Abstract
This paper provides an asymptotic expansion of the posterior based on pairwise likelihood instead of the regular likelihood. The celebrated Bernstein-von Mises theorem is derived as a special case. A multiparameter version of the asymptotic expansion is also given involving nuisance parameters. As a direct application of these expansions, one can obtain moment matching priors and quantile matching priors with or without nuisance parameters. A simulation study is provided verifying this agreement between frequentist quantiles and Bayesian quantiles using quantile matching priors. One of the major tools used in this paper is strong consistency of the maximum pairwise likelihood estimator (MPLE).
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References
Berger J.O. and Bernardo J.M. (1989). Estimating a product of means: Bayesian, analysis with reference priors. Journal of the American Statistical Association 84, 405, 200–207.
Berger J.O. and Bernardo J.M. (1992). Ordered group reference priors with application to the multinomial problem. Biometrika 79, 1, 25–37.
Bernardo J.M. (1979). Reference posterior distributions for Bayesian inference. Journal of the Royal Statistical Society. Series B (Methodological) 41, 2, 113–147.
Cox D.R. and Reid N. (1987). Parameter orthogonality and approximate conditional inference . Journal of the Royal Statistical Society. Series B (Methodological) 49, 1, 1–39.
Cox D.R. and Reid N. (2004). A note on pseudolikelihood constructed from marginal densities . Biometrika 91, 3, 729–737.
Datta G.S. and Ghosh, M. (1995). Some remarks on noninformative priors . Journal of the American Statistical Association 90, 432, 1357–1363.
Datta G.S. and Mukerjee R. 2004 Probability matching priors: higher order asymptotics, Springer.
Geys H., Molenberghs G. and Ryan L.M. (1999). Pseudolikelihood modeling of multivariate outcomes in developmental toxicology. Journal of the American Statistical Association 94, 447, 734–745.
Ghosh M. and Liu, R. (2011). Moment matching priors. Sankhya 73-A, 2, 185–201.
Johnson R.A. (1970). Asymptotic expansions associated with posterior distributions. The Annals of Mathematical Statistics 41, 3, 851–864.
Kent J.T. (1982). Robust properties of likelihood ratio tests. Biometrika 69, 1, 19–27.
Le Cessie S. and Van Houwelingen J. (1994). Logistic regression for correlated binary data. Applied Statistics 43, 1, 95–108.
Lindsay B.G. (1988). Composite likelihood methods. Contemporary Mathematics 80, 221–239.
Molenberghs G. and Verbeke V. 2005 Models for discrete longitudinal data. Springer.
Pauli F., Racugno W. and Ventura L. (2011). Bayesian composite marginal likelihoods. Statistica Sinica 21, 149–164.
Ribatet M., Cooley D. and Davison A.C. (2012). Bayesian inference from composite likelihoods, with an application to spatial extremes. Statistica Sinica, 813–845.
Rubin H. (1956). Uniform convergence of random functions with applications to statistics. The Annals of Mathematical Statistics 27, 1, 200–203.
Ruli E., Sartori N. and Ventura L. (2013). Approximate Bayesian computation with composite score functions. Statistics and Computing, 1–14.
Ruli E. and Ventura L. 2015 Higher-order Bayesian approximations for pseudo-posterior distributions , Communications in Statistics – Simulation and Computation, to appear.
Smith E.L. and Stephenson A.G. (2009). An extended Gaussian, max-stable process model for spatial extremes . Journal of Statistical Planning and Inference 139, 1266–1275.
Staicu A.-M. and Reid N. M. (2008). On probability matching priors . The Canadian Journal of Statistics 36, 4, 613–622.
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.
Tierney L., Kass R.E. and Kadane J.B. (1989). Fully exponential Laplace, approximations to expectations and variances of nonpositive functions . Journal of the American Statistical Association 84, 407, 710–716.
Varin C. (2008). On composite marginal likelihoods. AStA Advances in Statistical Analysis 92, 1, 1–28.
Varin C., Reid N. and Firth D. (2011). An overview of composite likelihood methods . Statistica Sinica 21, 5–42.
Xu X. and Reid N. (2011). On the robustness of maximum composite likelihood estimate. Journal of Statistical Planning and Inference 141, 3047–3054.
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Wu, Y., Ghosh, M. Asymptotic Expansion of the Posterior Based on Pairwise Likelihood. Sankhya A 79, 39–75 (2017). https://doi.org/10.1007/s13171-016-0094-y
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DOI: https://doi.org/10.1007/s13171-016-0094-y
Keywords and phrases
- Asymptotic theory
- Pseudo likelihood
- Strong consistency
- Taylor expansion
- Moment matching priors
- Quantile matching priors