Bayesian inference with optimal maps

Tarek El Moselhy and Youssef Marzouk just arXived a paper entitled `Bayesian inference with optimal maps‘. The idea of a map is to find a transform f that maps the prior into the posterior. This is a radically novel idea, with the core identity (involving the Jacobian of f)

setting the constraint on the transform f in terms of the prior p and the likelihood L(x|D). (Plus a non-negligible monotonicity constraint on f…) In practice, the idea however seems delicate to implement the idea, even when all terms in the above fixed point equation are known. The authors suggest using a functional basis for the construction of f, e.g., polynomials. However, the amount of approximation due to this choice seems hard to ascertain, the variance of the difference

being the stopping criterion advanced by the authors. (Imposing monotonicity on high degree polynomials is an additional difficulty.) The models illustrating the methods are nonlinear ODEs and high-dimensional PDEs whose approximation mostly escapes my intuition. Therefore, I remain intrigued by the idea if perplex about its applicability to less regular problems. (Fixed point equations like the above are also hinting at Markov solutions, even though the optimisation in f is not operating in the usual space…)