Conjugate variables in continuous maximum-entropy inference

For a continuous maximum-entropy distribution (obtained from an arbitrary number of simultaneous constraints), we derive a general relation connecting the Lagrange multipliers and the expectation values of certain particularly constructed functions of the states of the system. From this relation, an estimator for a given Lagrange multiplier can be constructed from derivatives of the corresponding constraining function. These estimators sometimes lead to the determination of the Lagrange multipliers by way of solving a linear system, and, in general, they provide another tool to widen the applicability of Jaynes's formalism. This general relation, especially well suited for computer simulation techniques, also provides some insight into the interpretation of the hypervirial relations known in statistical mechanics and the recently derived microcanonical dynamical temperature. We illustrate the usefulness of these new relations with several applications in statistics.