MMSE Estimation - Fusion of 2 Measurements Let's say I have 2 measurements of the same phenomenon (for example current temperature) and I want to find the MMSE (minimum mean square error) estimator, i.e to minimize the MSE (mean square error). The measurements are independent and the noise in each one is Gaussian. The mathematical formulation is:

$$Y_1[n]=X[n]+V[n]$$

$$Y_2[n]=X[n]+W[n]$$
where:


$X[n]$ is the real temperature at time $n$.
$Y_1[n]$ and $Y_2[n]$ are the measurements at time $n$.
$V[n]$ and $W[n]$ are zero-mean normally distributed random variables which represent the noise, and have variances $\sigma_v^2$ and $\sigma_w^2$, respectively.


I need a clue or reference for how to solve it.
*STOP! If you only want a hint and not the complete solution please see Stanley P.'s or Peter K.'s answers. *

Since you do not specify if there is model for the temperature evolving over time $n$, I will derive an estimator which is a combination of $Y_1$ and $Y_2$ for each fixed $n$. Let $\alpha \in (0,1)$ and suppose the estimate of $X$ can be written as $\hat X = \alpha Y_1 + (1-\alpha)Y_2$. The MSE is given by
\begin{eqnarray}
E[(\hat X-X)^2] &amp;=&amp; E[(\alpha Y_1 + (1-\alpha)Y_2 - X)^2]\\
&amp;=&amp; E[(\alpha X + (1-\alpha)X + \alpha V + (1-\alpha)W - X)^2]\\
&amp;=&amp; E[(\alpha V + (1-\alpha)W)^2]\\
&amp;=&amp; \alpha^2 \sigma_v^2 + (1-\alpha)^2\sigma_w^2.
\end{eqnarray}
To minimize this MSE, take the derivative with respect to $\alpha$ and set it to zero. After some algebra you get $\alpha = \frac{\sigma_w^2}{\sigma_v^2+\sigma_w^2}$ and so
$$
\hat X = \frac{\sigma_w^2}{\sigma_v^2+\sigma_w^2} Y_1 + \frac{\sigma_v^2}{\sigma_v^2+\sigma_w^2 }Y_2.
$$

Note that if the variances are unequal (say $\sigma_w^2 &gt; \sigma_v^2$), then the noisier observation ($Y_2$) is given less weight than the less noisy observation ($Y_1)$. If the noise variances are equal, the two observations are weighted equally and we get just the arithmetic mean of the two observations.

If you do have a model for temperature evolution over time you should use a Kalman filter which is guaranteed to produce the MMSE estimate.