Statistics and Computing

, Volume 28, Issue 3, pp 549–561 | Cite as

Langevin incremental mixture importance sampling

  • Matteo FasioloEmail author
  • Flávio Eler de Melo
  • Simon Maskell


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 (Biometrics 66(4):1162–1173, 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 to construct the mean vectors and covariance matrices of the mixture components by numerically solving certain differential equations, whose solution depends on the local shape 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 two synthetic non-Gaussian densities, one being defined on up to eighty dimensions, and on a Bayesian logistic regression model, using the Sonar dataset. The Julia code implementing the importance sampler proposed here can be found at


Importance sampling Langevin diffusion Mixture density Optimal importance distribution Local approximation Kalman-Bucy filter 



The authors would like to thank Samuel Livingstone and two anonymous referees for providing useful comments on an earlier version of this paper.

Supplementary material

11222_2017_9747_MOESM1_ESM.pdf (68 kb)
Supplementary material 1 (pdf 68 KB)


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK
  2. 2.School of Electrical Engineering, Electronics and Computer ScienceUniversity of LiverpoolLiverpoolUK

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