  We present a method to estimate the probabilities of outcomes of a quantum
observable, its mean value, and higher moments by measuring any other
observable. This method is general and can be applied to any quantum system. In
the case of estimating the mean energy of an isolated system, the estimate can
be further improved by measuring the other observable at different times.
Intuitively, this method uses interplay and correlations between the measured
observable, the estimated observable, and the state of the system. We provide
two bounds: one that is looser but analytically computable and one that is
tighter but requires solving a non-convex optimization problem. The method can
be used to estimate expectation values and related quantities such as
temperature and work in setups where performing measurements in a highly
entangled basis is difficult, finding use in state-of-the-art quantum
simulators. As a demonstration, we show that in Heisenberg and Ising models of
ten sites in the localized phase, performing two-qubit measurements excludes
97.5% and 96.7% of the possible range of energies, respectively, when
estimating the ground state energy.
