I was reading about marginalization on Wikipedia, specifically I read:
$$p_X(x) = \int_y p_{X\mid Y}(x\mid y)p_Y(y)\,dy$$
I was wondering if the following is true
$$\int_y p_{X\mid YZ}(x\mid y,z)p_Y(y) \, dy = p_{X\mid Z}(x\mid z)$$
$X, Y$ and $Z$ are random variables, with pdf-s $p_X$, p$_Y$ and $p_Z$ respectively.
Only if ${p}_{\lower{0.5ex}{Y}}(y)={p}_{\lower{0.5ex}{Y\mid Z}}(y\mid z)$
By the Law of Total Probability:
$${p}_{\lower{0.5ex}{X\mid Z}}(x\mid z) =\int_{\Bbb R} {p}_{\lower{0.5ex}{X\mid Y,Z}}(x\mid y,z)~{p}_{\lower{0.5ex}{Y\mid Z}}(y\mid z) \,\mathrm{d}y$$
However if $Y$ and $Z$ are pairwise independent, then indeed: $${p}_{\lower{0.5ex}{X\mid Z}}(x\mid z) =\int_{\Bbb R} {p}_{\lower{0.5ex}{X\mid Y,Z}}(x\mid y,z)~{p}_{\lower{0.5ex}{Y}}(y) \,\mathrm{d}y$$
