I have a question if the following relations on conditional probabilities hold for independent random variables?
$$P_{X \mid Y, G(Y)}(x_1)=P_{X \mid \{Y\}}(x_2)$$ where $G$ is not necessarily invertible.
Also can we say $$P_{X \mid Y, H(Z)}(x_3)=P_{X \mid Y, \{Z\}}(x_4)$$ where $H$ is not necessarily invertible.
Finally can we say $$P_{X \mid Y, U}(x_5)=P_{X \mid Y,\{V\}}(x_6)$$ where $U=g(V)$ is not necessarily invertible.
The values $x_1,x_2 $ can be equal or related through some function. same goes to $x_3, x_4$ and $x_5,x_6$.If the mappings $H$, $G$ are invertible does the answer change? Please explain the answer or provide a reference which I can find online.
My guess is they hold with $G$, $H$, $g$ being invertible or not.
Example: For continuous and independent random variables $X,N$ I think we can derive $$Y=X+N$$ $$F_{Y\mid X}(y)=F_N(y-x)$$ where $F$ is the distribution function. So the above relations are a generalization.
This seems awfully related to your previous question, no?
First question: the conditional distribution of $X$ conditionally on $(Y,G(Y))=(y,G(y))$ is the same as the conditional distribution of $X$ conditionally on $Y=y$, whether $G$ is injective or not.
Second question: the conditional distribution of $X$ conditionally on $(Y,H(Z))=(y,H(z))$ is not necessarily the same as the conditional distribution of $X$ conditionally on $(Y,Z)=(y,z)$, except if $H$ is injective.
Third question: this is a duplicate of the second question.