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Let $Y_i$ be a normal random variable given by noisy measurements of a normal random variable $X$ with have zero mean and variance $\sigma_X^2$ such that

$$Y_i = X + W_i$$

The noise $W_i$ are mutually independent, independent of $X$, and zero mean with variance $\sigma_W^2$. I am trying to find the minimum mean square error estimate of $X$ and the maximum a posterior estimate of $X$, but my question is how do I calculate the conditional density or joint density of $X$ and $Y$ since they are not independent. The MMSE and MAP will be straightforward if I can properly calculate the conditional and joint densities.

If $X$ and $Y$ were independent, I could simply calculate $f_{x,y}(x,y)=f_x(x)f_y(y)$ then calculate $f_{x|y}(x|y) = \frac{f_{x,y}(x,y)}{f_y(y)}$.

Or, if I could calculate $f_{y|x}(y|x)$, then I could simply calculate $f_{x|y}(x|y) = \frac{f_{y|x}(y|x)f_x(x)}{f_y(y)}$.

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Noting that if $\mathbf{X}$ and $\mathbf{W}$ are independent then $f_{w|x} = f_w$. Also $f(y|x) = f_{w|x}(y-x) = f_{w}(y-x)$.

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  • $\begingroup$ Why does $f_{y|x}(y|x) = f_{w|x}(y-x)$? $\endgroup$ Commented Nov 10, 2016 at 18:40
  • $\begingroup$ Please explain, I don't understand why these are equivalent. $\endgroup$ Commented Nov 10, 2016 at 19:08
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    $\begingroup$ It follows from the fact that $f_{y,x}(y,x) = f_{w,x}(y-x,x)$. For details I would recommend Probability, Random Variables and Stochastic Process by Athanasios Papoulis and S.U. Pillai. Specifically, have a look at example 6-24 of the fourth edition. $\endgroup$ Commented Nov 10, 2016 at 21:24

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