  We develop a Bayesian framework for sensing which adapts the sensing time
and/or basis functions to the instantaneous sensing quality measured in terms
of the expected posterior mean-squared error. For sparse Gaussian sources a
significant reduction in average sensing time and/or mean-squared error is
achieved in comparison to non-adaptive sensing. For compression ratio 3, a
sparse 10% Gaussian source and equal average sensing times, the proposed method
gains about 2 dB over the performance bound of optimum compressive sensing,
about 3 dB over non-adaptive 3-fold oversampled orthogonal sensing and about 6
to 7 dB to LASSO-based recovery schemes while enjoying polynomial time
complexity.
  We utilize that in the presence of Gaussian noise the mean-squared error
conditioned on the current observation is proportional to the derivative of the
conditional mean estimate with respect to this observation.
