(interacting) MCMC for multimodal posterior

Question

I am trying to sample from a posterior having many modes particularly far from each others using MCMC. It appears that in most cases, only one of these modes contains the 95% hpd I am looking for. I tried to implement solutions based on tempered simulating but this does not provide satisfactory results as in practice going from one "capture range" to another is to costly.

As a consequence, it appears to me that a more efficient solution would be to run many simple MCMCs from different starting points and to dive into the dominant solution by making the MCMCs interact each other. Do you know if there is some proper way to implement such an idea ?

Note: I found that paper http://lccc.eecs.berkeley.edu/Papers/dmcmc_short.pdf (Distributed Markov chain Monte Carlo, Lawrence Murray) that looks close to what I am looking for but I really do not understand the design of the function $R_i$.

[EDIT]: the lack of answers seems to indicate that there is no obvious solution to my initial problem (making several MCMCs sampling from the same target distribution from different starting point interact each other). Is that true ? why is it so complicated ? Thanks

Answer 1

Thirst of all I would recommend to look for a better method, or at least a method with more in-depth description, since the "Distributed Markov chain Monte Carlo" from the paper that you are refering doesn't seem to be clearly stated. The advantages and disadvantages are not well explored. There is a method, that showed up in arxiv quite recently called "Wormhole Hamiltonian Monte Carlo", I would recommend to check it.

Going back to the paper that you gave reference, the remote proposal $R_{i}(\theta_{i})$ is very vaguely discribed. In the application part it is described as "maximum likelihood Gaussian over the preceeding t/2 samples". Maybe this means that you average the last t/2 values of the $i^{th}$ chain? A bit hard to guess with the poor description given in the reference.

[UPDATE:] Interaction between several chains and the application of this idea to sample from posterior distribution can be found in parallel MCMC methods, for example here. However, running several chains and forcing them to interact may not fit for the multimodal posterior: for example, if there is a very pronounced region where most of the posterior distribution is concentrated the interaction of the chains may even worsten things by sticking to that specific region and not exploring other, less pronounced, regions/modes. So, I would strongly recommend to look for MCMC specifically designed for multimodal problems. And if you want to create another/new method, then after you know what is available in the "market", you can create more efficient method.

Answer 2

You should try multinest: https://arxiv.org/pdf/0809.3437.pdf https://github.com/JohannesBuchner/MultiNest It's a bayesian inference engine that will give you parameter samples for a multimodal distribution.

The github link contains multinest source code that you compile and install as per instructions. it also has a python wrapper that's easier to use. The example codes have a prior section which serves to constrain your parameters, and a likelihood section which contains your likelihood. the settings file contains all your settings, and chains folder multinest output after fitting. it will give you samples of your parameters

Answer 3

This appears to be a difficult and ongoing problem in computational stats. However, there are a couple of less state-of-the-art methods that should work ok.

Say you have already found several distinct modes of the posterior and you are happy that these are the most important modes, and if the posterior around these modes is reasonably normal. Then you can calculate the hessian at these modes (say, using optim in R with hessian=T) and you can approximate the posterior as a mixture of normals (or t distributions). See p318-319 in Gelman et al. (2003) "Bayesian Data Analysis" for details. Then you can use the normal/t-mixture approximation as the proposal distribution in an independence sampler to obtain samples from the full posterior.

Another idea, that I haven't tried, is Annealed Importance Sampling (Radford Neal, 1998, link here).

Answer 4

What about trying a new MCMC method for multimodality, a repulsive-attractive Metropolis algorithm (http://arxiv.org/abs/1601.05633)? This multimodal sampler works with a single tuning parameter like a Metropolis algorithm and is simple to implement.