  Markov chain Monte Carlo methods are primarily used for sampling from a given
probability distribution and estimating multi-dimensional integrals based on
the information contained in the generated samples. Whenever it is possible,
more accurate estimates are obtained by combining Monte Carlo integration and
integration by numerical quadrature along particular coordinates. We show that
this variance reduction technique, referred to as conditioning in probability
theory, can be advantageously implemented in \emph{expanded ensemble}
simulations. These simulations aim at estimating thermodynamic expectations as
a function of an external parameter that is sampled like an additional
coordinate. Conditioning therein entails integrating along the external
coordinate by numerical quadrature. We prove variance reduction with respect to
alternative standard estimators and demonstrate the practical efficiency of the
technique by estimating free energies and characterizing a structural phase
transition between two solid phases.
