Convolution of two probabilities (non-independent)

I was wondering if the following is true, where * denotes convolution:

p(A|C,D) = p(A|B) * p(B|C,D).

Basically, I want to know if convolution will marginalize out B. My intuition says this is correct, but I'm not entirely certain.

Thank you.



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To simplify things let's just consider C,D as a single event. So for * representing convolution we're looking at

P(A|C) = P(A|B) * P(B|C)

(this was mainly for my benefit for thinking about it)
Thank you, Dason. So my original equation is correct? Just want to be sure. I have limited probability training, but sometimes delve into it with my work.