  We propose a Monte Carlo method in which trial configurations are generated
by iterating a finite number of times an initial guess, sampled from a prior
distribution, using Markov updates. We define an extended configurational space
comprising all possible ensembles of such updates leading to a given
configuration in a finite number of steps. By enforcing detailed balance
between extended configurations, we validate two prototypical schemes that
sample desired target distributions while generating new configurations through
truncated Markov Chains. Using representative systems, we identify the
important factors controlling the quality of the sampling of this type of
algorithms. Applications of the proposed methodology range from the design of
new generative models to the improvement of the portability of Monte Carlo
algorithms, like Configurational Bias schemes, relying on exact sampling of a
subset of terms entering the Hamiltonian while generating new configurations.
