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A series about Monte Carlo methods and generating samples from probability distributions.
This is a series about Monte Carlo methods and sampling from statistical models. For a long time I avoided “stepping down” into the “cellar” of Bayesian statistics and tried to stay at high level tools like Stan or PyMC3 until I realized that (too) often I needed a finer grained approach.
The purpose of this blog post series is to give you a working understanding of fine-grained composable abstractions (FCA) that you can use to build adapted solutions for your problems. Along the way you might also develop a deeper understanding of statistical modelling. At least I benefited a lot from the deeper understanding you gain from implementing the whole process end-to-end.
In principle there are only 2 core building blocks for Monte Carlo simulation:
- Importance Sampling and its variations / synonyms like Sequential Importance Sampling (SIS), Sequential Monte Carlo (SMC), Particle Filters, …
- Markov chain Monte Carlo (MCMC) and its variations like Metropolis-Hastings (MH), Hamiltonian (or Hybrid) Monte Carlo (HMC, NUTS), Gibbs sampling, …
- Monte Carlo: Fundamental Concepts
- … more to be published soon …
For a general overview of the Bayesian approach to statistics I recommend the following books:
- 2014: Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan by John Kruschke
- 2020: Statistical Rethinking: A Bayesian Course with Examples in R and STAN by Richard McElreath
For a text book with a clear and comprehensive presentation of Monte Carlo / simulation methods I’d suggest to start with:
- 2020: Machine Learning: A Bayesian and Optimization Perspective by Sergios
The author is very careful and diligent with notation and gives good algorithm descriptions as pseudo code.
The following two books are more specialized on Importance Sampling / Sequential Monte Carlo and Markov Chain Monte Carlo respectively:
- 2001: Sequential Monte Carlo Methods in Practice by Arnaud Doucet, Nando de Freitas, Neil Gordon
- 2011: Handbook of Markov Chain Monte Carlo by Steve Brooks, Andrew Gelman, Galin Jones, Xiao-Li Meng
I also regularly encounter pointers to the following book, but did not read it myself yet:
- 2004: Monte Carlo Statistical Methods by Christian Robert, George Casella
I put the publishing year in front of the above references, because this is a fast-moving field and while the underlying core principles remain the same the state-of-the-art is evolving.