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Modified Cholesky Riemann Manifold Hamiltonian Monte Carlo: exploiting sparsity for fast sampling of high-dimensional targets

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Abstract

Riemann manifold Hamiltonian Monte Carlo (RMHMC) has the potential to produce high-quality Markov chain Monte Carlo output even for very challenging target distributions. To this end, a symmetric positive definite scaling matrix for RMHMC is proposed. The scaling matrix is obtained by applying a modified Cholesky factorization to the potentially indefinite negative Hessian of the target log-density. The methodology is able to exploit the sparsity of the Hessian, stemming from conditional independence modeling assumptions, and thus admit fast implementation of RMHMC even for high-dimensional target distributions. Moreover, the methodology can exploit log-concave conditional target densities, often encountered in Bayesian hierarchical models, for faster sampling and more straightforward tuning. The proposed methodology is compared to alternatives for some challenging targets and is illustrated by applying a state-space model to real data.

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References

  • Aït-Sahalia, Y.: Testing continuous-time models of the spot interest rate. Rev. Financ. Stud. 9(2), 385–426 (1996)

    Article  Google Scholar 

  • Andrieu, C., Doucet, A., Holenstein, R.: Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 72(3), 269–342 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Beskos, A., Pillai, N., Roberts, G., Sanz-Serna, J.-M., Stuart, A.: Optimal tuning of the hybrid Monte Carlo algorithm. Bernoulli 19(5A), 1501–1534 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Betancourt, M.: A general metric for Riemannian manifold Hamiltonian Monte Carlo. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information, Volume 8085 of Lecture Notes in Computer Science, vol. 8085, pp. 327–334. Springer, Berlin (2013a)

    Google Scholar 

  • Betancourt, M.: Identifying the Optimal Integration Time in Hamiltonian Monte Carlo. arXiv:1601.00225 (2016)

  • Betancourt, M., Byrne, S., Livingstone, S., Girolami, M.: The Geometric Foundations of Hamiltonian Monte Carlo. Bernoulli 23(4A), 2257–2298 (2017). doi:10.3150/16-BEJ810

    Article  MathSciNet  MATH  Google Scholar 

  • Betancourt, M.J.: Generalizing the No-u-turn Sampler to Riemannian Manifolds. arXiv:1304.1920 (2013b)

  • Betancourt, M.J., Girolami, M.: Hamiltonian Monte Carlo for Hierarchical Models. arXiv:1312.0906 (2013)

  • Calderhead, B.: A general construction for parallelizing Metropolis–Hastings algorithms. Proc Natl Acad Sci 111(49), 17408–17413 (2014)

    Article  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)

    Article  Google Scholar 

  • Chan, K.C., Karolyi, G.A., Longstaff, F.A., Sanders, A.B.: An empirical comparison of alternative models of the short-term interest rate. J. Finance 47(3), 1209–1227 (1992)

    Article  Google Scholar 

  • Davis, T.A.: Direct methods for sparse linear systems. In: Fundamentals of Algorithms 2. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2006)

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

    Article  Google Scholar 

  • Flury, T., Shephard, N.: Bayesian inference based only on simulated likelihood: particle filter analysis of dynamic economic models. Econom. Theory 27(Special Issue 05), 933–956 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.: Bayesian Data Analysis, 3rd edn. CRC Press, Boca Raton (2014)

    MATH  Google Scholar 

  • Geweke, J., Tanizaki, H.: On Markov chain Monte Carlo methods for nonlinear and non-gaussian state-space models. Commun. Stat. Simul. Comput. 28(4), 867–894 (1999)

    Article  MATH  Google Scholar 

  • Geweke, J., Tanizaki, H.: Note on the sampling distribution for the Metropolis–Hastings algorithm. Commun. Stat. Theory Methods 32(4), 775–789 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Geyer, C.J.: Practical Markov chain Monte Carlo. Stat. Sci. 7(4), 473–483 (1992)

    Article  Google Scholar 

  • Gill, P., Murray, W.: Newton-type methods for unconstrained and linearly constrained optimization. Math. Program. 7, 311–350 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  • Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Academic Press, London (1981)

    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 

  • Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  • Griewank, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia (2000)

    MATH  Google Scholar 

  • Grothe, O., Kleppe, T.S., Liesenfeld, R.: The Gibbs Sampler with Particle Efficient Importance Sampling for State-space Models. arXiv:1601.01125 (2016)

  • Guerrera, T., Rue, H., Simpson, D.: Discussion of Riemann manifold Langevin and Hamiltonian Monte Carlo by Girolami and Calderhead. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 73(2), 123–214 (2011)

    Article  MathSciNet  Google Scholar 

  • Haario, H., Saksman, E., Tamminen, J.: Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Stat. 14(3), 375–395 (1999)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Hogan, R .J.: Fast reverse-mode automatic differentiation using expression templates in C++. ACM Trans. Math. Softw. 40(4), 26:1–26:16 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Jasra, A., Singh, S.: Discussion of Riemann manifold Langevin and Hamiltonian Monte Carlo by Girolami and Calderhead. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 73(2), 123–214 (2011)

    Article  MathSciNet  Google Scholar 

  • Kleppe, T.S.: Adaptive step size selection for Hessian-based manifold Langevin samplers. Scand. J. Stat. 43(3), 788–805 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Kristensen, K., Nielsen, A., Berg, C., Skaug, H., Bell, B.: Tmb: automatic differentiation and Laplace approximation. J. Stat. Softw. 70(1), 1–21 (2016)

    Google Scholar 

  • Lan, S., Stathopoulos, V., Shahbaba, B., Girolami, M.: Markov chain Monte Carlo from Lagrangian dynamics. J. Comput. Graph. Stat. 24(2), 357–378 (2015)

    Article  MathSciNet  Google Scholar 

  • Leimkuhler, B., Reich, S.: Simulating Hamiltonian dynamics. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  • Lindgren, F., Rue, H., Lindström, J.: An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J. R. Stat. Soc. Ser. B 73(4), 423–498 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Lindsten, F., Jordan, M.I., Schön, T.B.: Particle Gibbs with ancestor sampling. J. Mach. Learn. Res. 15, 2145–2184 (2014)

    MathSciNet  MATH  Google Scholar 

  • Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. Springer, Berlin (2001)

    MATH  Google Scholar 

  • Mannseth, J., Kleppe, T.S., Skaug, H.J.: On the application of improved symplectic integrators in Hamiltonian Monte Carlo. Commun. Stat. Simul. Comput. 1–10 (2017). doi:10.1080/03610918.2017.1283703

  • Martin, J., Wilcox, L., Burstedde, C., Ghattas, O.: A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion. SIAM J. Sci. Comput. 34(3), A1460–A1487 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Neal, R.M.: Bayesian Learning for Neural Networks. Number 118 in Lecture Notes in Statistics. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  • Neal, R.M.: Slice sampling. Ann. Stat. 31(3), 705–767 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Neal, R. M.: MCMC using Hamiltonian dynamics. In: Brooks, S., Gelman, A., Jones, G. & Meng, X.-L. (eds) Handbook of Markov chain Monte Carlo, pp. 113–162. Chapman & Hall/CRC, Boca Raton (2010)

  • Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  • Pitt, M.K., dos Santos Silva, R., Giordani, P., Kohn, R.: On some properties of Markov chain Monte Carlo simulation methods based on the particle filter. J. Econom. 171(2), 134–151 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Qi, Y., and Minka, T.P.:Hessian-based Markov Chain Monte-Carlo algorithms. Unpublished manuscript (2002). https://www.cs.purdue.edu/homes/alanqi/papers/qi-minka-HMH-AMIT-02.pdf

  • Robert, C.P., Casella, G.: Monte Carlo Statistical Methods, 2nd edn. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  • Roberts, G., Stramer, O.: Langevin diffusions and Metropolis–Hastings algorithms. Methodol. Comput. Appl. Probab. 4(4), 337–357 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Rue, H.: Fast sampling of Gaussian Markov random fields. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 63(2), 325–338 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Rue, H., Held, L.: Gaussian Markov Random Fields: Theory and Application. Chapman and Hall-CRC Press, Boca Raton (2005)

    Book  MATH  Google Scholar 

  • Rue, H., Martino, S., Chopin, N.: Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 71(2), 319–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Scharth, M., Kohn, R.: Particle efficient importance sampling. J. Econom. 190(1), 133–147 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Schnabel, R., Eskow, E.: A new modified Cholesky factorization. SIAM J. Sci. Stat. Comput. 11(6), 1136–1158 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Schnabel, R., Eskow, E.: A revised modified Cholesky factorization algorithm. SIAM J. Optim. 9(4), 1135–1148 (1999)

  • Shephard, N., Pitt, M.K.: Likelihood analysis of non-Gaussian measurement time series. Biometrika 84, 653–667 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Skaug, H., Fournier, D.: Automatic approximation of the marginal likelihood in non-Gaussian hierarchical models. Comput. Stat. Data Anal. 56, 699–709 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Xifara, T., Sherlock, C., Livingstone, S., Byrne, S., Girolami, M.: Langevin diffusions and the Metropolis-adjusted Langevin algorithm. Stat. Probab. Lett. 91, 14–19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics and Natural Sciences, University of Stavanger, 4036, Stavanger, Norway

    Tore Selland Kleppe

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  1. Tore Selland Kleppe
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Correspondence to Tore Selland Kleppe.

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The author would like to thank the Editor, the Associate Editor, two referees, Michael Betancourt, Hans J. Skaug and Anders Tranberg for comments that have sparked many improvements.

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Kleppe, T.S. Modified Cholesky Riemann Manifold Hamiltonian Monte Carlo: exploiting sparsity for fast sampling of high-dimensional targets. Stat Comput 28, 795–817 (2018). https://doi.org/10.1007/s11222-017-9763-5

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