  We study the problem of recovering a block-sparse signal from under-sampled
observations. The non-zero values of such signals appear in few blocks, and
their recovery is often accomplished using a $\ell_{1,2}$ optimization problem.
In applications such as DNA micro-arrays, some prior information about the
block support, i.e., blocks containing non-zero elements, is available. A
typical way to consider the extra information in recovery procedures is to
solve a weighted $\ell_{1,2}$ problem. In this paper, we consider a block
sparse model, where the block support has intersection with some given subsets
of blocks with known probabilities. Our goal in this work is to minimize the
number of required linear Gaussian measurements for perfect recovery of the
signal by tuning the weights of a weighted $\ell_{1,2}$ problem. For this goal,
we apply tools from conic integral geometry and derive closed-form expressions
for the optimal weights. We show through precise analysis and simulations that
the weighted $\ell_{1,2}$ problem with optimal weights significantly
outperforms the regular $\ell_{1,2}$ problem. We further examine the
sensitivity of the optimal weights to the mismatch of block probabilities, and
conclude stability under small probability deviations.
