I have problem in proving $H(Y \mid X) \leq H(Y \mid f(X))$, where $f$ is a function of $X$. In the textbooks, they already proved $I(Y; X) \geq I(Y;f(X))$, and $H(f(X)) \leq H(X)$ but I can't relate those with the question problem.
Besides, there is one other question that I am concerned about. If $H(Y \mid X) = H(Y \mid f(X))$, what is the conditions? Is that $p_Y(\cdot \mid X) = p_Y(\cdot \mid f(X))$?
We already have the case $H(Z) = H(W) \rightarrow p_Z(\cdot) = p_W(\cdot)$, that is wrong. However I doubt it may be right for $H(Y \mid X) = H(Y \mid f(X)) \rightarrow p_Y(\cdot \mid X) = p_Y(\cdot \mid f(X))$.
Can anyone help? Thanks.