  This work proposes a novel method through which local information about the
target density can be used to construct an efficient importance sampler. The
backbone of the proposed method is the Incremental Mixture Importance Sampling
(IMIS) algorithm of Raftery and Bao (2010), which builds a mixture importance
distribution incrementally, by positioning new mixture components where the
importance density lacks mass, relative to the target. The key innovation
proposed here is that the mixture components used by IMIS are local
approximations to the target density. In particular, their mean vectors and
covariance matrices are constructed by numerically solving certain differential
equations, whose solution depends on the gradient field of the target
log-density. The new sampler has a number of advantages: a) it provides an
extremely parsimonious parametrization of the mixture importance density, whose
configuration effectively depends only on the shape of the target and on a
single free parameter representing pseudo-time; b) it scales well with the
dimensionality of the target; c) it can deal with targets that are not log-
concave. The performance of the proposed approach is demonstrated on a
synthetic non-Gaussian multimodal density, defined on up to eighty dimensions,
and on a Bayesian logistic regression model, using the Sonar data-set. The
Julia code implementing the importance sampler proposed here can be found at
https:/github.com/mfasiolo/LIMIS.
