**Motivation**

Then the usual stochastic filtering problem says that: $$ \operatorname{argmin}_{Z \in L^2(\mathscr{G}_t)}\,\mathbb{E}[(Y_t-Z_t)^2], $$ where $\mathscr{G}_t$ is the $\sigma$-algebra generated by $Y$ up to time $t$ is solved by $$ \hat{Y}_t\triangleq \mathbb{E}[Z|\mathscr{G}_t]. $$

**Question:**

My question is, if I fix an injective Borel-measurable $\phi:\mathbb{R}^d\mapsto \mathbb{R}^{d'}$ and let $Y_t$ be a stochastic process with values in $\mathbb{R}^d$ (say d'\geq d).

Is it still true that the minimizer of: $$ \operatorname{argmin}_{Z \in L^2(\mathscr{G}_t)}\,\mathbb{E}[(\phi(Y_t)-\phi(Z_t))^2], $$ is $$ \hat{Y}_t\triangleq \mathbb{E}[Z|\mathscr{G}_t]? $$

If not, what additional assumptions do I need to make, probably on $\phi$?