Let $X_t=f(Y_t,Z_t)$ be a stochastic process depending on $Y_t$ and on $Z_t$; all of which are Markovian. If I know $g,h$ where $$ E[X_t|Y_t]=h(Z_t,Y_t), $$ and $$ E[X_t|Z_t]=g(Z_t,Y_t) $$ then can I determine what $X_t$ is at time $t$?
2026-03-29 09:54:08.1774778048
Reconstructing From Conditionals
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No you can't.
Let $Y,Z$ be independent copies of the two state symmetric Markov chain starting with the invariant distribution. Let $X=1_{Y=Z}-1_{Y\neq Z}$. It is easy to check $$ \mathbb{E}[X\mid Y]=\mathbb{E}[X\mid Z]=0, $$ but of course $X\neq 0$.