Lifted samplers for partially ordered discrete state-spaces
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A technique called lifting is employed in practice for avoiding that the Markov chains simulated for sampling backtrack too often. It consists in lifting the state-space to include direction variables for guiding these chains. Its implementation is direct when the probability mass function targeted is defined on a totally ordered set, such as that of a univariate random variable taking values on the integers. In this paper, we adapt this technique to the situation where only a partial order can be established and explore its benefits. Important applications include simulation of systems formed from binary variables, such as those described by the Ising model, and variable selection when the marginal model probabilities can be evaluated, up to a normalising constant. To accommodate for the situation where one does not have access to these marginal model probabilities, a lifted trans-dimensional sampler for partially ordered model spaces is introduced. We show through theoretical analyses and empirical experiments that the lifted samplers outperform their non-lifted counterparts in some situations, and this at no extra computational cost. The code to reproduce all experiments is available online.
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