  We study covariance matrix estimation for the case of partially observed
random vectors, where different samples contain different subsets of vector
coordinates. Each observation is the product of the variable of interest with a
$0-1$ Bernoulli random variable. We analyze an unbiased covariance estimator
under this model, and derive an error bound that reveals relations between the
sub-sampling probabilities and the entries of the covariance matrix. We apply
our analysis in an active learning framework, where the expected number of
observed variables is small compared to the dimension of the vector of
interest, and propose a design of optimal sub-sampling probabilities and an
active covariance matrix estimation algorithm.
