MCMC with strings and branes

branes“Roughly speaking, the idea [of MCMC] is that the thermal fluctuations of a particle moving in an energy landscape provides a conceptually elegant way to sample from a target distribution.”

A short version of their earlier long paper was arXived last week by Heckman et al. The starting point of the paper is to consider simultaneously M parallel samplers by envisioning the M-uple as a single object. This reminded me of the attempt we had made in our 1995 pinball sampler paper with Kerrie Mengersen, where we introduced a repulsive scheme to keep the particles apart and individually stationary against the same target. The joint target being defined as the product of the individual targets. As a single Markov chain, the MCMC sampler can take advantage of the parallel chains to possibly improve the efficiency when compared with running M parallel chains. Possibly only, because the target has moved to a space that is M times larger…

“…although the physics of point particles underlies much of our modern understanding of natural phenomena, it has proven fruitful to consider objects such as strings and branes with finite extent in p spatial dimensions.”

The details of the proposal are somewhat unclear in that the notion of brane remains a mystery to me. It sounds like a sort of random graph over the indices of the particles, but endowed with further (magical?) physical properties. As defined in the paper the suburban algorithm picks a random (neighbourhood) graph at each iteration that is used in the proposal over the particle system (or ensemble). As justified by the authors, the fact that this choice is independent of the current state of the Markov chain implies stationarity. If not efficiency, when compared with the independent parallel MCMC scheme. And because there are so many ways in taking into account the neighbours. (I did not see (11) as a particularly relevant implementation of the algorithm, mixing a random walk move with another random walk on the time differences.) Actually, I have somewhat of a worry about the term “nearest neighbour” which may be defined by the graph (which is fine) or by the configuration of the particle system at time t (which is not fine).

“The effective dimension rather than the overall topology of the grid plays the dominant role in the performance of the algorithm.”

The limited influence of the grid topology is quite understandable in that the chain targets an iid sample, so there is no reason one index value is more relevant than another. All particles in the vector are interchangeable in this respect and in the long run only the number of connected particles should matter. A more interesting feature is that the suburban sampler seems to perform best for strings, i.e. when the number of connected particles is larger than two. This somewhat agrees with my initial remark that dealing with the M particles as a single object should slow down convergence because of the dimension increase. This is one reason why we did not pursue our pinball sampler any further, as it seemed to converge quite slowly.

“To summarize: With too few friends one drifts into oblivion, but with too many friends one becomes a boring conformist.”

The notion of generating a sample all at once is quite appealing, especially because of the iid nature of the target, but I am not convinced the approach followed in this paper is sufficiently involved for this purpose. Alex¬†Shestopaloff and Radford Neal’s recent work on ensemble MCMC is certainly pushing things further in the analysis of efficient moves. I also wonder if a repulsive component shouldn’t be added to the target as in pinball sampling, borrowing maybe from determinental processes, towards a more thorough exploration of the target. Even though it remains unclear that this will be more efficient than running M parallel chains.

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