Hamiltonian Monte Carlo (2021)

https://bjlkeng.github.io/posts/hamiltonian-monte-carlo/

by KqAmJQ7 • 3 years ago

96 17 3 years ago

bjlkeng.github.io

Impressive how this post references back to a post six years before as the starting point. Shows the value of tenacity and sticking with it, as learning compounds over time.

nvrmnd3• 3 years ago

I did a PhD on Monte Carlo methods, and I remember really enjoying how elegant the HMC is. But seeing this here I feel I should comment, in the event that anyone happens to read this and thinks it would be a useful algorithm to implement. Words to heed:

"Monte Carlo is an extremely bad method; it should only be used when all alternative methods are worse." -- Alan Sokal

The reality of the situation is that you would only bother understanding and implementing such a complex algorithm if you had a rather difficult distribution to sample. Unfortunately, that means that you are very unlikely to know whether the answers you are getting are at all correct. You may have a bug in your code (highly likely), or the method's hyper parameters may be poorly tuned (likelier still), or the MCMC chain is not mixing rapidly leading to undetectable approximation error (almost certainly).

Almost always I would recommend attempting to simplify whatever model you are approximating and to use a method where approximation quality can be assured, rather than attempting to approximate complex (high dimensional) distributions with a more efficient algorithm like HMC.

Of course, sometimes you must approximate such models, and for this you should be glad the method exists, and good luck to you.

ailun4• 3 years ago

Why wouldn’t you just use Stan or Pymc or one of the other dozen existing implementations with something like HMC? They use it as the default sampling algorithm.

nvrmnd3• 3 years ago

Yes by all means, though I do hope anyone who is using this default algorithm is familiar with the other pitfalls I mention.

versteegen3• 3 years ago

I also did a PhD on MCMC (for sampling images), and agree fully with everything you said. I'll just add that I never felt I could trust HMC at all (I hardly used it) because it was too abstract, I couldn't form much intuition about why it mightn't be mixing. On the other hand I could easily understand the deficiencies of Gibbs sampling (in my case, changing one pixel at a time) and work around them (somewhat). I came away thinking that a 'feed-forward' approximation with a fixed amount of computation (e.g. a probabilistic neural net) would have been a much superior method to MCMC.

boulos• 3 years ago

superior method to MCMC

Superior in what ways?

I could imagine simplicity for many problems. Roughly "here, replace your complex simulation with this magic function approximator". But for problems where you're calculating a physical quantity with a true answer, I don't see NNs being "much superior".

nextos• 3 years ago

Unfortunately, that means that you are very unlikely to know whether the answers you are getting are at all correct.

For large multilevel GLMs I have found Rhat and posterior predictive checks to be sufficient to diagnose HMC sampling issues.

But, of course, these might not fall within the category of difficult distributions you mentioned.

cfcf14• 3 years ago

Amazing post, agree with everything you've said. I've always felt that the problems with advanced MCMC methods (HMC, RM-MC, etc) are even more painful when one looks at approximate bayesian methods - ADVI (variational approximation), SGLD (langevin dynamics), and so on. My grad research was on ABC-SMC methods, probably the last resort of all last resorts.

There was a period a few years ago when it was all the rage to take arbitrary probabilistic programming models and just toss ADVI at it blindly with fancy tools like Pymc3 or stan. I feel like everybody eventually came to the conclusion that if the model was simple enough that you could guarantee that ADVI was actually correct, you didn't need it, and if it wasn't, you couldn't possibly verify you were approximating anything close to the true posterior.

At least with HMC it would generally explode when dealing with pathological geometry (multimodality, non-identifiability, whatever), whereas lots of the approximate methods will 'converge' to completely incorrect answers. I know there's been some work on determining if things have gone off the rails (https://arxiv.org/pdf/1802.02538.pdf), but I couldn't ever find a place where it felt safe to use this stuff blindly.

syntaxers• 3 years ago

Here's a really cool interactive visualization of the Hamiltonian Monte Carlo algorithm, where you can choose different target distributions and play around with the algorithm parameters: https://chi-feng.github.io/mcmc-demo/app.html?algorithm=Hami...

Source code: https://github.com/chi-feng/mcmc-demo/blob/master/algorithms...

It also has visualizations for other flavors of Hamiltonian Monte Carlo, including the No-U-Turn Sampler (NUTS) used by Stan.

michaericalribo• 3 years ago

I did a PhD in statistics, and spent a lot of that time doing applied modeling.

My perspective is that MCMC is one of the great scientific discoveries of the 20th century. There are pros and cons to different samplers, lots of caveats, concerns over convergence—all valid. Still, MCMC opens up a whole new world of model building, letting you easily try a whole slew of models. It provides flexibility, even creativity. As a concept it’s clever. As a technical achievement it’s amazing.

Hamiltonian Monte Carlo (2021)

Related Stories