  A classical approach for approximating expectations of functions w.r.t.
partially known distributions is to compute the average of function values
along a trajectory of a Metropolis-Hastings (MH) Markov chain. A key part in
the MH algorithm is a suitable acceptance/rejection of a proposed state, which
ensures the correct stationary distribution of the resulting Markov chain.
However, the rejection of proposals causes highly correlated samples. In
particular, when a state is rejected it is not taken any further into account.
In contrast to that we consider a MH importance sampling estimator which
explicitly incorporates all proposed states generated by the MH algorithm. The
estimator satisfies a strong law of large numbers as well as a central limit
theorem, and, in addition to that, we provide an explicit mean squared error
bound. Remarkably, the asymptotic variance of the MH importance sampling
estimator does not involve any correlation term in contrast to its classical
counterpart. Moreover, although the analyzed estimator uses the same amount of
information as the classical MH estimator, it can outperform the latter in
scenarios of moderate dimensions as indicated by numerical experiments.
