---

Remember that the posterior distribution, $f_{X|Y}(x|y)$, contains all the knowledge that we have about the unknown quantity $X$. Therefore, to find a point estimate of $X$, we can just choose a summary statistic of the posterior such as its mean, median, or mode. If we choose the mode (the value of $x$ that maximizes $f_{X|Y}(x|y)$), we obtain the MAP estimate of $X$. Another option would be to choose the posterior mean, i.e., \begin{align} \hat{x}=E[X|Y=y]. \end{align} We will show that $E[X|Y=y]$ will give us the best estimate of $X$ in terms of the mean squared error. For this reason, the conditional expectation is called the minimum mean squared error (MMSE) estimate of $X$. It is also called the least mean squares (LMS) estimate or simply the Bayes' estimate of $X$.

---

Example
Let $X$ be a continuous random variable with the following PDF \begin{align} \nonumber f_X(x) = \left\{ \begin{array}{l l} 2x & \quad \textrm{if }0 \leq x \leq 1 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{align} We also know that \begin{align} \nonumber f_{Y|X}(y|x) = \left\{ \begin{array}{l l} 2xy-x+1 & \quad \textrm{if }0 \leq y \leq 1 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{align} Find the MMSE estimate of $X$, given $Y=y$ is observed.

---