$\newcommand{\Var}{\operatorname{Var}}$Suppose $X,Y,W$ form a Markov chain $X \to Y \to W$. Can we simplify the following expression? \begin{align*} E [ \Var ( \Var (X\mid Y) \mid W)] \end{align*} Because we have a Markov chain, that is $p(x\mid y)=p(x\mid y,w)$, we have that \begin{align*} E[X\mid Y]&=E[X\mid Y,W]\\ \Var(X\mid Y)&=\Var(X\mid Y,W)\\ \end{align*}
If we were to replace $\Var$ with expected value we can use towering property of expected value that is \begin{align*} E \left[ E \left(E [X\mid Y] \mid W\right)\right]&=E \left[ E \left(E [X\mid Y,W] \mid W\right)\right]\\ &=E[E[X\mid W]]=E[X] \end{align*}
Can we do something similar for $E \left[ \Var \left( \Var (X\mid Y) \mid W\right)\right]$?
One thing we can show is \begin{align*} E \left[ \Var \left( \Var (X\mid Y) \mid W\right)\right]=E \left[ \Var \left( \Var (X\mid Y,W) \mid W\right)\right] \end{align*} but I am not sure if this helps at all.
Would be grateful for any ideas. Thank you very much
