from the new conditional distribution . (Note that often the later expected value is called the conditional relative entropy (or conditional Kullback-Leibler divergence) and denoted by ) This is minimized if over the whole support of ; and we note that this result incorporates Bayes' theorem, if the new distribution is in fact a δ function representing certainty that has one particular value. MDI can be seen as an extension of Laplace's Principle of Insufficient Reason, and the Principle of Maximum Entropy of E.T. Jaynes. In particular, it is the natural extension of the principle of maximum entropy from discrete to continuous distributions, for which Shannon entropy ceases to be so useful (see differential entropy), but the relative entropy continues to be just as relevant. In the engineering literature, MDI is sometimes called the Principle of Minimum Cross-Entropy (MCE) or Minxent for short. Minimising relative entropy from to with respect to is equivalent to minimizing the cross-entropy of and , since which is appropriate if one is trying to choose an adequate approximation to . However, this is just as often not the task one is trying to achieve. Instead, just as often it is that is some fixed prior reference measure, and that one is attempting to optimise by minimising subject to some constraint. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by redefining cross-entropy to be , rather than . Relationship to available work Surprisals add where probabilities multiply. The surprisal for an event of probability is defined as . If is then surprisal is in nats, bits, or so that, for instance, there are bits of surprisal for landing all "heads" on a toss of coins. Best-guess states (e.g. for atoms in a gas) are inferred by maximizing the average surprisal (entropy) for a given set of control parameters (like pressure or volume ). This constrained entropy maximization, both classically and quantum mechanically, minimizes Gibbs availability in entropy units where is a constrained multiplicity or partition function. When temperature is fixed, free energy () is also minimized. Thus if and number of molecules are constant, the Helmholtz free energy (where is energy and is entropy) is minimized as a system "equilibrates." If and are held constant (say during processes in your body), the Gibbs free energy is minimized instead. The change in free energy under these conditions is