Given the correlation matrix, estimate the value of a random variable based on the value of other random variables.

Question

A process generates $N$ random variables $(X_i \mid 1 \leq i \leq N)$.

The process is run $K$ times, and the values of each random variable $X_i$ is observed.

Based on this data, the following things are computed:

• The Means: $\mu_i$ for each random variable
• The Variances: $\sigma^2_i$ for each random variable
• Correlation Matrix $M_{N \times N}$: such that $m_{ij} = \text{corr}(X_i, X_j)$

Another observation is made and values of all but the first random variable are observed. Given those values $x_2 \ldots x_n$, how can we estimate the value of $x_1$ for that observation (in terms of $x_2 \ldots x_n, \mu_i, \sigma^2_i, M$)?

Answer 1

Since all you have are the means and the covariances, the most natural thing to do would seem to be to assume a multivariate Gaussian with those parameters, plug in $x_2,\ldots,x_n$ and normalise to get the conditional distribution for $x_1$; the estimate would then be the mean of that distribution (which is the same as its mode and median, since it's normal).