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Zero variance differential geometric Markov chain Monte Carlo algorithms. (English) Zbl 1327.60140

Summary: Differential geometric Markov Chain Monte Carlo (MCMC) strategies exploit the geometry of the target to achieve convergence in fewer MCMC iterations at the cost of increased computing time for each of the iterations. Such computational complexity is regarded as a potential shortcoming of geometric MCMC in practice. This paper suggests that part of the additional computing required by Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms produces elements that allow concurrent implementation of the zero variance reduction technique for MCMC estimation. Therefore, zero variance geometric MCMC emerges as an inherently unified sampling scheme, in the sense that variance reduction and geometric exploitation of the parameter space can be performed simultaneously without exceeding the computational requirements posed by the geometric MCMC scheme alone. A MATLAB package is provided, which implements a generic code framework of the combined methodology for a range of models.

MSC:

60J22 Computational methods in Markov chains
60D05 Geometric probability and stochastic geometry
62-04 Software, source code, etc. for problems pertaining to statistics
62F15 Bayesian inference
62J02 General nonlinear regression

Software:

Matlab; BayesDA; TFad
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Adler, S. L. (1981). “Over-Relaxation Method for the Monte Carlo Evaluation of the Partition Function for Multiquadratic Actions.” Physical Review D , 23: 2901-2904.
[2] Albert, J. H. and Chib, S. (1993). “Bayesian Analysis of Binary and Polychotomous Response Data.” Journal of the American Statistical Association , 88(422): 669-679. · Zbl 0774.62031
[3] Andradóttir, S., Heyman, D. P., and Ott, T. J. (1993). “Variance Reduction Through Smoothing and Control Variates for Markov Chain Simulations.” ACM Transactions on Modeling and Computer Simulation , 3(3): 167-189. · Zbl 0846.60071
[4] Assaraf, R. and Caffarel, M. (1999). “Zero-Variance Principle for Monte Carlo Algorithms.” Physical Review Letters , 83: 4682-4685.
[5] Atchadé, Y. F. and Perron, F. (2005). “Improving on the Independent Metropolis-Hastings Algorithm.” Statistica Sinica , 15(1): 3-18. · Zbl 1059.62086
[6] Barone, P. and Frigessi, A. (1990). “Improving Stochastic Relaxation for Gaussian Random Fields.” Probability in the Engineering and Informational Sciences , 4(03): 369-389. · Zbl 1134.60347
[7] Barone, P., Sebastiani, G., and Stander, J. (2001). “General Over-Relaxation Markov Chain Monte Carlo Algorithms for Gaussian Densities.” Statistics and Probability Letters , 52(2): 115-124. · Zbl 0987.60087
[8] Calderhead, B. and Girolami, M. (2009). “Estimating Bayes factors via thermodynamic integration and population MCMC.” Computational Statistics and Data Analysis , 53(12): 4028 -4045. · Zbl 1453.62055
[9] Calderhead, B., Girolami, M., and Lawrence, N. (2009). “Accelerating Bayesian Inference over Nonlinear Differential Equations with Gaussian Processes.” Advances in Neural Information Processing Systems, MIT Press , 21.
[10] Chan, P. S. (1993). “A Statistical Study of Log-Gamma Distribution.” Ph.D. thesis, McMaster University.
[11] Craiu, R. V. and Lemieux, C. (2007). “Acceleration of the Multiple-Try Metropolis Algorithm Using Antithetic and Stratified Sampling.” Statistics and Computing , 17(2): 109-120.
[12] Dellaportas, P. and Kontoyiannis, I. (2012). “Control Variates for Estimation Based on Reversible Markov Chain Monte Carlo Samplers.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 74(1): 133-161.
[13] Diaconis, P., Holmes, S., and Neal, R. M. (2000). “Analysis of a Nonreversible Markov Chain Sampler.” The Annals of Applied Probability , 10(3): pp. 726-752. · Zbl 1083.60516
[14] Duane, S., Kennedy, A. D., Pendleton, B. J., and Roweth, D. (1987). “Hybrid Monte Carlo.” Physics Letters B , 195(2): 216-222.
[15] Dyk, D. A. v. and Meng, X.-L. (2001). “The Art of Data Augmentation.” Journal of Computational and Graphical Statistics , 10(1): pp. 1-50. · Zbl 04565162
[16] Finney, D. J. (1947). “The Estimation from Individual Records of the Relationship Between Dose and Quantal Response.” Biometrika , 34(3-4): 320-334. · Zbl 0036.09701
[17] Flury, B. and Riedwyl, H. (1988). Multivariate Statistics . Chapman and Hall. · Zbl 0495.62057
[18] Fort, G., Moulines, E., Roberts, G. O., and Rosenthal, J. S. (2003). “On the Geometric Ergodicity of Hybrid Samplers.” Journal of Applied Probability , 40(1): pp. 123-146. · Zbl 1028.65002
[19] Gay, D. M. (2006). Semiautomatic Differentiation for Efficient Gradient Computations , volume 50, chapter 13, 147-158. Berlin/Heidelberg: Springer-Verlag. · Zbl 1270.65012
[20] Gelfand, A. E. and Smith, A. F. M. (1990). “Sampling-Based Approaches to Calculating Marginal Densities.” Quarterly of applied mathematics , 85(410): 398-409. · Zbl 0702.62020
[21] Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2004). Bayesian Data Analysis . Chapman and Hall. · Zbl 1039.62018
[22] Geyer, C. J. (1992). “Practical Markov Chain Monte Carlo.” Statistical Science , 7(4): 473-483.
[23] Geyer, C. J. and Mira, A. (2000). “On Non-Reversible Markov chains.” In Institute Communications, Volume 26: Monte Carlo Methods , 93-108. American Mathematical Society. · Zbl 0969.60071
[24] Girolami, M. and Calderhead, B. (2011). “Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 73(2): 123-214.
[25] Green, P. J. and Mira, A. (2001). “Delayed Rejection in Reversible Jump Metropolis-Hastings.” Biometrika , 88(4): pp. 1035-1053. · Zbl 1099.60508
[26] Hammer, H. and Tjemeland, H. (2008). “Control Variates for the Metropolis-Hastings Algorithm.” Scandinavian Journal of Statistics , 35(3): 400-414. · Zbl 1199.65018
[27] Henderson, S. G. (1997). “Variance Reduction via an Approximating Markov Process.” Ph.D. thesis, Stanford University.
[28] Mira, A., Solgi, R., and Imparato, D. (2012). “Zero Variance Markov Chain Monte Carlo for Bayesian Estimators.” Statistics and Computing , 1-10. · Zbl 1322.62212
[29] Mira, A. and Tierney, L. (2002). “Efficiency and Convergence Properties of Slice Samplers.” Scandinavian Journal of Statistics , 29(1): pp. 1-12. · Zbl 1018.91030
[30] Naumann, U. (2008). “Optimal Jacobian accumulation is NP-complete.” Mathematical Programming , 112(2): 427-441. · Zbl 1158.68013
[31] Philippe, A. and Robert, C. P. (2001). “Riemann Sums for MCMC Estimation and Convergence Monitoring.” Statistics and Computing , 11(2): 103-115.
[32] Pregibon, D. (1981). “Logistic Regression Diagnostics.” The Annals of Statistics , 9(4): pp. 705-724. · Zbl 0478.62053
[33] Ramsay, J. O., Hooker, G., Campbell, D., and Cao, J. (2007). “Parameter Estimation for Differential Equations: a Generalized Smoothing Approach.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 69(5): 741-796.
[34] Ripley, B. (1987). Stochastic Simulation . John Wiley & Sons. · Zbl 0613.65006
[35] Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods . Springer Texts in Statistics, 2nd edition. · Zbl 1096.62003
[36] Roberts, G. O. and Stramer, O. (2002). “Langevin Diffusions and Metropolis-Hastings Algorithms.” Methodology and Computing in Applied Probability , 4: 337-357. · Zbl 1033.65003
[37] Siskind, J. M. and Pearlmutter, B. A. (2008). “Nesting Forward-Mode AD in a Functional Framework.” Higher Order and Symbolic Computation , 21(4): 361-376. · Zbl 1175.68104
[38] Smith, S. P. (1995). “Differentiation of the Cholesky Algorithm.” Journal of Computational and Graphical Statistics , 4(2): 134-147.
[39] Solgi, R. and Mira, A. (2013). “A Bayesian Semiparametric Multiplicative Error Model with an Application to Realized Volatility.” Journal of Computational and Graphical Statistics , 22(3): 558-583.
[40] Swendsen, R. H. and Wang, J.-S. (1987). “Nonuniversal Critical Dynamics in Monte Carlo Simulations.” Physical Review Letters , 58: 86-88.
[41] Tierney, L. and Mira, A. (1999). “Some Adaptive Monte Carlo Methods for Bayesian Inference.” Statistics in Medicine , 18: 2507-2515.
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