PHYS366: Statistical Methods in Astrophysics

Lesson 4: Inference in Practice: Sampling Techniques

Goals for this session:

• Brainstorm, implement, and test methods of accelerating the convergence of Metropolis-based MCMC.
• Explore the common and uncommon failure modes of MCMC.
• Learn about the advanced techniques that address these problems (or don't!), and the available software packages implementing them.

Related reading:

• Ivezic section 5.8
• MacKay chapters 29-30
• Gelman chapters 11 and 13
• Online resources (reading and software)
  • Notes on CosmoMC by Antony Lewis: recent and less recent
  • MultiNest paper by Feroz et al. (for python)
  • emcee paper by Foreman-Makey et al. (for python)
  • pymc, which is, yes, for python
  • BUGS and JAGS
  • kombine for python

About this session

By now, you know that the main practical challenge of doing Bayesian inference is obtaining samples from the posterior distribution. Today's class is all about the nuts and bolts of sampling from probability distributions that, in general, we don't know very much about ahead of time. We'll start out by playing around with a case where we know the right answer analytically, namely the posterior from fitting a linear model to data with Gaussian uncertainites.

1. recap generative model
2. generate some data
3. review likelihood function
4. review analytic posterior

Accelerating Metropolis sampling

Difficult densities

More advanced sampling techniques