This week, Andrea Rizzi will present a paper titled: "A Common Derivation for Markov Chain Monte Carlo Algorithms with Tractable and Intractable Targets". The paper was suggested to us by Lee Zamparo from Christina Leslie's few months ago. The paper lays down a general framework for MCMC algorithms, and subsumes as special cases
2) Gibbs sampling
3) Metropolis-Hastings withing Gibbs sampling
4) Slice sampling
5) Directional sampling
6) Directional slice sampling
7) Langevin and Hamiltonian Monte Carlo sampling
8) Elliptical Hamiltonian Slice sampling
9) Pseudo Marginal Metropolis–Hastings
10) Pseudo Marginal Hamiltonian Slice sampling
We very likely won't have time to go through all of them, but let us know if there are some in which you are particularly interested!
Markov chain Monte Carlo is a class of algorithms for drawing Markovian samples from high dimensional target densities to approximate the numerical integration associated with computing statistical expectation, especially in Bayesian statistics. However, many Markov chain Monte Carlo algorithms do not seem to share the same theoretical support and each algorithm is proven in a different way. This incurs a large amount of terminologies and ancillary concepts, which makes Markov chain Monte Carlo literature seems to be scattered and intimidating to researchers from many other fields, including new researchers of Bayesian statistics.
A generalised version of the Metropolis–Hastings algorithm is constructed with a random number generator and a self–reverse mapping. This formulation admits many other Markov chain Monte Carlo algorithms as special cases. A common derivation for many Markov chain Monte Carlo algorithms is useful in drawing connections and comparisons between these algorithms. As a result, we now can construct many novel combinations of multiple Markov chain Monte Carlo algorithms that amplify the efficiency of each individual algorithm. Specifically, we reinterpret slice sampling as a special case of Metropolis–Hastings and then propose two novel sampling schemes that combine slice sampling with directional or Hamiltonian sampling. Our Hamiltonian slice sampling scheme is also applicable in the pseudo marginal context where the target density is intractable but can be unbiasedly estimated, e.g. using particle filtering.