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Bayesian Model Averaging

  • David FletcherEmail author
Chapter
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Abstract

We provide an overview of Bayesian model averaging (BMA), starting with a summary of the mathematics associated with classical BMA, including the calculation of posterior model probabilities and the choice of priors for both the models and the model parameters. We also consider prediction-based approaches to BMA and argue that these are preferable to the classical approach. Use of BMA is illustrated by two examples involving real data. We finish with a discussion of the advantages and disadvantages of BMA.

References

  1. 1.
    Aitkin, M.: Posterior Bayes factors. J. Roy. Stat. Soc. B. Methodol. 53, 111–142 (1991)zbMATHGoogle Scholar
  2. 2.
    Amini, S.M., Parmeter, C.F.: Bayesian model averaging in R. J. Econ. Soc. Meas. 36, 253–287 (2011)Google Scholar
  3. 3.
    Anandalingam, G., Chen, L.: Linear combination of forecasts: a general Bayesian model. J. Forecasting 8, 199–214 (1989)CrossRefGoogle Scholar
  4. 4.
    Ando, T., Tsay, R.: Predictive likelihood for Bayesian model selection and averaging. Int. J. Forecasting 26, 744–763 (2010)CrossRefGoogle Scholar
  5. 5.
    Andrieu, C., Doucet, A., Robert, C.P.: Computational advances for and from Bayesian analysis. Stat. Sci. 19, 118–127 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Annest, A., Bumgarner, R.E., Raftery, A.E., Yeung, K.Y.: Iterative Bayesian model averaging: a method for the application of survival analysis to high-dimensional microarray data. BMC Bioinform. 10, 72 (2009)CrossRefGoogle Scholar
  7. 7.
    Barbieri, M.M., Berger, J.O.: Optimal predictive model selection. Ann. Stat. 32, 870–897 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Barker, R.J., Link, W.A.: Bayesian multimodel inference by RJMCMC: a Gibbs sampling approach. Am. Stat. 67, 150–156 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bartolucci, F., Scaccia, L., Mira, A.: Efficient Bayes factor estimation from the reversible jump output. Biometrika 93, 41–52 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Berger, J.O., Pericchi, L.R.: The intrinsic Bayes factor for model selection and prediction. J. Am. Stat. Assoc. 91, 109–122 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Berger, J.O., Ghosh, J.K., Mukhopadhyay, N.: Approximations and consistency of Bayes factors as model dimension grows. J. Stat. Plan. Infer. 112, 241–258 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Berger, J.O., Molina, G.: Posterior model probabilities via pathbased pairwise priors. Stat. Neerl. 59, 3–15 (2005)zbMATHCrossRefGoogle Scholar
  13. 13.
    Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, New York (1994)zbMATHCrossRefGoogle Scholar
  14. 14.
    Bottolo, L., Richardson, S.: Evolutionary stochastic search for Bayesian model exploration. Bayesian Anal. 5, 583–618 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Brooks, S.P.: Discussion of Spiegelhalter, D.J., Best, N.G., Carlin, B.P., Van Der Linde, A.: Bayesian measures of model complexity and fit. J. R. Stat. Soc. 64, 616–618 (2002)Google Scholar
  16. 16.
    Brown, P.J., Vannucci, M., Fearn, T.: Bayes model averaging with selection of regressors. J. Roy. Stat. Soc. B Methodol. 64, 519–536 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Bunn, D.W.A.: Bayesian approach to the linear combination of forecasts. Oper. Res. Quart. 26, 325–329 (1975)zbMATHCrossRefGoogle Scholar
  18. 18.
    Burnham, K.P., Anderson, D.R.: Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd edn. Springer, New York (2002)Google Scholar
  19. 19.
    Carlin, B.P., Chib, S.: Bayesian model choice via Markov chain Monte Carlo methods. J. Roy. Stat. Soc. B. Methodol. 57, 473–484 (1995)zbMATHGoogle Scholar
  20. 20.
    Castillo, I., Schmidt-Hieber, J., Van der Vaart, A.: Bayesian linear regression with sparse priors. Ann. Stat. 43, 1986–2018 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Cefalu, M., Dominici, F., Arvold, N., Parmigiani, G.: Model averaged double robust estimation. Biometrics 73, 410–421 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Celeux, G., Forbes, F., Robert, C.P., Titterington, D.M.: Deviance information criteria for missing data models. Bayesian Anal. 1, 651–673 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Chen, M.-H., Shao, Q.-M.: On Monte Carlo methods for estimating ratios of normalizing constants. Ann. Stat. 25, 1563–1594 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Chen, M.-H., Shao, Q.-M., Ibrahim, J.G.: Monte Carlo Methods in Bayesian Computation. Springer, New York (2000)zbMATHCrossRefGoogle Scholar
  25. 25.
    Chen, M.-H., Ibrahim, J.G.: Conjugate priors for generalized linear models. Stat. Sinica. 13, 461–476 (2003)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Chib, S.: Marginal likelihood from the Gibbs output. J. Am. Stat. Assoc. 90, 1313–1321 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Chib, S.: Monte Carlo methods and Bayesian computation: overview. In: Smelser, N.J., Baltes, P.B. (eds.) International Encyclopedia of the Social and Behavioral Sciences: Statistics. Elsevier Science, Oxford (2001)Google Scholar
  28. 28.
    Chickering, D.M., Heckerman, D.: Efficient approximations for the marginal likelihood of Bayesian networks with hidden variables. Mach. Learn. 29, 181–212 (1997)zbMATHCrossRefGoogle Scholar
  29. 29.
    Ching, J., Chen, Y.-C.: Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging. J. Eng. Mech. 133, 816–832 (2007)CrossRefGoogle Scholar
  30. 30.
    Chipman, H.: Bayesian variable selection with related predictors. Can. J. Stat. 24, 17–36 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Chipman, H., George, E.I., McCulloch, M., Clyde, D.P.F., Stine, R.A.: The practical implementation of Bayesian model selection. Inst. Math. S. 38, 65–134 (2001)MathSciNetGoogle Scholar
  32. 32.
    Claeskens, G., Hjort, N.L.: Model Selection and Model Averaging, vol. 330. Cambridge University Press, Cambridge (2008)Google Scholar
  33. 33.
    Clarke, B.: Comparing Bayes model averaging and stacking when model approximation error cannot be ignored. J. Mach. Learn. Res. 4, 683–712 (2003)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Clyde, M., Desimone, H., Parmigiani, G.L.: Prediction via orthogonalized model mixing. J. Am. Stat. Assoc. 91, 1197–1208 (1996)zbMATHCrossRefGoogle Scholar
  35. 35.
    Clyde, M.: Model uncertainty and health effect studies for particulate matter. Environmetrics 11, 745–763 (2000)CrossRefGoogle Scholar
  36. 36.
    Clyde, M.: Model averaging. In: Press, S.J. (ed.) Subjective and Objective Bayesian Statistics, 2nd edn. Wiley-Interscience, New Jersey (2003)Google Scholar
  37. 37.
    Clyde, M., George, E.I.: Model uncertainty. Stat. Sci. 19, 81–94 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Clyde, M.A., Ghosh, J., Littman, M.L.: Bayesian adaptive sampling for variable selection and model averaging. J. Comput. Graph. Stat. 20, 80–101 (2011)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Clyde, M., Iversen, E.S.: Bayesian model averaging in the M-open framework. In: Damien, P., Dellaportas, P., Polson, N.G., Stephens, D.A. (eds.) Bayesian Theory and Applications. Oxford University Press, Oxford (2013)Google Scholar
  40. 40.
    Congdon, P.: Bayesian model choice based on Monte Carlo estimates of posterior model probabilities. Comput. Stat. Data Anal. 50, 346–357 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Congdon, P.: Model weights for model choice and averaging. Stat. Methodol. 4, 143–157 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Consonni, G., Fouskakis, D., Liseo, B., Ntzoufras, I.: Prior distributions for objective Bayesian analysis. Bayesian Anal. 13, 627–679 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Corani, G., Zaffalon, M.: Credal model averaging: an extension of Bayesian model averaging to imprecise probabilities. In: Joint European Conference on Machine Learning and Knowledge Discovery in Databases. Springer, Heidelberg (2008)Google Scholar
  44. 44.
    Corani, G., Antonucci, A.: Credal ensembles of classifiers. Comput. Stat. Data Anal. 71, 818–831 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Corani, G., Mignatti, A.: Credal model averaging for classification: representing prior ignorance and expert opinions. Int. J. Approx. Reason. 56, 264–277 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Corani, G., Mignatti, A.: Robust Bayesian model averaging for the analysis of presence-absence data. Environ. Ecol. Stat. 22, 513–534 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Cox, D.R.: Principles of Statistical Inference. Cambridge University Press, Cambridge (2006)zbMATHCrossRefGoogle Scholar
  48. 48.
    Cuaresma, J.C., Grün, B., Hofmarcher, P., Humer, S., Moser, M.: Unveiling covariate inclusion structures in economic growth regressions using latent class analysis. Eur. Econ. Rev. 81, 189–202 (2016)CrossRefGoogle Scholar
  49. 49.
    Datta, G.S., Mukerjee, R.: Probability Matching Priors: Higher Order Asymptotics. Springer, New York (2004)zbMATHCrossRefGoogle Scholar
  50. 50.
    DiCiccio, T.J., Kass, R.E., Raftery, A., Wasserman, L.: Computing Bayes factors by combining simulation and asymptotic approximations. J. Am. Stat. Assoc. 92, 903–915 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Diebold, F.X., Pauly, P.: The use of prior information in forecast combination. Int. J. Forecasting 6, 503–508 (1990)CrossRefGoogle Scholar
  52. 52.
    Domingos, P.: Why does bagging work? A Bayesian account and its implications. In: Proceedings of the Third International Conference on Knowledge Discovery and Data Mining, pp. 155–158 (1997)Google Scholar
  53. 53.
    Domingos, P.: Bayesian averaging of classifiers and the overfitting problem. In: Proceedings of the Seventeenth International Conference on Machine Learning, pp. 223–230 (2000)Google Scholar
  54. 54.
    Doppelhofer, G., Weeks, M.: Jointness of growth determinants. J. Appl. Econ. 24, 209–244 (2009)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Doppelhofer, G., Weeks, M.: Jointness of growth determinants: reply to comments by Rodney Strachan, Eduardo Ley and Mark FJ Steel. J. Appl. Econ. 24, 252–256 (2009)CrossRefGoogle Scholar
  56. 56.
    Drachal, K.: Comparison between Bayesian and information-theoretic model averaging: fossil fuels prices example. Energ. Econ. 74, 208–251 (2018)CrossRefGoogle Scholar
  57. 57.
    Draper, D.: Model uncertainty yes, discrete model averaging maybe. Stat. Sci. 14, 405–409 (1999)Google Scholar
  58. 58.
    Eicher, T.S., Papageorgiou, C., Raftery, A.E.: Default priors and predictive performance in BMA, with application to growth determinants. J. Appl. Econ. 26, 30–55 (2011)CrossRefGoogle Scholar
  59. 59.
    Eklund, J., Karlsson, S.: Forecast combination and model averaging using predictive measures. Econ. Rev. 26, 329–363 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Ellison, A.M.: Bayesian inference in ecology. Ecol. Lett. 7, 509–520 (2004)CrossRefGoogle Scholar
  61. 61.
    Fan, T.-H., Wang, G.-T., Yu, J.-H.: A new algorithm in Bayesian model averaging in regression models. Commun. Stat. Simul. 43, 315–328 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Farnsworth, M.L., Hoeting, J.A., Thompson Hobbs, N., Miller, M.W.: Linking chronic wasting disease to mule deer movement scales: a hierarchical Bayesian approach. Ecol. Appl. 16, 1026–1036 (2006)CrossRefGoogle Scholar
  63. 63.
    Feldkircher, M.: Forecast combination and BMA: a prior sensitivity analysis. J. Forecasting 31, 361–376 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Forte, A., Garcia-Donato, G., Steel, M.F.J.: Methods and tools for Bayesian variable selection and model averaging in normal linear regression. Department of Statistics working paper, University of Warwick (2017)Google Scholar
  65. 65.
    Fragoso, T.M., Bertoli, W., Louzada, F.: Bayesian model averaging: a systematic review and conceptual classification. Int. Stat. Rev. (2017).  https://doi.org/10.1111/insr.12243MathSciNetCrossRefGoogle Scholar
  66. 66.
    Garthwaite, P.H., Mubwandarikwa, E.: Selection of weights for weighted model averaging. Aust. NZ. J. Stat. 52, 363–382 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Geisser, S., Eddy, W.F.: A predictive approach to model selection. J. Am. Stat. Assoc. 74, 153–160 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Gelfand, A.E., Dey, D.K., Chang, H.: Model determination using predictive distributions with implementation via sampling-based methods. Technical report 462. Department of Statistics, Stanford University (1992)Google Scholar
  69. 69.
    Gelfand, A., Dey, D.K.: Bayesian model choice: asymptotics and exact calculations. J. R. Stat. Soc. B. Methodol. 56, 501–514 (1994)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Gelfand, A.E.: Model determination using sampling-based methods. In: Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (eds.) Markov Chain Monte Carlo in Practice, pp. 145–162. Chapman and Hall (1996) In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics, vol. 6, pp. 175–177. Oxford University Press (1999)Google Scholar
  71. 71.
    Gelman, A., Meng, X.-L.: Simulating normalizing constants: from importance sampling to bridge sampling to path sampling. Stat. Sci. 13, 163–185 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Gelman, A., Hwang, J., Vehtari, A.: Understanding predictive information criteria for Bayesian models. Stat. Comput. 24, 997–1016 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B.: Bayesian Data Analysis. CRC Press, Boca Raton (2014)zbMATHGoogle Scholar
  74. 74.
    George, E.I., McCulloch, R.E.: Variable selection via Gibbs sampling. J. Am. Stat. Assoc. 88, 881–889 (1993)CrossRefGoogle Scholar
  75. 75.
    George, E.I., McCulloch, R.E.: Approaches for Bayesian variable selection. Stat. Sin. 7, 339–373 (1997)zbMATHGoogle Scholar
  76. 76.
    George, E.I., Discussion of Clyde, M.A.: BMA and model search strategies. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics, vol. 6, pp. 175–177. Oxford University Press (1999)Google Scholar
  77. 77.
    Ghosh, J., Ghattas, A.E.: Bayesian variable selection under collinearity. Am. Stat. 69, 165–173 (2015)MathSciNetCrossRefGoogle Scholar
  78. 78.
    Gneiting, T., Raftery, A.E.: Strictly proper scoring rules, prediction, and estimation. J. Am. Stat. Assoc. 102, 359–378 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Godsill, S.J.: On the relationship between Markov chain Monte Carlo methods for model uncertainty. J. Comput. Graph. Stat. 10, 230–248 (2001)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Gutiérrez-Peña, E., Walker, S.G.: Statistical decision problems and Bayesian nonparametric methods. Int. Stat. Rev. 73, 309–330 (2005)zbMATHCrossRefGoogle Scholar
  82. 82.
    Han, C., Carlin, B.P.: Markov chain Monte Carlo methods for computing Bayes factors. J. Am. Stat. Assoc. 96, 1122–1132 (2001)CrossRefGoogle Scholar
  83. 83.
    Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, vol. 2, no. 1. Springer, New York (2009)Google Scholar
  84. 84.
    Hernández, B., Raftery, A.E., Pennington, S.R., Parnell, A.C.: Bayesian additive regression trees using Bayesian model averaging. Stat. Comput. 28, 869–890 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Hjort, N.L., Claeskens, G.: Frequentist model average estimators. J. Am. Stat. Assoc. 98, 879–945 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Hoegh, A., Maiti, D., Leman, S.: Multiset model selection. J. Comput. Graph. Stat. (2018).  https://doi.org/10.1080/10618600.2017.1379408MathSciNetCrossRefGoogle Scholar
  87. 87.
    Hoeting, J.A., Madigan, D., Raftery, A.E., Volinsky, C.T.: Bayesian model averaging: a tutorial. Stat. Sci. 14, 382–401 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Hofmarcher, P., Cuaresma, J.C., Grün, B., Humer, S., Moser, M.: Bivariate jointness measures in Bayesian model averaging: solving the conundrum. J. Macroecon. 57, 150–165 (2018)CrossRefGoogle Scholar
  89. 89.
    Hooten, M.B., Thompson Hobbs, N.: A guide to Bayesian model selection for ecologists. Ecol. Monogr. 85, 3–28 (2015)CrossRefGoogle Scholar
  90. 90.
    Hubin, A., Storvik, G.: Mode jumping MCMC for Bayesian variable selection in GLMM. Comput. Stat. Data Anal. 127, 281–297 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Jeffreys, H.: Theory of Probability, 3rd edn. Oxford University Press, Oxford (1961)zbMATHGoogle Scholar
  92. 92.
    Jiao, Y., Reid, K., Smith, E.: Model selection uncertainty and BMA in fisheries recruitment modeling. In: Beamish, R.J., Rothschild, B.J. (eds.) The Future of Fisheries Science in North America, pp. 505–524. Springer, Dordrecht (2009)Google Scholar
  93. 93.
    Kadane, J.B., Lazar, N.A.: Methods and criteria for model selection. J. Am. Stat. Assoc. 99, 279–290 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  94. 94.
    Kamary, K., Mengersen, K., Robert, C.P., Rousseau, J.: Testing hypotheses via a mixture estimation model (2014). arXiv preprint: arXiv:1412.2044
  95. 95.
    Kapetanios, G., Labhard, V., Price, S.P.: Forecasting using Bayesian and information-theoretic model averaging: an application to UK inflation. J. Bus. Econ. Stat. 26, 33–41 (2008)CrossRefGoogle Scholar
  96. 96.
    Kashyap, R.L.: Optimal choice of AR and MA parts in autoregressive moving average models. IEEE Trans. Pattern Anal. 4, 99–104 (1982)zbMATHCrossRefGoogle Scholar
  97. 97.
    Kass, R.E.: Bayes factors in practice. J. Roy. Stat. Soc. D Stat. 42, 551–560 (1993)Google Scholar
  98. 98.
    Kass, R.E., Raftery, A.E.: Bayes factors. J. Am. Stat. Assoc. 90, 773–795 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Kiartzis, S., Kehagias, A., Bakirtzis, A., Petridis, V.: Short term load forecasting using a Bayesian combination method. Int. J. Electr. Power 19, 171–177 (1997)CrossRefGoogle Scholar
  100. 100.
    Kim, H.-C., Ghahramani Z.: Bayesian classifier combination. In: Proceedings the 15th International Conference Artificial Intelligence and Statistics, pp. 619–627 (2012)Google Scholar
  101. 101.
    King, R., Brooks, S.P.: On the Bayesian analysis of population size. Biometrika 88, 317–336 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    Le, T., Clarke, B.: A Bayes interpretation of stacking for M-complete and M-open settings. Bayesian Anal. 12, 807–829 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    Lee, H.K.H.: Model selection for neural network classification. J. Classif. 18, 227–243 (2001)MathSciNetzbMATHGoogle Scholar
  104. 104.
    Ley, E., Steel, M.F.J.: Jointness in Bayesian variable selection with applications to growth regression. J. Macroecon. 29, 476–493 (2007)CrossRefGoogle Scholar
  105. 105.
    Ley, E., Steel, M.F.J.: On the effect of prior assumptions in Bayesian model averaging with applications to growth regression. J. Appl. Economet. 24, 651–674 (2009)MathSciNetCrossRefGoogle Scholar
  106. 106.
    Ley, E., Steel, M.F.J.: Comments on Jointness of growth determinants. J. Appl. Economet. 24, 248–251 (2009)CrossRefGoogle Scholar
  107. 107.
    Ley, E., Steel, M.F.J.: Mixtures of g-priors for Bayesian model averaging with economic applications. J. Economet. 171, 251–266 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 108.
    Li, G., Shi, J., Zhou, J.: Bayesian adaptive combination of short-term wind speed forecasts from neural network models. Renew. Energ. 36, 352–359 (2011)CrossRefGoogle Scholar
  109. 109.
    Li, Y., Clyde, M.A.: Mixtures of g-priors in generalized linear models. J. Am. Stat. Assoc. (2018).  https://doi.org/10.1080/01621459.2018.1469992
  110. 110.
    Liang, F., Wong, W.H.: Evolutionary Monte Carlo: applications to C\(_\text{p}\) model sampling and change point problem. Stat. Sin. 317–342 (2000)Google Scholar
  111. 111.
    Liang, F., Paulo, R., Molina, G., Clyde, M.A., Berger, J.O.: Mixtures of g priors for Bayesian variable selection. J. Am. Stat. Assoc. 103, 410–423 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    Liddle, A.R.: Information criteria for astrophysical model selection. Mon. Not. R. Astron. Soc. 377, L74–L78 (2007)CrossRefGoogle Scholar
  113. 113.
    Lindley, D.V.: A statistical paradox. Biometrika 44, 187–192 (1957)zbMATHCrossRefGoogle Scholar
  114. 114.
    Link, W., Barker, R.: Model weights and the foundations of multimodel inference. Ecology 87, 2626–2635 (2006)CrossRefGoogle Scholar
  115. 115.
    Link, W.A., Barker, R.J.: Bayesian Inference: With Ecological Applications. Academic Press, New York (2010)Google Scholar
  116. 116.
    Lu, D., Ye, M., Neuman, S.P.: Dependence of Bayesian model selection criteria and Fisher information matrix on sample size. Math. Geosci. 43, 971–993 (2011)zbMATHCrossRefGoogle Scholar
  117. 117.
    Lumley, T., Scott, A.: AIC and BIC for modeling with complex survey data. J. Surv. Stat. Methodol. 3, 1–18 (2015)CrossRefGoogle Scholar
  118. 118.
    Madigan, D., Raftery, A.E.: Model selection and accounting for model uncertainty in graphical models using Occam’s window. J. Am. Stat. Assoc. 89, 1535–1546 (1994)zbMATHCrossRefGoogle Scholar
  119. 119.
    Madigan, D., York, J., Allard, D.: Bayesian graphical models for discrete data. Int. Stat. Rev. 63, 215–232 (1995)zbMATHCrossRefGoogle Scholar
  120. 120.
    Maruyama, Y., George, E.I.: Fully Bayes factors with a generalized g-prior. Ann. Stat. 39, 2740–2765 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  121. 121.
    Meng, X.-L., Wong, W.H.: Simulating ratios of normalizing constants via a simple identity: a theoretical exploration. Stat. Sin. 6, 831–860 (1996)MathSciNetzbMATHGoogle Scholar
  122. 122.
    Millar, R.B.: Comparison of hierarchical Bayesian models for overdispersed count data using DIC and Bayes’ factors. Biometrics 65, 962–969 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  123. 123.
    Min, C.-K., Zellner, A.: Bayesian and non-Bayesian methods for combining models and forecasts with applications to forecasting international growth rates. J. Econ. 56, 89–118 (1993)zbMATHCrossRefGoogle Scholar
  124. 124.
    Minka, T.: Bayesian model averaging is not model combination. MIT Media Lab Note, December 2000Google Scholar
  125. 125.
    Mohammadi, A., Wit, E.C.: Bayesian structure learning in sparse Gaussian graphical models. Bayesian Anal. 10, 109–138 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  126. 126.
    Monteith, K., Carroll, J.L., Seppi, K., Martinez, T.: Turning BMA into Bayesian model combination. In: International Joint Conference on Neural Networks (2011)Google Scholar
  127. 127.
    Moore, J.E., Barlow, J.: Bayesian statespace model of fin whale abundance trends from a 1991–2008 time series of linetransect surveys in the California Current. J. Appl. Ecol. 48, 1195–1205 (2011)CrossRefGoogle Scholar
  128. 128.
    Moral-Benito, E.: Determinants of economic growth: a Bayesian panel data approach. Rev. Econ. Stat. 94, 566–579 (2012)CrossRefGoogle Scholar
  129. 129.
    Moral-Benito, E.: Model averaging in economics: an overview. J. Econ. Surv. 29, 46–75 (2015)CrossRefGoogle Scholar
  130. 130.
    Müller, S., Scealy, J.L., Welsh, A.H.: Model selection in linear mixed models. Stat. Sci. 28, 135–167 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  131. 131.
    Nelder, J.A., Wedderburn, R.W.M.: Generalized linear models. J. Roy. Stat. Soc. A. Stat. 135, 370–384 (1972)CrossRefGoogle Scholar
  132. 132.
    Newton, M.A., Raftery, A.E.: Approximate Bayesian inference with the weighted likelihood bootstrap. J. Roy. Stat. Soc. B. Methodol. 56, 3–48 (1994)MathSciNetzbMATHGoogle Scholar
  133. 133.
    Nott, D.J., Kohn, R.: Adaptive sampling for Bayesian variable selection. Biometrika 92, 747–763 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  134. 134.
    O’Hagan, A.: Discussion of Aitkin, M.: Posterior Bayes factors. J. Roy. Stat. Soc. B. Methodol. 53, 136 (1991)Google Scholar
  135. 135.
    O’Hagan, A.: Fractional Bayes factors for model comparison. J. Roy. Stat. Soc. B. Methodol. 57, 99–138 (1995)MathSciNetzbMATHGoogle Scholar
  136. 136.
    O’Hara, R.B., Mikko, J.S.: A review of Bayesian variable selection methods: what, how and which. Bayesian Anal. 4, 85–117 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  137. 137.
    Parry, M.: Extensive scoring rules. Electron. J. Stat. 10, 1098–1108 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  138. 138.
    Pauler, D.K.: The Schwarz criterion and related methods for normal linear models. Biometrika 85, 13–27 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  139. 139.
    Pauler, D.K., Wakefield, J.C., Kass, R.E.: Bayes factors and approximations for variance component models. J. Am. Stat. Assoc. 94, 1242–1253 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  140. 140.
    Pérez, J.M., Berger, J.O.: Expected-posterior prior distributions for model selection. Biometrika 89, 491–511 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  141. 141.
    Pole, A., West, M., Harrison, J.: Applied Bayesian Forecasting and Time Series Analysis. CRC Press, Boca Raton (1994)zbMATHCrossRefGoogle Scholar
  142. 142.
    Pooley, C.M., Marion, G.: Bayesian model evidence as a practical alternative to deviance information criterion. Roy. Soc. Open Sci. 5, 171519 (2018)MathSciNetCrossRefGoogle Scholar
  143. 143.
    Price, M.J., Welton, N.J., Briggs, A.H., Ades, A.E.: Model averaging in the presence of structural uncertainty about treatment effects: influence on treatment decision and expected value of information. Value Health 14, 205–218 (2011)CrossRefGoogle Scholar
  144. 144.
    R Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2017). https://www.R-project.org/
  145. 145.
    Raftery, A.E.: Bayesian model selection in social research. Sociol. Methodol. 25, 111–164 (1995)CrossRefGoogle Scholar
  146. 146.
    Raftery, A.E.: Approximate Bayes factors and accounting for model uncertainty in generalised linear models. Biometrika 83, 251–266 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  147. 147.
    Raftery, A.E., Madigan, D., Hoeting, J.A.: Bayesian model averaging for linear regression models. J. Am. Stat. Assoc. 92, 179–191 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  148. 148.
    Raftery, A.E., Zheng, Y.: Discussion of Hjort, N.L., Claeskens, G.: Frequentist model average estimators. J. Am. Stat. Assoc. 98, 931–938 (2003)Google Scholar
  149. 149.
    Raftery, A.E., Káný, M., Ettler, P.: Online prediction under model uncertainty via dynamic model averaging: application to a cold rolling mill. Technometrics 52, 52–66 (2010)MathSciNetCrossRefGoogle Scholar
  150. 150.
    Rissanen, J.: A universal prior for integers and estimation by minimum description length. Ann. Stat. 11, 416–431 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  151. 151.
    Robert, C.P., Marin, J.-M.: On some difficulties with a posterior probability approximation technique. Bayesian Anal. 3, 427–441 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  152. 152.
    Rossell, D., Telesca, D.: Nonlocal priors for high-dimensional estimation. J. Am. Stat. Assoc. 112, 254–265 (2017)MathSciNetCrossRefGoogle Scholar
  153. 153.
    Sabanés Bové, D., Held, L.: Bayesian fractional polynomials. Stat. Comput. 21, 309–324 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  154. 154.
    Sabanés Bové, D., Held, L.: Hyper-\(g\) priors for generalized linear models. Bayesian Anal. 6, 387–410 (2011)MathSciNetzbMATHGoogle Scholar
  155. 155.
    Sabanés Bové, D., Held, L., Kauermann, G.: Objective Bayesian model selection in generalized additive models with penalized splines. J. Comput. Graph. Stat. 24, 394–415 (2015)MathSciNetCrossRefGoogle Scholar
  156. 156.
    Sala-i-Martin, X., Doppelhofer, G., Miller, R.: Determinants of long-term growth: a Bayesian averaging of classical estimates (BACE) approach. Am. Econ. Rev. 94, 813–835 (2004)CrossRefGoogle Scholar
  157. 157.
    Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  158. 158.
    Scott, J.G., Berger, J.O.: Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem. Ann. Stat. 38, 2587–2619 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  159. 159.
    Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A.: Bayesian measures of model complexity and fit. J. R. Stat. Soc. B. Methodol. 64, 583–639 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  160. 160.
    Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A.: The deviance information criterion: 12 years on. J. R. Stat. Soc. B. Methodol. 76, 485–493 (2014)MathSciNetCrossRefGoogle Scholar
  161. 161.
    Steel, M.F.J.: Bayesian model averaging and forecasting. Bull. EU US Inflation Macroecon. Anal. 200, 30–41 (2011)Google Scholar
  162. 162.
    Steel, M.F.J.: Model averaging and its use in economics (2017). arXiv preprint: arXiv:1709.08221
  163. 163.
    Stock, J.H., Watson, M.W.: Forecasting with many predictors. In: Elliott, C.G.G., Timmermann, A. (eds.) Handbook of Economic Forecasting. Elsevier (2006)Google Scholar
  164. 164.
    Stone, M.: Cross-validatory choice and assessment of statistical predictions. J. Roy. Stat. Soc. B. Methodol. 36, 111–147 (1974)MathSciNetzbMATHGoogle Scholar
  165. 165.
    Stone, M.: Comments on model selection criteria of Akaike and Schwarz. J. Roy. Stat. Soc. B. Methodol. 41, 276–278 (1979)Google Scholar
  166. 166.
    Strachan, R.W.: Comment on Jointness of growth determinants by Gernot Doppelhofer and Melvyn Weeks. J. Appl. Economet. 24, 245–247 (2009)MathSciNetCrossRefGoogle Scholar
  167. 167.
    Thogmartin, W.E., Knutson, M.G., Sauer, J.R.: Predicting regional abundance of rare grassland birds with a hierarchical spatial count model. Condor 108, 25–46 (2006)CrossRefGoogle Scholar
  168. 168.
    Vehtari, A., Gelman, A., Gabry, J.: Pareto smoothed importance sampling (2017). arXiv preprint: arxiv:1507.02646
  169. 169.
    Vehtari, A., Gelman, A., Gabry, J.: Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Stat. Comput. 27, 1413–1432 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  170. 170.
    Villa, C., Walker, S.: An objective Bayesian criterion to determine model prior probabilities. Scand. J. Stat. 42, 947–966 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  171. 171.
    Volinsky, C.T., Raftery, A.E.: Bayesian information criterion for censored survival models. Biometrics 56, 256–262 (2000)zbMATHCrossRefGoogle Scholar
  172. 172.
    Walker, S.G., Gutiérrez-Peña, E., Muliere, P.: A decision theoretic approach to model averaging. J. Roy. Stat. Soc. D Stat. 50, 31–39 (2001)CrossRefGoogle Scholar
  173. 173.
    Wang, C., Dominici, F., Parmigiani, G., Zigler, C.M.: Accounting for uncertainty in confounder and effect modifier selection when estimating average causal effects in generalized linear models. Biometrics 71, 654–665 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  174. 174.
    Watanabe, S.: Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. J. Mach. Learn. Res. 11, 3571–3594 (2010)MathSciNetzbMATHGoogle Scholar
  175. 175.
    Watanabe, S.: A widely applicable Bayesian information criterion. J. Mach. Learn. Res. 14, 867–897 (2013)MathSciNetzbMATHGoogle Scholar
  176. 176.
    Wei, Y., McNicholas, P.D.: Mixture model averaging for clustering. Adv. Data Anal. Classi. 9, 197–217 (2015)MathSciNetCrossRefGoogle Scholar
  177. 177.
    Wilberg, M.J., Bence, J.R.: Performance of deviance information criterion model selection in statistical catch-at-age analysis. Fish. Res. 93, 212–221 (2008)CrossRefGoogle Scholar
  178. 178.
    Wong, H., Clarke, B.: Improvement over Bayes prediction in small samples in the presence of model uncertainty. Can. J. Stat. 32, 269–283 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  179. 179.
    Xie, W., Lewis, P.O., Fan, Y., Kuo, L., Chen, M.-H.: Improving marginal likelihood estimation for Bayesian phylogenetic model selection. Syst. Biol. 60, 150–160 (2010)CrossRefGoogle Scholar
  180. 180.
    Yang, Y.: Can the strengths of AIC and BIC be shared? A conflict between model indentification and regression estimation. Biometrika 92, 937–950 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  181. 181.
    Yao, Y., Vehtari, A., Simpson, D., Gelman, A.: Using stacking to average Bayesian predictive distributions. Bayesian Anal. (2018).  https://doi.org/10.1214/17-BA1091MathSciNetzbMATHCrossRefGoogle Scholar
  182. 182.
    Yao, Y., Vehtari, A., Simpson, D., Gelman, A.: Rejoinder to the Discussion of Yao, Y., Vehtari, A., Simpson, D., Gelman, A.: Using stacking to average Bayesian predictive distributions. Bayesian Anal. (2018).  https://doi.org/10.1214/17-BA1091MathSciNetzbMATHCrossRefGoogle Scholar
  183. 183.
    Ye, M., Meyer, P.D., Neuman, S.P.: On model selection criteria in multimodel analysis. Water Resour. Res. 44, W03428 (2008)Google Scholar
  184. 184.
    Yeung, K.Y., Bumgarner, R.E., Raftery, A.E.: Bayesian model averaging: development of an improved multi-class, gene selection and classification tool for microarray data. Bioinformatics 21, 2394–2402 (2005)CrossRefGoogle Scholar
  185. 185.
    Zaffalon, M.: The naive credal classifier. J. Stat. Plan. Infer. 105, 5–21 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  186. 186.
    Zellner, A., Siow, A.: Posterior odds ratios for selected regression hypotheses. In: Bernardo, J.M., DeGroot, M.H., Lindley, D.V., Smith, A.F.M. (eds.) Bayesian Statistics: Proceedings of the First International Meeting held in Valencia, Spain, pp. 585–603. University Press (1980)Google Scholar
  187. 187.
    Zellner, A.: On assessing prior distributions and Bayesian regression analysis with g-prior distributions. In: Goel, P.K., Zellner, A. (eds.) Bayesian Inference and Decision Techniques: Essays in Honor of Bruno De Finetti, pp. 233–243. Elsevier Science, Oxford (1986)zbMATHGoogle Scholar
  188. 188.
    Zhao, J., Jin, L., Shi, L.: Mixture model selection via hierarchical BIC. Comput. Stat. Data Anal. 88, 139–153 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  189. 189.
    Zigler, C.M., Dominici, F.: Uncertainty in propensity score estimation: Bayesian methods for variable selection and model-averaged causal effects. J. Am. Stat. Assoc. 109, 95–107 (2014)MathSciNetCrossRefGoogle Scholar

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© The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OtagoDunedinNew Zealand

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