Skip to main content
Log in

Langevin incremental mixture importance sampling

  • Published:
Statistics and Computing Aims and scope Submit manuscript


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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others


  1. Note that, of course, most of the conventional SMC samplers and particle filters are based on IS.


  • Ascher, U.M, Petzold, L.R.: Computer methods for ordinary differential equations and differential-algebraic equations. Soc. Ind. Appl. Math. 73–78 (1998)

  • Bates, S.: Bayesian inference for deterministic simulation models for environmental assessment. PhD Thesis, University of Washington (2001)

  • Bezanson, J., Karpinski, S., Shah, V.B., Edelman, A.: Julia: a fast dynamic language for technical computing. arXiv:1209.5145 (2012)

  • Brent, R.P.: Algorithms for Minimization Without Derivatives. Courier Corporation, North Chelmsford (2013)

    MATH  Google Scholar 

  • Bucy, R.S., Joseph, P.D.: Filtering for stochastic processes with applications to guidance. Am. Math. Soc. 43–55 (1987)

  • Bunch, P., Godsill, S.: Approximations of the optimal importance density using Gaussian particle flow importance sampling. J. Am. Stat. Assoc. 111(514), 748–762 (2016)

    Article  MathSciNet  Google Scholar 

  • Cappé, O., Douc, R., Guillin, A., Marin, J.M., Robert, C.P.: Adaptive importance sampling in general mixture classes. Stat. Comput. 18(4), 447–459 (2008)

    Article  MathSciNet  Google Scholar 

  • Carpenter, B., Gelman, A., Hoffman, M., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., Riddell, A.: Stan: a probabilistic programming language. J. Stat. Softw. 76(1), 1–32 (2017)

  • Daum, F., Huang, J.: Particle flow for nonlinear filters with log-homotopy. In: SPIE Defense and Security Symposium, International Society for Optics and Photonics, pp. 696,918–696,918 (2008)

  • Duane, S., Kennedy, A.D., Pendleton, B.J., Roweth, D.: Hybrid monte carlo. Phys. Lett. B 195(2), 216–222 (1987)

    Article  Google Scholar 

  • Faes, C., Ormerod, J.T., Wand, M.P.: Variational Bayesian inference for parametric and nonparametric regression with missing data. J. Am. Stat. Assoc. 106(495), 959–971 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Girolami, M., Calderhead, B.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 73(2), 123–214 (2011)

    Article  MathSciNet  Google Scholar 

  • Givens, G.H., Raftery, A.E.: Local adaptive importance sampling for multivariate densities with strong nonlinear relationships. J. Am. Stat. Assoc. 91(433), 132–141 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Gorman, R.P., Sejnowski, T.J.: Analysis of hidden units in a layered network trained to classify sonar targets. Neural Netw. 1(1), 75–89 (1988)

    Article  Google Scholar 

  • Haario, H., Saksman, E., Tamminen, J.: An adaptive Metropolis algorithm. Bernoulli 7(2), 223–242 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Hoffman, M.D., Gelman, A.: The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mac. Learn. Res. 15(1), 1593–1623 (2014)

    MathSciNet  MATH  Google Scholar 

  • Ionides, E.L.: Truncated importance sampling. J. Comput. Graph. Stat. 17(2), 295–311 (2008)

    Article  MathSciNet  Google Scholar 

  • Kong, A., Liu, J.S., Wong, W.H.: Sequential imputations and Bayesian missing data problems. J. Am. Stat. Assoc. 89(425), 278–288 (1994)

    Article  MATH  Google Scholar 

  • Lichman, M.: UCI machine learning repository. URL (2013)

  • Raftery, A.E., Bao, L.: Estimating and projecting trends in HIV/AIDS generalized epidemics using incremental mixture importance sampling. Biometrics 66(4), 1162–1173 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Roberts, G.O., Tweedie, R.L.: Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2(4), 341–363 (1996)

  • Roberts, G.O., Rosenthal, J.S., et al.: Optimal scaling for various Metropolis-Hastings algorithms. Stat. Sci. 16(4), 351–367 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Schuster I.: Gradient importance sampling. arXiv:1507.05781. (2015)

  • Sim, A., Filippi, S., Stumpf, M.P.: Information geometry and sequential Monte Carlo. arXiv:1212.0764. (2012)

  • Süli, E., Mayers, D.F.: An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  • West, M.: Modelling with mixtures. In: Berger, J., Bernardo, J., Dawid, A., Smith, A. (eds.) Bayesian Statistics, pp. 503–525. Oxford University Press, Oxford (1992)

    Google Scholar 

Download references


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

Author information

Authors and Affiliations


Corresponding author

Correspondence to Matteo Fasiolo.

Additional information

This work was partially funded by the Defence Science and Technology Laboratory through projects WSTC0058 and CDE36610.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 68 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fasiolo, M., de Melo, F.E. & Maskell, S. Langevin incremental mixture importance sampling. Stat Comput 28, 549–561 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: