Measure Transport with Kernel Stein Discrepancy
This work addresses the need for more flexible posterior approximation methods in Bayesian statistics, though it appears incremental as it builds on existing measure transport techniques.
The paper tackles the problem of posterior approximation in Bayesian inference by proposing to minimize kernel Stein discrepancy (KSD) instead of Kullback-Leibler divergence (KLD) for measure transport, resulting in a more flexible method that is competitive with KLD-based approaches.
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.