This chain pump is not a good metaphor for how a Markov chain Monte Carlo sampler works, but it does correctly evoke the effort involved.
Despite studying within this area, I have nothing to say about MCMC broadly, but I do have some things I wish to keep notes on.
Hamiltonian Monte Carlo
Connection to variational inference
Deep. See (Salimans, Kingma, and Welling 2015)
Adaptive MCMC
See Adaptive MCMC.
Langevin Monte carlo
See log concave distributions for now.
Tempering
e.g. Ge, Lee, and Risteski (2020); Syed et al. (2020). Saif Syed can explain this quite well. Or, as Lee and Risteski explain:
The main idea is to create a meta-Markov chain (the simulated tempering chain) which has two types of moves: change the current “temperature” of the sample, or move “within” a temperature. The main intuition behind this is that at higher temperatures, the distribution is flatter, so the chain explores the landscape faster (see the figure below).
Mixing rates
Debiasing via coupling
Pierre E. Jacob, John O’Leary, Yves Atchadé, crediting Glynn and Rhee (2014), made MCMC estimators without finite-time-bias, which is nice for parallelisation (Jacob, O’Leary, and Atchadé 2017).
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