Consider two Markov Chains $X-Y_1-Z_1$ and $X-Y_2-Z_2$ defined on same alphabet space $\mathcal{X}$, such that $Z_1= g_1(Y_1)$ and $Z_2=g_2(Y_2)$ for some functions $g_1,g_2$. Assume further that given $X$, $Y_1,Y_2$ are conditionally independent random variables.

For brevity, let us think that $Y_1$ and $Y_2$ are independent estimates of $X$ and each estimate is independent of the other conditioned that $X$ was observed.

I am interested in knowing what could be the functions $g_1,g_2$ such that the following conditions are satisfied.

For some real valued loss function $l:\mathcal{X} \times \mathcal{X}\to \mathbb{R}^+$ and for some $\rho, \rho'>0$, I wish to have $l(Z_1=g_1(Y_1),Z_2=g_2(Y_2))< \rho$ and $l(Z_1,X)<\rho'$.

Intuitively, I am looking for maps $g_1,g_2$ such that these maps extract some common estimates from their previous estimates (here $Y_1,Y_2$) such that the new common estimates (here $Z_1,Z_2$) agree on an error of $\rho$ also provided that the estimates ($Z_1$ or $Z_2$) can't be far away from the true $X$ which were observed upto an error of $\rho'$.

One such function could be a quantization function. I am interested to know if there are better functions than that?

Any help on the ideas or literature is highly appreciated.

Thanks in advance


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