Adaptive radial-based direction sampling: some flexible and robust Monte Carlo integration methods. (English) Zbl 1085.62028
Summary: Adaptive radial-based direction sampling (ARDS) algorithms are specified for Bayesian analysis of models with non-elliptical, possibly multimodal target distributions. A key step is a radial-based transformation to directions and distances. After the transformation a Metropolis-Hastings method or, alternatively, an importance sampling method is applied to evaluate generated directions. Next, distances are generated from the exact target distribution. An adaptive procedure is applied to update the initial location and covariance matrix in order to sample directions in an efficient way. The ARDS algorithms are illustrated on a regression model with scale contamination and a mixture model for economic growth of the USA.
MSC:
| 62F15 | Bayesian inference |
| 65C60 | Computational problems in statistics (MSC2010) |
| 62P20 | Applications of statistics to economics |
| 65C40 | Numerical analysis or methods applied to Markov chains |
References:
| [1] | Bauwens, L., Bos, C.S., Van Dijk, H.K., Van Oest, R.D., 2002. Adaptive polar sampling: a class of flexible and robust Monte Carlo integration methods. Working paper, Econometric Institute Erasmus University Rotterdam.; Bauwens, L., Bos, C.S., Van Dijk, H.K., Van Oest, R.D., 2002. Adaptive polar sampling: a class of flexible and robust Monte Carlo integration methods. Working paper, Econometric Institute Erasmus University Rotterdam. · Zbl 1085.62028 |
| [2] | Box, G. E.P.; Muller, M. E., A note on the generation of random normal deviates, Annals of Mathematical Statistics, 29, 610-611 (1958) · Zbl 0085.13720 |
| [3] | Brownlee, K. A., Statistical Theory and Methodology in Science and Engineering (1965), Wiley: Wiley New York · Zbl 0136.39203 |
| [4] | Chen, M.-H.; Shao, Q.-M.; Ibrahim, J. G., Monte Carlo Methods in Bayesian Computation (2000), Springer: Springer New York · Zbl 0949.65005 |
| [5] | Frühwirth-Schnatter, S., Markov chain Monte Carlo estimation of classical and dynamic switching models, Journal of the American Statistical Association, 96, 194-209 (2001) · Zbl 1015.62022 |
| [6] | Geweke, J., Bayesian inference in econometric models using Monte Carlo integration, Econometrica, 57, 1317-1339 (1989) · Zbl 0683.62068 |
| [7] | Geweke, J., Using simulation methods for Bayesian econometric modelsinference, development, and communication, Econometric Reviews, 18, 1-73 (1999) · Zbl 0930.62105 |
| [8] | Gilks, W.R., Roberts, G.O., 1996. Strategies for improving MCMC. In: Gilks, W.R., Richardson, S., Spiegelhalter, D.J., (Eds.), Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC, New York, Boca Raton.; Gilks, W.R., Roberts, G.O., 1996. Strategies for improving MCMC. In: Gilks, W.R., Richardson, S., Spiegelhalter, D.J., (Eds.), Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC, New York, Boca Raton. · Zbl 0844.62100 |
| [9] | Gilks, W. R.; Roberts, G. O.; George, E. I., Adaptive direction sampling, The Statistician, 43, 179-189 (1994) · Zbl 0806.62052 |
| [10] | Givens, G. H.; Raftery, A. E., Local adaptive importance sampling for multivariate densities with strong nonlinear relationships, Journal of the American Statistical Association, 91, 132-141 (1996) · Zbl 0869.62025 |
| [11] | Hobert, J. P.; Casella, G., The effect of improper priors on Gibbs sampling in hierarchical linear mixed models, Journal of the American Statistical Association, 91, 1461-1473 (1996) · Zbl 0882.62020 |
| [12] | Hop, J. P.; Van Dijk, H. K., SISAM and MIXINTwo algorithms for the computation of posterior moments and densities using Monte Carlo integration, Computer Science in Economics and Management, 5, 183-220 (1992) · Zbl 0753.65104 |
| [13] | Justel, A.; Peña, D., Gibbs sampling will fail in outlier problems with strong masking, Journal of Computational & Graphical Statistics, 5, 176-189 (1996) |
| [14] | Kleibergen, F. R.; Van Dijk, H. K., On the shape of the likelihood/posterior in cointegration models, Econometric Theory, 10, 514-551 (1994) |
| [15] | Kleibergen, F. R.; Van Dijk, H. K., Bayesian simultaneous equations analysis using reduced rank structures, Econometric Theory, 14, 701-743 (1998) |
| [16] | Koop, G.; Van Dijk, H. K., Testing for integration using evolving trend and seasonal modelsa Bayesian approach, Journal of Econometrics, 97, 261-291 (2000) · Zbl 1079.62557 |
| [17] | Monahan, J.; Genz, A., Spherical-radial integration rules for Bayesian computation, Journal of the American Statistical Association, 92, 664-674 (1997) · Zbl 0889.62019 |
| [18] | Muirhead, R. J., Aspects of Multivariate Statistical Theory (1982), Wiley: Wiley New York · Zbl 0556.62028 |
| [19] | Oh, M. S.; Berger, J. O., Adaptive importance sampling in Monte Carlo integration, Journal of Statistical Computation and Simulation, 41, 143-168 (1992) · Zbl 0781.65016 |
| [20] | Rubinstein, R., Simulation and the Monte Carlo Method (1981), Wiley: Wiley New York · Zbl 0529.68076 |
| [21] | Schmeiser, B., Chen, M.-H., 1991. General hit-and-run Monte Carlo sampling for evaluating multidimensional integrals. Working paper, School of Industrial Engineering Purdue University.; Schmeiser, B., Chen, M.-H., 1991. General hit-and-run Monte Carlo sampling for evaluating multidimensional integrals. Working paper, School of Industrial Engineering Purdue University. |
| [22] | Smith, A. F.M.; Roberts, G. O., Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, Journal of the Royal Statistical Society, Series B, 55, 3-23 (1993) · Zbl 0779.62030 |
| [23] | Van Dijk, H. K.; Kloek, T., Further experience in Bayesian analysis using Monte Carlo integration, Journal of Econometrics, 14, 307-328 (1980) · Zbl 0465.62106 |
| [24] | Van Dijk, H.K., Kloek, T., 1984. Experiments with some alternatives for simple importance sampling in Monte Carlo integration. In: Bernardo, J.M., Degroot, M., Lindley, D., Smith, A.F.M. (Eds.), Bayesian Statistics, Vol. 2. Amsterdam, North Holland.; Van Dijk, H.K., Kloek, T., 1984. Experiments with some alternatives for simple importance sampling in Monte Carlo integration. In: Bernardo, J.M., Degroot, M., Lindley, D., Smith, A.F.M. (Eds.), Bayesian Statistics, Vol. 2. Amsterdam, North Holland. |
| [25] | Van Dijk, H. K.; Kloek, T.; Boender, C. G.E., Posterior moments computed by mixed integration, Journal of Econometrics, 29, 3-18 (1985) · Zbl 0585.62195 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
