Monte Carlo (MC) sampling methods are widely applied in Bayesian inference, system simulation and optimization problems. The Markov Chain Monte Carlo (MCMC) algorithms are a well-known class of MC methods which generate a Markov chain with the desired invariant distribution. In this document, we focus on the Metropolis-Hastings (MH) sampler, which can be considered as the atom of the MCMC techniques, introducing the basic notions and different properties. We describe in details all the elements involved in the MH algorithm and the most relevant variants. Several improvements and recent extensions proposed in the literature are also briefly discussed, providing a quick but exhaustive overview of the current Metropolis-based sampling's world.
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Buy a book on modern data science, avoid statistics textbooks re-labeled as data science like plague: they will lead you to nowhere. Any public-domain stuff that's been invented 50 years ago will lead to a job that will eventually be replaced by a robot - we are working on this to make it happen. If you have an analytic background, my book is a good start. Older versions are still available for free, but the Wiley version is much more organized and easy to read, and costs less than $25. Other books can be found in the reference section below.