Minimum cross-entropy estimation with inaccurate side information

Given a prior estimate of a probability, q, and a constraint Sigma p(i)x(i) = a, one well-known way of estimating p is to minimize the cross-entropy I (p:q) subject to the constraint. A modification to this method is proposed for use when the value a is only approximately known. The modification is b ased on the penalty function method in constrained optimization. It has an interpretation in differential geometry methods in statistics and it someti mes gives a maximum-likelihood estimate.