This post is the second of a two-part series on early Monte Carlo methods from the 1940s and 1950s. In my previous post I gave an overview of Monte Carlo methods in the 40s and focused on the 1949 conference in Los Angeles. In this post I’ll go over the classic paper Equation of State Calculations by Fast Computing Machines (1953) by Nick Metropolis and co-authors. I’ll give an overview of the paper and its main results and then give some context about how it was written. I’ll then delve into some details of how the MANIAC computer worked to give an idea of what it must have been like to write algorithms such as MCMC on it.

This post is the first of a two-part series on early Monte Carlo methods from the 1940s and 1950s. Although Monte Carlo methods had been previously hinted at (for example Buffon’s needle), these methods only started getting serious attention in the 1940s with the development of fast computers. In 1949 a conference on Monte Carlo was held in Los Angeles in which the participants - many of them legendary mathematicians, physicists, and statisticians - shared their work involving Monte Carlo methods. It can be enlightening to review this research today to understand what issues these researchers were grappling with. In this post I’ll give a brief overview on computing and Monte Carlo in the 1940s, and then discuss some of the talks from this 1949 conference. In the next post I’ll go over the classic 1953 paper on the Metropolis sampler and give some details of the MANIAC computer.

A few years ago Bob Carpenter wrote a fantastic post on how why ensemble methods cannot work in high dimensions. He explains how high dimensions proposals that interpolate and extrapolate among samples are unlikely to fall in the typical set, causing the sampler to fail. During my PhD I worked on sampling problems with infinite-dimension parameters (ie: functions) and intractible gradients. After reading Bob’s post I always wondered if there was a way around this problem.

JAX allows you to write optimisers and samplers which are really fast if you use the scan or fori_loop functions. However if you write them in this way it’s not obvious how to add progress bar for your algorithm. This post explains how to make a progress bar using Python’s print function as well as using tqdm. After briefly setting up the sampler, we first go over how to create a basic version using Python’s print function, and then show how to create a nicer version using tqdm. You can find the code for the basic version here and the code for the tqdm version here.

This post goes over 3 ways to write a sampler using JAX. I found that although there are a bunch of tutorials about learning the basics of JAX, it was not clear to me what was the best way to write a sampler in JAX. In particular, how much of the sampler should you write in JAX? Just the log-posterior (or the loss in the case of optimisation), or the entire loop? This blog post tries to answer this by going over 3 ways to write a sampler while focusing on the speed of each sampler.