Bayesian methods are useful for statistical inference. However, real-world problems can be challenging using Bayesian methods when the data analyst has only limited prior knowledge. In this paper we consider a class of problems, called partial Bayes problems, in which the prior information is only partially available. Taking the recently proposed inferential model approach, we develop a general inference framework for partial Bayes problems, and derive both exact and efficient solutions. In addition to the theoretical investigation, numerical results and real applications are used to demonstrate the superior performance of the proposed method.
References
Carlin, B. P. and Gelfand, A. E. (1990). Approaches for Empirical Bayes Confidence Intervals., Journal of the American Statistical Association 85 105-114. MR1137356 0731.62080 10.1080/01621459.1990.10475312Carlin, B. P. and Gelfand, A. E. (1990). Approaches for Empirical Bayes Confidence Intervals., Journal of the American Statistical Association 85 105-114. MR1137356 0731.62080 10.1080/01621459.1990.10475312
Deely, J. J. and Lindley, D. V. (1981). Bayes empirical bayes., Journal of the American Statistical Association 76 833–841. 0495.62009 10.1080/01621459.1981.10477731Deely, J. J. and Lindley, D. V. (1981). Bayes empirical bayes., Journal of the American Statistical Association 76 833–841. 0495.62009 10.1080/01621459.1981.10477731
Efron, B. (2010)., Large-scale inference: empirical Bayes methods for estimation, testing, and prediction. Cambridge University Press. 1277.62016Efron, B. (2010)., Large-scale inference: empirical Bayes methods for estimation, testing, and prediction. Cambridge University Press. 1277.62016
Efron, B. and Morris, C. (1971). Limiting the Risk of Bayes and Empirical Bayes Estimators – Part I: The Bayes Case., Journal of the American Statistical Association 66 807-815. MR0323014 0229.62003Efron, B. and Morris, C. (1971). Limiting the Risk of Bayes and Empirical Bayes Estimators – Part I: The Bayes Case., Journal of the American Statistical Association 66 807-815. MR0323014 0229.62003
Efron, B. and Morris, C. (1972a). Empirical Bayes on vector observations: An extension of Stein’s method., Biometrika 59 335–347. 0238.62072 10.1093/biomet/59.2.335Efron, B. and Morris, C. (1972a). Empirical Bayes on vector observations: An extension of Stein’s method., Biometrika 59 335–347. 0238.62072 10.1093/biomet/59.2.335
Efron, B. and Morris, C. (1972b). Limiting the Risk of Bayes and Empirical Bayes Estimators – Part II: The Empirical Bayes Case., Journal of the American Statistical Association 67 130-139. MR0323015 0231.62013Efron, B. and Morris, C. (1972b). Limiting the Risk of Bayes and Empirical Bayes Estimators – Part II: The Empirical Bayes Case., Journal of the American Statistical Association 67 130-139. MR0323015 0231.62013
Efron, B. and Morris, C. (1973). Stein’s Estimation Rule and its Competitors – An Empirical Bayes Approach., Journal of the American Statistical Association 68 117-130. MR0388597 0275.62005Efron, B. and Morris, C. (1973). Stein’s Estimation Rule and its Competitors – An Empirical Bayes Approach., Journal of the American Statistical Association 68 117-130. MR0388597 0275.62005
Efron, B. and Morris, C. (1975). Data Analysis Using Stein’s Estimator and its Generalizations., Journal of the American Statistical Association 70 311-319. 0319.62018 10.1080/01621459.1975.10479864Efron, B. and Morris, C. (1975). Data Analysis Using Stein’s Estimator and its Generalizations., Journal of the American Statistical Association 70 311-319. 0319.62018 10.1080/01621459.1975.10479864
Goggin, E. M. (1994). Convergence in Distribution of Conditional Expectations., Ann. Probab. 22 1097–1114. 0805.60017 10.1214/aop/1176988743 euclid.aop/1176988743Goggin, E. M. (1994). Convergence in Distribution of Conditional Expectations., Ann. Probab. 22 1097–1114. 0805.60017 10.1214/aop/1176988743 euclid.aop/1176988743
Laird, N. M. and Louis, T. A. (1987). Empirical Bayes Confidence Intervals Based on Bootstrap Samples., Journal of the American Statistical Association 82 739-750. 0643.62018 10.1080/01621459.1987.10478490Laird, N. M. and Louis, T. A. (1987). Empirical Bayes Confidence Intervals Based on Bootstrap Samples., Journal of the American Statistical Association 82 739-750. 0643.62018 10.1080/01621459.1987.10478490
Lambert, D. and Duncan, G. T. (1986). Single-parameter inference based on partial prior information., Canadian Journal of Statistics 14 297–305. 0629.62006 10.2307/3315187Lambert, D. and Duncan, G. T. (1986). Single-parameter inference based on partial prior information., Canadian Journal of Statistics 14 297–305. 0629.62006 10.2307/3315187
Lindley, D. V. and Smith, A. F. M. (1972). Bayes Estimates for the Linear Model., Journal of the Royal Statistical Society. Series B (Methodological) 34 1-41. MR0415861 0246.62050 10.1111/j.2517-6161.1972.tb00885.xLindley, D. V. and Smith, A. F. M. (1972). Bayes Estimates for the Linear Model., Journal of the Royal Statistical Society. Series B (Methodological) 34 1-41. MR0415861 0246.62050 10.1111/j.2517-6161.1972.tb00885.x
Martin, R. (2015). Plausibility Functions and Exact Frequentist Inference., Journal of the American Statistical Association 110 1552-1561. 1373.62027 10.1080/01621459.2014.983232Martin, R. (2015). Plausibility Functions and Exact Frequentist Inference., Journal of the American Statistical Association 110 1552-1561. 1373.62027 10.1080/01621459.2014.983232
Martin, R. and Liu, C. (2013). Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference., Journal of the American Statistical Association 108 301-313. 06158344 10.1080/01621459.2012.747960Martin, R. and Liu, C. (2013). Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference., Journal of the American Statistical Association 108 301-313. 06158344 10.1080/01621459.2012.747960
Martin, R. and Liu, C. (2015a). Conditional inferential models: combining information for prior-free probabilistic inference., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 195–217.Martin, R. and Liu, C. (2015a). Conditional inferential models: combining information for prior-free probabilistic inference., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 195–217.
Martin, R. and Liu, C. (2015b). Marginal Inferential Models: Prior-Free Probabilistic Inference on Interest Parameters., Journal of the American Statistical Association 110 1621-1631. 1373.62028 10.1080/01621459.2014.985827Martin, R. and Liu, C. (2015b). Marginal Inferential Models: Prior-Free Probabilistic Inference on Interest Parameters., Journal of the American Statistical Association 110 1621-1631. 1373.62028 10.1080/01621459.2014.985827
Meaux, L., Seaman Jr, J. and Young, D. (2002). Statistical inference with partial prior information based on a Gauss-type inequality., Mathematical and computer modelling 35 1483–1488. 1023.62004 10.1016/S0895-7177(02)00099-7Meaux, L., Seaman Jr, J. and Young, D. (2002). Statistical inference with partial prior information based on a Gauss-type inequality., Mathematical and computer modelling 35 1483–1488. 1023.62004 10.1016/S0895-7177(02)00099-7
Moreno, E., Bertolino, F. and Racugno, W. (2003). Bayesian inference under partial prior information., Scandinavian Journal of Statistics 30 565–580. 1034.62021 10.1111/1467-9469.00349Moreno, E., Bertolino, F. and Racugno, W. (2003). Bayesian inference under partial prior information., Scandinavian Journal of Statistics 30 565–580. 1034.62021 10.1111/1467-9469.00349
Morris, C. N. (1983). Parametric Empirical Bayes Inference: Theory and Applications., Journal of the American Statistical Association 78 47-55. 0506.62005 10.1080/01621459.1983.10477920Morris, C. N. (1983). Parametric Empirical Bayes Inference: Theory and Applications., Journal of the American Statistical Association 78 47-55. 0506.62005 10.1080/01621459.1983.10477920
Robbins, H. (1956). An Empirical Bayes Approach to Statistics. In, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 157–163. 0074.35302Robbins, H. (1956). An Empirical Bayes Approach to Statistics. In, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 157–163. 0074.35302
Sweeting, T. J. (1989). On conditional weak convergence., Journal of Theoretical Probability 2 461–474. 0695.60036 10.1007/BF01051878Sweeting, T. J. (1989). On conditional weak convergence., Journal of Theoretical Probability 2 461–474. 0695.60036 10.1007/BF01051878
Xie, M., Singh, K. and Strawderman, W. E. (2011). Confidence Distributions and a Unifying Framework for Meta-Analysis., Journal of the American Statistical Association 106 320-333. 1396.62051 10.1198/jasa.2011.tm09803Xie, M., Singh, K. and Strawderman, W. E. (2011). Confidence Distributions and a Unifying Framework for Meta-Analysis., Journal of the American Statistical Association 106 320-333. 1396.62051 10.1198/jasa.2011.tm09803
Xie, M., Liu, R. Y., Damaraju, C. V. and Olson, W. H. (2013). Incorporating external information in analyses of clinical trials with binary outcomes., The Annals of Applied Statistics 7 342-368. 06171275 10.1214/12-AOAS585 euclid.aoas/1365527202Xie, M., Liu, R. Y., Damaraju, C. V. and Olson, W. H. (2013). Incorporating external information in analyses of clinical trials with binary outcomes., The Annals of Applied Statistics 7 342-368. 06171275 10.1214/12-AOAS585 euclid.aoas/1365527202
Xiong, S. and Li, G. (2008). Some results on the convergence of conditional distributions., Statistics & Probability Letters 78 3249 - 3253. 05380111 10.1016/j.spl.2008.06.026Xiong, S. and Li, G. (2008). Some results on the convergence of conditional distributions., Statistics & Probability Letters 78 3249 - 3253. 05380111 10.1016/j.spl.2008.06.026