Abstract
We explore the use of importance sampling based on exponentially tilted signed root log-likelihood ratios for Bayesian computation. Approximations based on exponentially tilted signed root log-likelihood ratios are used in two distinct ways; firstly, to define an importance function with antithetic variates and, secondly, to define suitable control variates for variance reduction. These considerations give rise to alternative simulation-consistent schemes to other importance sampling techniques (for example, conventional and/or adaptive importance sampling) for Bayesian computation in moderately parameterized regular problems. The schemes based on control variates can also be viewed as usefully supplementing computations based on asymptotic approximations by supplying external estimates of error. The methods are illustrated by a censored regression model and a more challenging 12-parameter nonlinear repeated measures model for bacterial clearance.
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References
Bugallo MF, Elvira V, Martino L, Luengo D, Miguez J, Djuric PM (2017) Adaptive importance sampling: the past, the present, and the future. IEEE Signal Process Mag 34(4):60–79
Cerquetti A (2007) A note on Bayesian nonparametric priors derived from exponentially tilted Poisson–Kingman models. Stat Probab Lett 77:1705–1711
Crawford DE (1970) Analysis of incomplete life test data on motorettes. Insul Circuits 16:43–48
Efron B (1981) Nonparameteric standard errors and confidence intervals. Can J Stat 9:139–58
Elvira V, Martino L, Luengo D, Bugallo MF (2016) Heretical multiple importance sampling. IEEE Signal Process Lett 23(10):1474–1478
Evans M, Swartz T (1995) Methods for approximating integrals in statistics with special emphasis on Bayesian integration problems. Stat Sci 10:254–272
Evans M, Swartz T (2000) Approximating integrals via Monte Carlo and deterministic methods. Oxford University Press, Oxford
Fuh CD, Teng HW, Wang RH (2013) Efficient importance sampling for rare event simulation with applications. arXiv:1302.0583 [stat.ME]
Hammersley JM, Handscomb DC (1964) Monte Carlo methods. Methuen, London
Hesterberg T (1995) Weighted average importance sampling and defensive mixture distributions. Technometrics 37:185–194
Kharroubi SA, Sweeting TJ (2010) Posterior simulation via signed root log-likelihood ratios. Bayesian Anal 5(4):787–816
Kharroubi SA, Sweeting TJ (2016) Exponential tilting in Bayesian asymptotics. Biometrika 103(2):337–349
Owen A, Zhou Y (2000) Safe and effective importance sampling. J Am Stat Assoc 95(449):135–143
Pitman J (2003) Poisson–Kingman partitions. In: Goldstein DR (ed) A festschrift for terry speed, IMS lecture notes monograph series, vol 40. Institute of Mathematical Statistics, Hayward, pp 1–34
Ripley B (1987) Stochastic simulation. Wiley, New York
Schennach SM (2005) Bayesian exponentially tilted empirical likelihood. Biometrika 92:31–46
Schennach SM (2007) Point estimation with exponentially tilted empirical likelihood. Ann Stat 35:634–672
Schmee J, Hahn GJ (1979) A simple method for regression analysis with censored data. Technometrics 21:417–432
Smith AM, Rahman FZ, Hayee BH, Graham SJ, Marks DJB, Sewell GW, Palmer CD, Wilde J, Foxwell BMJ, Gloger IS, Sweeting T, Marsh M, Walker AP, Bloom SL, Segal AW (2009) Disordered macrophage cytokine secretion underlies impaired acute inflammation and bacterial clearance in Crohn’s disease. J Exp Med 206:1883–1897
Sweeting TJ (1996) Approximate Bayesian computation based on signed roots of log-density ratios (with discussion). In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics, vol 5. Oxford University Press, Oxford, pp 427–444
Sweeting TJ, Kharroubi SA (2003) Some new formulae for posterior expectations and Bartlett corrections. Test 12(497–521):2003
Sweeting TJ, Kharroubi SA (2005) Application of a predictive distribution formula to Bayesian computation for incomplete data models. Stat Comput 15:167–178
Tierney L, Kadane J (1986) Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81:82–86
Van Dijk HK, Kloek T, Louter AS (1986) An algorithm for the computation of posterior moments and densities using simple importance sampling. Statistician 35:83–90
Vehtari A, Gelman A, Gabry J (2016) Pareto smoothed importance sampling. arXiv:1507.02646
Acknowledgements
The author would particularly like to thank Professor Trevor J Sweeting for all his continual support, useful guidance and invaluable insights during his time working on this manuscript.
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Kharroubi, S.A. Posterior simulation via the exponentially tilted signed root log-likelihood ratio. Comput Stat 33, 213–234 (2018). https://doi.org/10.1007/s00180-017-0772-9
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DOI: https://doi.org/10.1007/s00180-017-0772-9