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Markov chain Monte Carlo is a class of algorithms for drawing Markoviansamples from high-dimensional target densities to approximate the numericalintegration associated with computing statistical expectation, especially inBayesian statistics. However, many Markov chain Monte Carlo algorithms do notseem to share the same theoretical support and each algorithm is proven in adifferent way. This incurs many terminologies and ancillary concepts, whichmakes Markov chain Monte Carlo literature seems to be scattered andintimidating to researchers from many other fields, including new researchersof Bayesian statistics.  A generalised version of the Metropolis-Hastings algorithm is constructedwith a random number generator and a self-reverse mapping. This formulationadmits many other Markov chain Monte Carlo algorithms as special cases. Acommon derivation for many Markov chain Monte Carlo algorithms is useful indrawing connections and comparisons between these algorithms. As a result, wenow can construct many novel combinations of multiple Markov chain Monte Carloalgorithms that amplify the efficiency of each individual algorithm.Specifically, we propose two novel sampling schemes that combine slice samplingwith directional or Hamiltonian sampling. Our Hamiltonian slice sampling schemeis also applicable in the pseudo-marginal context where the target density isintractable but can be unbiasedly estimated, e.g. using particle filtering.

A Common Derivation for Markov Chain Monte Carlo Algorithms with Tractable and Intractable Targets