zbMATH — the first resource for mathematics

Theoretical properties of quasi-stationary Monte Carlo methods. (English) Zbl 1408.60072
Summary: This paper gives foundational results for the application of quasi-stationarity to Monte Carlo inference problems. We prove natural sufficient conditions for the quasi-limiting distribution of a killed diffusion to coincide with a target density of interest. We also quantify the rate of convergence to quasi-stationarity by relating the killed diffusion to an appropriate Langevin diffusion. As an example, we consider in detail a killed Ornstein-Uhlenbeck process with Gaussian quasi-stationary distribution.

MSC:
 60J60 Diffusion processes 65C05 Monte Carlo methods 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Full Text:
References:
 [1] Bardenet, R., Doucet, A. and Holmes, C. (2017). On Markov chain Monte Carlo methods for tall data. J. Mach. Learn. Res.18 Paper No. 47, 43. · Zbl 1433.68394 [2] Benaim, M., Cloez, B. and Panloup, F. (2018). Stochastic approximation of quasi-stationary distributions on compact spaces and applications. Ann. Appl. Probab.28 2370–2416. · Zbl 06974754 [3] Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli12 1077–1098. · Zbl 1129.60073 [4] Blanchet, J., Glynn, P. and Zheng, S. (2016). Analysis of a stochastic approximation algorithm for computing quasi-stationary distributions. Adv. in Appl. Probab.48 792–811. · Zbl 1352.60106 [5] Braverman, M., Milatovich, O. and Shubin, M. (2002). Essential selfadjointness of Schrödinger-type operators on manifolds. Uspekhi Mat. Nauk57 3–58. [6] Champagnat, N. and Villemonais, D. (2016). Exponential convergence to quasi-stationary distribution and $$Q$$-process. Probab. Theory Related Fields164 243–283. · Zbl 1334.60015 [7] Collet, P., Martínez, S. and San Martín, J. (2013). Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems. Springer, Heidelberg. [8] Dalalyan, A. S. (2017). Theoretical guarantees for approximate sampling from smooth and log-concave densities. J. R. Stat. Soc. Ser. B. Stat. Methodol.79 651–676. [9] Davies, E. B. (1995). Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics42. Cambridge Univ. Press, Cambridge. · Zbl 0893.47004 [10] Demuth, M. and van Casteren, J. A. (2000). Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional Integration Approach. Birkhäuser, Basel. · Zbl 0980.60005 [11] Diaconis, P. and Miclo, L. (2015). On quantitative convergence to quasi-stationarity. Ann. Fac. Sci. Toulouse Math. (6) 24 973–1016. · Zbl 1335.60142 [12] Durmus, A. and Moulines, É. (2017). Nonasymptotic convergence analysis for the unadjusted Langevin algorithm. Ann. Appl. Probab.27 1551–1587. · Zbl 1377.65007 [13] Kolb, M. and Steinsaltz, D. (2012). Quasilimiting behavior for one-dimensional diffusions with killing. Ann. Probab.40 162–212. · Zbl 1278.60121 [14] Mandl, P. (1961). Spectral theory of semi-groups connected with diffusion processes and its application. Czechoslovak Math. J.11 (86) 558–569. · Zbl 0115.13503 [15] Metafune, G., Pallara, D. and Priola, E. (2002). Spectrum of Ornstein–Uhlenbeck operators in $$L^{p}$$ spaces with respect to invariant measures. J. Funct. Anal.196 40–60. · Zbl 1027.47036 [16] Ouhabaz, E. M. (2005). Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series31. Princeton Univ. Press, Princeton, NJ. · Zbl 1082.35003 [17] Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics45. Cambridge Univ. Press, Cambridge. · Zbl 0858.31001 [18] Pinsky, R. G. (2009). Explicit and almost explicit spectral calculations for diffusion operators. J. Funct. Anal.256 3279–3312. · Zbl 1173.34052 [19] Pollett, P. K. (2015). Quasi-stationary distributions: A bibliography. http://www.maths.uq.edu.au/ pkp/papers/qsds/qsds.html. [20] Pollock, M., Fearnhead, P., Johansen, A. M. and Roberts, G. O. (2016). The scalable Langevin exact algorithm: Bayesian inference for big data. Preprint. Available at arXiv:1609.03436. [21] Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York. · Zbl 0401.47001 [22] Roberts, G. O. and Tweedie, R. L. (1996). Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli2 341–363. · Zbl 0870.60027 [23] Simon, B. (1993). Large time behavior of the heat kernel: On a theorem of Chavel and Karp. Proc. Amer. Math. Soc.118 513–514. · Zbl 0789.58075 [24] Tuominen, P. and Tweedie, R. L. (1979). Exponential decay and ergodicity of general Markov processes and their discrete skeletons. Adv. in Appl. Probab.11 784–803. · Zbl 0421.60065 [25] Tweedie, R. L. (1974). $$R$$-theory for Markov chains on a general state space. I. Solidarity properties and $$R$$-recurrent chains. Ann. Probab.2 840–864. · Zbl 0292.60097 [26] van der Vaart, A.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.