Summary
Sampling from probability density functions (pdfs) has become more and more important in many areas of applied science, and has therefore been the subject of great attention. Many sampling procedures proposed allow for approximate or asymptotic sampling. On the other hand, very few methods allow for exact sampling. Direct sampling of standard pdfs is feasible, but sampling of much more complicated pdfs is often required. Rejection sampling allows to exactly sample from univariate pdfs, but has the huge drawback of needing a case-by-case calculation of a comparison function that often reveals as a tremendous chore, whose results dramatically affect the efficiency of the sampling procedure. In this paper, we restrict ourselves to a pdf that is proportional to a product of standard distributions. From there, we show that an automated selection of both the comparison function and the upper bound is possible. Moreover, this choice is performed in order to optimize the sampling efficiency among a range of potential solutions. Finally, the method is illustrated on a few examples.
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Notes
1respectively defined as \(\frac{\|\hat{\mu}-\mu\|_2}{\|\mu\|_2}\) and \(\frac{\|\hat{\Sigma}-\Sigma\|_2}{\|\Sigma\|_2}\)
References
Dagpunar, J. (1988).Principles of Random Variate Generation. Oxford: Clarendon Press.
Devroye, L. (1986).Nonuniform Random Variate Generation (John Wiley and Sons ed.). Springer, New York.
Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin (1998).Bayesian Data Analysis. Texts in Statistical Science. Chapman & Hall, London.
Gilks, W. R. and P. Wild (1992). Adaptive rejection sampler for Gibbs sampling.Journal of Applied Statistics 41, 337–348.
Kinderman, A. J. and F. J. Monahan (1977). Computer generation of random variable using the ratio of uniform deviates.ACM Transactions on Mathematical Software 3(3), 257–260.
Leydold, J. (1998). A rejection technique for sampler from log-concave multivariate distributions.ACM Transactions on Modeling and Computer Simulation 8(3), 254–280.
Neal, R. M. (2000). Slice sampling. Technical report, Department of Statistics, University of Toronto.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (1992).Numerical Recipes in C: The Art of Science Computing. Cambridge University Press.
Ruanaidh, J. J. K. O. and W. J. Fitzgerald (1996).Numerical Bayesian Methods Applied to Signal Processing. Statistics and Computing. Springer, New York.
Tanner, M. A. (1994).Tools for Statistical Inference - Methods for the Exploration of Posterior Distributions and Likelihood Functions (2nd ed.). Springer Series in Statistics. Springer, New York.
Winkler, G. (1995).Image Analysis, Random Fields and Dynamic Monte Carlo Methods - A Mathematical Introduction. Number 27 in Applications of Mathematics. Springer, Berlin.
Acknowledgements
The authors are grateful to M. Pélégrini-Issac for her kind proofreading the paper. G. Marrelec is supported by the Fondation pour la Recherche Médicale.
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Marrelec, G., Benali, H. Automated rejection sampling from product of distributions. Computational Statistics 19, 301–315 (2004). https://doi.org/10.1007/BF02892062
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DOI: https://doi.org/10.1007/BF02892062