## Bernoulli

• Bernoulli
• Volume 24, Number 4B (2018), 3711-3750.

### Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution

Kengo Kamatani

#### Abstract

The purpose of this paper is to introduce a new Markov chain Monte Carlo method and to express its effectiveness by simulation and high-dimensional asymptotic theory. The key fact is that our algorithm has a reversible proposal kernel, which is designed to have a heavy-tailed invariant probability distribution. A high-dimensional asymptotic theory is studied for a class of heavy-tailed target probability distributions. When the number of dimensions of the state space passes to infinity, we will show that our algorithm has a much higher convergence rate than the pre-conditioned Crank–Nicolson (pCN) algorithm and the random-walk Metropolis algorithm.

#### Article information

Source
Bernoulli, Volume 24, Number 4B (2018), 3711-3750.

Dates
Revised: March 2017
First available in Project Euclid: 18 April 2018

https://projecteuclid.org/euclid.bj/1524038768

Digital Object Identifier
doi:10.3150/17-BEJ976

Mathematical Reviews number (MathSciNet)
MR3788187

Zentralblatt MATH identifier
06869890

#### Citation

Kamatani, Kengo. Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution. Bernoulli 24 (2018), no. 4B, 3711--3750. doi:10.3150/17-BEJ976. https://projecteuclid.org/euclid.bj/1524038768

#### References

• [1] Beskos, A., Roberts, G. and Stuart, A. (2009). Optimal scalings for local Metropolis–Hastings chains on nonproduct targets in high dimensions. Ann. Appl. Probab. 19 863–898.
• [2] Chen, L.H.Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Probability and Its Applications. Springer, Berlin.
• [3] Cotter, S.L., Roberts, G.O., Stuart, A.M. and White, D. (2013). MCMC methods for functions: Modifying old algorithms to make them faster. Statist. Sci. 28 424–446.
• [4] Eberle, A. (2014). Error bounds for Metropolis–Hastings algorithms applied to perturbations of Gaussian measures in high dimensions. Ann. Appl. Probab. 24 337–377.
• [5] Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes: Characterization and Convergence. New York: Wiley.
• [6] Geyer, C.J. (1992). Practical Markov chain Monte Carlo. Statist. Sci. 7 473–483.
• [7] Goto, F. (2017). An extension and practical performances of mpcn algorithm. Master’s thesis, Graduate School of Engineering Science, Osaka University.
• [8] Hairer, M., Stuart, A.M. and Vollmer, S.J. (2014). Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions. Ann. Appl. Probab. 24 2455–2490.
• [9] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Berlin: Springer.
• [10] Kamatani, K. (2014). Rate optimality of Random walk Metropolis algorithm in high-dimension with heavy-tailed target distribution. Preprint. Available at arXiv:1406.5392.
• [11] Kamatani, K. (2014). Local consistency of Markov chain Monte Carlo methods. Ann. Inst. Statist. Math. 66 63–74.
• [12] Kamatani, K. (2017). Ergodicity of Markov chain Monte Carlo with reversible proposal. J. Appl. Probab. 54 638–654.
• [13] Kamatani, K., Nogita, A. and Uchida, M. (2016). Hybrid multi-step estimation of the volatility for stochastic regression models. Bull. Inform. Cybernet. 48 19–35.
• [14] Kamatani, K. and Uchida, M. (2015). Hybrid multi-step estimators for stochastic differential equations based on sampled data. Stat. Inference Stoch. Process. 18 177–204.
• [15] Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. New York: Springer.
• [16] Kotz, S. and Nadarajah, S. (2004). Multivariate $t$ Distributions and Their Applications. Cambridge: Cambridge Univ. Press.
• [17] Neal, R.M. (1999). Regression and classification using Gaussian process priors. In Bayesian Statistics, 6 (Alcoceber, 1998) 475–501. New York: Oxford Univ. Press.
• [18] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
• [19] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus, from Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge: Cambridge Univ. Press.
• [20] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Berlin: Springer.
• [21] Pillai, N.S., Stuart, A.M. and Thiery, A.H. (2014). Optimal proposal design for random walk type Metropolis Algorithms with Gaussian random field priors. Preprint. Available at arXiv:1108.1494v2.
• [22] Plummer, M., Best, N., Cowles, K. and Vines, K. (2006). Coda: Convergence diagnosis and output analysis for mcmc. R News 6 7–11.
• [23] Robert, C.P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. New York: Springer.
• [24] Roberts, G.O., Gelman, A. and Gilks, W.R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7 110–120.
• [25] Roberts, G.O. and Rosenthal, J.S. (1998). Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 255–268.
• [26] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press.
• [27] Shigekawa, I. (1980). Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20 263–289.
• [28] Stroock, D.W. and Varadhan, S.R.S. (1979). Multidimensional Diffussion Processes. Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen Mit Besonderer Berücksichtigung der Anwendungsgebiete. Berlin: Springer.
• [29] Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701–1762.