Conditional density for random effects prediction in GLMM

Question

I am currently working on generalized linear mixed models (GLMM) and need some help concerning the prediction of the random effects. More specifically, I don't understand the given representation of the conditional expectation of the random effect, i.e. $$ \mathbb{E}[X | Y] = \int f_{X|Y} (x|y)\: dx = \int \frac{f_{Y|X}(y|x)f_{X}(x)}{\int f_{Y|X}(y|x)f_{X}(x)dx} dx$$(the second equation being the one I don't understand)

This is the representation in general. In a one way classification, $X$ would be the random effect, $Y$ the response group (vector) for the random effect.
Found in several papers and books of McCulloch et al.

Answer 1

This has nothing to do with GLMM's per se. All what is done here is using the definition $$f_{Y|X}(y|x):=\frac{f_{X,Y}(x,y)}{f_X(x)}$$ (if $f_X(x)\ne0$) to write $$f_{Y|X}(y|x)f_X(x)=f_{X,Y}(x,y),$$ so that
$$\int f_{Y|X}(y|x)f_X(x)\,dx=\int f_{X,Y}(x,y)\,dx=f_Y(y)$$ and hence $$\frac{f_{Y|X}(y|x)f_{X}(x)}{\int f_{Y|X}(y|x)f_{X}(x)\,dx} =\frac{f_{X,Y}(x,y)}{f_Y(y)} =f_{X|Y} (x|y).$$