  Lifted probabilistic inference algorithms have been successfully applied to a
large number of symmetric graphical models. Unfortunately, the majority of
real-world graphical models is asymmetric. This is even the case for relational
representations when evidence is given. Therefore, more recent work in the
community moved to making the models symmetric and then applying existing
lifted inference algorithms. However, this approach has two shortcomings.
First, all existing over-symmetric approximations require a relational
representation such as Markov logic networks. Second, the induced symmetries
often change the distribution significantly, making the computed probabilities
highly biased. We present a framework for probabilistic sampling-based
inference that only uses the induced approximate symmetries to propose steps in
a Metropolis-Hastings style Markov chain. The framework, therefore, leads to
improved probability estimates while remaining unbiased. Experiments
demonstrate that the approach outperforms existing MCMC algorithms.
