Time-Dependent Variational Principle for Predicting the Expectation Value of an Observable

For a given statistical state at time $t_0$, the expectation value of some observable at a later time $t_1$ is expressed in a variational form. Different trial choices for the quantities to be varied (state and observable) generate different approximations, in which the evolution of the state is optimally fitted to the measured quantity. Examples are given in the context of mean-field theories.