\int p(a|b,c)p(c|b)\mathrm{d}c$
but can not reach the result.
This formular is from Bishop's Pattern Recongnition p.31 (1.68), I just renamed and formalized the problem.