Implicit Sampling, with Application to Data Assimilation

There are many computational tasks in which it is necessary to sample a given probability density function (or pdf for short), i.e., to use a computer to construct a sequence of independent random vectors x i (i=1,2,…), whose histogram converges to the given pdf. This can be difficult because the sample space can be huge, and more importantly, because the portion of the space where the density is significant, can be very small, so that one may miss it by an ill-designed sampling scheme. Indeed, Markov-chain Monte Carlo, the most widely used sampling scheme, can be thought of as a search algorithm, where one starts at an arbitrary point and one advances step-by-step towards the high probability region of the space. This can be expensive, in particular because one is typically interested in independent samples, while the chain has a memory. The authors present an alternative, in which samples are found by solving an algebraic equation with a random right-hand side rather than by following a chain; each sample is independent of the previous samples. The construction is explained in the context of numerical integration, and it is then applied to data assimilation.