Conditional probability and independence ($Y \perp D$)

Question

Consider random variables Y, D , and X drawn from joint density $p(Y, D,X)$. Given $Y \perp D$, can I say that $p(Y|X) = p(Y|D,X)$?

Can someone demonstrate it?

Answer 1

Your conjecture is false, which can be shown by adapting the well-known counter-example for events which are pairwise independent but not mutually independent.

• Throw $2$ independent coins

• $Y= $ first coin is Heads

• $D = $ second coin is Heads

• $X = $ two coins have equal results

We have $Y \perp D$, and $Y \perp X \implies P(Y|X) = P(Y) = 1/2$. But conditioning on both $X$ and $D$ would fully determine $Y$ e.g. $P(Y=1|X=1,D=1) = 1$