Bayesian Analysis

Marginal Posterior Simulation via Higher-order Tail Area Approximations

Erlis Ruli, Nicola Sartori, and Laura Ventura

Full-text: Open access


A new method for posterior simulation is proposed, based on the combination of higher-order asymptotic results with the inverse transform sampler. This method can be used to approximate marginal posterior distributions, and related quantities, for a scalar parameter of interest, even in the presence of nuisance parameters. Compared to standard Markov chain Monte Carlo methods, its main advantages are that it gives independent samples at a negligible computational cost, and it allows prior sensitivity analyses under the same Monte Carlo variation. The method is illustrated by a genetic linkage model, a normal regression with censored data and a logistic regression model.

Article information

Bayesian Anal., Volume 9, Number 1 (2014), 129-146.

First available in Project Euclid: 24 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Asymptotic expansion Bayesian computation Inverse transform sampling Marginal posterior distribution MCMC Modified likelihood root Nuisance parameter Sensitivity analysis


Ruli, Erlis; Sartori, Nicola; Ventura, Laura. Marginal Posterior Simulation via Higher-order Tail Area Approximations. Bayesian Anal. 9 (2014), no. 1, 129--146. doi:10.1214/13-BA851.

Export citation


  • Barndorff-Nielsen, O. E. and Chamberlin, S. R. (1994). “Stable and invariant adjusted directed likelihoods.” Biometrika, 81(3): 485–499.
  • Barndorff-Nielsen, O. E. and Cox, D. R. (1994). Inference and Asymptotics. Monographs on Statistics and Applied Probability. Taylor & Francis.
  • Brazzale, A. R. and Davison, A. C. (2008). “Accurate parametric inference for small samples.” Statistical Science, 23(4): 465–484.
  • Brazzale, A. R., Davison, A. C., and Reid, N. (2007). Applied Asymptotics. Cambridge: Cambridge University Press.
  • Cox, D. R. and Wermuth, N. (1990). “An approximation to maximum likelihood estimates in reduced models.” Biometrika, 77(4): 747–761.
  • Davison, A. (2003). Statistical Models. Cambridge: Cambridge University Press.
  • Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2003). Bayesian Data Analysis. Chapman & Hall/CRC.
  • Gilbert, P. and Varadhan, R. (2012). numDeriv: Accurate Numerical Derivatives. R package version 2012.9-1. URL
  • Kass, R. E., Tierney, L., and Kadane, J. B. (1989). “Approximate methods for assessing influence and sensitivity in Bayesian analysis.” Biometrika, 76(4): 663–674.
  • — (1990). “The validity of posterior expansions based on Laplace’s method.” In Geisser, S., Hodges, J., Press, S., and Zellner, A. (eds.), Bayesian and Likelihood Methods in Statistics and Econometrics: Essays in Honor of George A. Barnard, 473–487. North Holland.
  • Kharroubi, S. A. and Sweeting, T. J. (2010). “Posterior simulation via the signed root log-likelihood ratio.” Bayesian Analysis, 5(4): 787–815.
  • Marin, J. and Robert, C. (2007). Bayesian Core: A Practical Approach to Computational Bayesian Statistics. New York: Springer.
  • Pace, L. and Salvan, A. (1997). Principles of statistical inference: from a Neo-Fisherian perspective. Singapore: World Scientific.
  • — (2006). “Adjustments of the profile likelihood from a new perspective.” Journal of Statistical Planning and Inference, 136: 3554–3564.
  • R Core Team (2012). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. URL
  • Reid, N. (1996). “Likelihood and Bayesian approximation methods.” In Bayesian Statistics, volume 5, 351–368. Oxford University Press.
  • — (2003). “Asymptotics and the theory of inference.” The Annals of Statistics, 31(6): 1695–1731.
  • Reid, N. and Sun, Y. (2010). “Assessing Sensitivity to Priors Using Higher Order Approximations.” Communications in Statistics–Theory and Methods, 39(8-9): 1373–1386.
  • Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods. New York: Springer, 2nd edition.
  • Severini, T. A. (2000). Likelihood Methods in Statistics. New York: Oxford University Press.
  • Sweeting, T. J. (1996). “Approximate Bayesian computation based on signed roots of log-density ratios (with discussion).” In Bayesian Statistics, volume 5, 427–444. Oxford University Press.
  • Tibshirani, R. (1989). “Noninformative priors for one parameter of many.” Biometrika, 76(3): 604–608.
  • Tierney, L. and Kadane, J. B. (1986). “Accurate approximations for posterior moments and marginal densities.” Journal of the American Statistical Association, 81(393): 82–86.
  • Ventura, L., Cabras, S., and Racugno, W. (2009). “Prior distributions from pseudo-likelihoods in the presence of nuisance parameters.” Journal of the American Statistical Association, 104(486): 768–774.
  • Ventura, L. and Racugno, W. (2011). “Recent advances on Bayesian inference for P(X<Y).” Bayesian Analysis, 6: 411–428.
  • Ventura, L., Sartori, N., and Racugno, W. (2013). “Objective Bayesian higher-order asymptotics in models with nuisance parameters.” Computational Statistics & Data Analysis, 60: 90–96.