Measure Transport with Kernel Stein Discrepancy

Abstract

Measure transport underpins several recent algorithms for posterior approximation in the Bayesian context, wherein a transport map is sought to minimise the Kullback-Leibler divergence (KLD) from the posterior to the approximation. The KLD is a strong mode of convergence, requiring absolute continuity of measures and placing restrictions on which transport maps can be permitted. Here we propose to minimise a kernel Stein discrepancy (KSD) instead, requiring only that the set of transport maps is dense in an L 2 sense and demonstrating how this condition can be validated. The consistency of the associated posterior approximation is established and empirical results suggest that KSD is competitive and more flexible alternative to KLD for measure transport.

Stein Variational Gradient Descent: many-particle and long-time asymptotics

“…a quantity that has natural links with the cotangent space construction to be introduced in Section 3.2. Let us also note that I k Stein (ρ) is known in other contexts as the kernelised Stein discrepancy KSD(ρ|π) and has found various applications in scenarios where ρ needs to be compared to an unnormalised 4 distribution π, see [14,25,34]. In fact, the kernelised Stein discrepancy lies as the heart of the original derivation of SVGD, see [46].…”

“…Note that the chain rule, i.e. the last condition of (38), is expected to hold at a formal level, combining (25) and (35). Since ( 39) is always non-negative, the proposition continues to hold if '=' is replaced by '≤'; analogues of (39) are therefore often called energy-dissipation inequalities in the literature.…”

“…, where the derivative in P(R d ) is understood in the sense of (25). See [79, Theorem 2.2.1] for the general Novikov condition (in finite dimensions) and [63] for an even more general condition.…”