# Breaking NS & SMC

This is the rolling blog page for the second part of this project: to come up with some examples of practical applications that (i) break nested sampling without breaking SMC, and (ii) break SMC without breaking nested sampling.

Topic subpages can be found via the dropdown list above: Nested sampling -> Breaking NS & SMC -> Etc.

Q. Why SMC (Sequential Monte Carlo)?
A. Although nested sampling is by far the most popular tool for marginal likelihood estimation in astronomy/cosmology, and is also highly popular in physics, it is very rarely considered by applied statisticians outside of these fields, who would (generally) prefer to use their favourite implementation of SMC (also called population Monte Carlo) for this purpose.  Also, as noted in the ‘convergence proof’ section of this project, a theoretical understanding of nested sampling with MCMC may require some of the theory for particle systems already developed by SMC researchers.

Q. What do you mean by ‘break’?
A. To first order I mean ‘find a problem for which nested sampling [SMC] is demonstrably more efficient than SMC [nested sampling] at estimating the marginal likelihood (i.e., requires far fewer likelihood function evaluations to achieve a given level of accuracy in mean squared error).  Of course, one would hope for a sensible example problem such as might convince a pragmatist currently using only one of these techniques to consider learning to use the other.  Moreover, one would hope for an example requiring little fine tuning of the SMC algorithm (noting here that there are many flavours of SMC so we will need to make precise this point at a later stage; and further noting that a strength of SMC should be that it’s automatically adaptive) for the same reason.

To second order I am interested in the question of whether the ‘thermodynamic catastrophe’ or ‘change of phase’ problem described by Skilling (2006) (Section 3.8)—in which $L(X)$ features both concave and convex segments—does in fact ‘break’ SMC in the sense of drastically reducing the accuracy of SMC without compromising that of nested sampling.