Exploiting Symmetries to Construct Efficient MCMC Algorithms With an Application to SLAM

The Metropolis-Hastings (MH) algorithm is a flexible method to generate samples from a target distribution, a key problem in probabilistic inference. In this paper we propose a variation of the MH algorithm based on group moves, where the next state is obtained by first choosing a random transformation of the state space and then applying this transformation to the current state. This adds much-needed flexibility to the “textbook” MH algorithm where all measures involved must be given in terms of densities with respect to a common reference measure. Under mild conditions, our main result extends the acceptance probability formula of the textbook algorithm to MH algorithms with group moves. We work out how the new algorithms can be used to exploit a problem’s natural symmetries and apply the technique to the simultaneous localization and mapping (SLAM) problem, obtaining the first fully rigorous justification of a previous MCMCbased SLAM method. New experimental results comparing our method to existing state-of-theart specialized methods on a standard range-only SLAM benchmark problem validate the strength of the approach.