  Minimum mean square error (MMSE) estimation of block sparse signals from
noisy linear measurements is considered. Unlike in the standard compressive
sensing setup where the non-zero entries of the signal are independently and
uniformly distributed across the vector of interest, the information bearing
components appear here in large mutually dependent clusters. Using the replica
method from statistical physics, we derive a simple closed-form solution for
the MMSE obtained by the optimum estimator. We show that the MMSE is a version
of the Tse-Hanly formula with system load and MSE scaled by parameters that
depend on the sparsity pattern of the source. It turns out that this is equal
to the MSE obtained by a genie-aided MMSE estimator which is informed in
advance about the exact locations of the non-zero blocks. The asymptotic
results obtained by the non-rigorous replica method are found to have an
excellent agreement with finite sized numerical simulations.
