##
**Benchmark priors for Bayesian model averaging.**
*(English)*
Zbl 1091.62507

Summary: In contrast to a posterior analysis given a particular sampling model, posterior model probabilities in the context of model uncertainty are typically rather sensitive to the specification of the prior. In particular, ‘diffuse’ priors on model-specific parameters can lead to quite unexpected consequences. Here we focus on the practically relevant situation where we need to entertain a (large) number of sampling models and we have (or wish to use) little or no subjective prior information. We aim at providing an ‘automatic’ or ‘benchmark’ prior structure that can be used in such cases. We focus on the normal linear regression model with uncertainty in the choice of regressors. We propose a partly non-informative prior structure related to a natural conjugate g-prior specification, where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter \(g_{0j}\). The consequences of different choices for \(g_{0j}\) are examined. We investigate theoretical properties, such as consistency of the implied Bayesian procedure. Links with classical information criteria are provided. More importantly, we examine the finite sample implications of several choices of \(g_{0j}\)in a simulation study. The use of the MC\(^3\) algorithm of D. Madigan and J. York [Int. Stat. Rev. 63, No. 2, 215–232 (1995; Zbl 0834.62003)], combined with efficient coding in Fortran, makes it feasible to conduct large simulations. In addition to posterior criteria, we shall also compare the predictive performance of different priors. A classic example concerning the economics of crime will also be provided and contrasted with results in the literature. The main findings of the paper will lead us to propose a ‘benchmark’ prior specification in a linear regression context with model uncertainty.

### MSC:

62F15 | Bayesian inference |

62P20 | Applications of statistics to economics |

65C60 | Computational problems in statistics (MSC2010) |

### Citations:

Zbl 0834.62003
PDF
BibTeX
XML
Cite

\textit{C. Fernández} et al., J. Econom. 100, No. 2, 381--427 (2001; Zbl 1091.62507)

Full Text:
DOI

### References:

[1] | Akaike, H.: Likelihood of a model and information criteria. Journal of econometrics 16, 3-14 (1981) · Zbl 0457.62032 |

[2] | Amemiya, T.: Advanced econometrics. (1985) |

[3] | Atkinson, A. C.: Likelihood ratios, posterior odds and information criteria. Journal of econometrics 16, 15-20 (1981) |

[4] | Bauwens, L.: The pathology of the natural conjugate prior density in the regression model. Annales d’economie et de statistique 23, 49-64 (1991) |

[5] | Becker, G. S.: Crime and punishment: an economic approach. Journal of political economy 76, 169-217 (1968) |

[6] | Berger, J. O.; Pericchi, L. R.: The intrinsic Bayes factor for model selection and prediction. Journal of the American statistical association 91, 109-122 (1996) · Zbl 0870.62021 |

[7] | Bernardo, J. M.: Expected information as expected utility. The annals of statistics 7, 686-690 (1979) · Zbl 0407.62002 |

[8] | Bernardo, J. M.: A Bayesian analysis of classical hypothesis testing (with discussion). Bayesian statistics, 605-618 (1980) |

[9] | Box, G. E. P.: Sampling and Bayes inference in scientific modelling and robustness (with discussion). Journal of the royal statistical society, series A 143, 383-430 (1980) · Zbl 0471.62036 |

[10] | Chib, S.; Greenberg, E.: Understanding the metropolis-Hastings algorithm. The American statistician 49, 327-335 (1995) |

[11] | Chipman, H.: Bayesian variable selection with related predictors. Canadian journal of statistics 24, 17-36 (1996) · Zbl 0849.62032 |

[12] | Chow, G. C.: A comparison of the information and posterior probability criteria for model selection. Journal of econometrics 16, 21-33 (1981) · Zbl 0457.62033 |

[13] | Clyde, M.; Desimone, H.; Parmigiani, G.: Prediction via orthogonalized model mixing. Journal of the American statistical association 91, 1197-1208 (1996) · Zbl 0880.62026 |

[14] | Cornwell, C.; Trumbull, W. N.: Estimating the economic model of crime with panel data. Review of economics and statistics 76, 360-366 (1994) |

[15] | Dawid, A. P.: Statistical theory: the prequential approach. Journal of the royal statistical society, series A 147, 278-292 (1984) · Zbl 0557.62080 |

[16] | Dawid, A. P.: Probability forecasting. Encyclopedia of statistical sciences, vol. 7 7, 210-218 (1986) |

[17] | Draper, D.: Assessment and propagation of model uncertainty (with discussion). Journal of the royal statistical society, series B 57, 45-97 (1995) · Zbl 0812.62001 |

[18] | Ehrlich, I.: Participation in illegitimate activities: a theoretical and empirical investigation. Journal of political economy 81, 521-567 (1973) |

[19] | Ehrlich, I.: The deterrent effect of capital punishment: a question of life and death. American economic review 65, 397-417 (1975) |

[20] | Foster, D. P.; George, E. I.: The risk inflation criterion for multiple regression. The annals of statistics 22, 1947-1975 (1994) · Zbl 0829.62066 |

[21] | Freedman, D. A.: A note on screening regressions. The American statistician 37, 152-155 (1983) |

[22] | Geisser, S.; Eddy, W. F.: A predictive approach to model selection. Journal of the American statistical association 74, 153-160 (1979) · Zbl 0401.62036 |

[23] | Gelfand, A. E.; Dey, D. K.: Bayesian model choice: asymptotics and exact calculations. Journal of the royal statistical society, series B 56, 501-514 (1994) · Zbl 0800.62170 |

[24] | George, E.I., 1999. Bayesian model selection, Encyclopedia of Statistical Sciences Update, Vol. 3. (eds.) S. Kotz, C. Read and D.L. Banks Wiley, New York. |

[25] | George, E.I., Foster, D.P., 1997. Calibration and empirical Bayes variable selection. Mimeo, University of Texas, Austin. · Zbl 1029.62008 |

[26] | George, E. I.; Mcculloch, R. E.: Variable selection via Gibbs sampling. Journal of the American statistical association 88, 881-889 (1993) |

[27] | George, E. I.; Mcculloch, R. E.: Approaches for Bayesian variable selection. Statistica sinica 7, 339-373 (1997) · Zbl 0884.62031 |

[28] | Geweke, J.: Variable selection and model comparison in regression. Bayesian statistics 5, 609-620 (1996) |

[29] | Good, I. J.: Rational decisions. Journal of the royal statistical society, series B 14, 107-114 (1952) |

[30] | Green, P. J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711-732 (1995) · Zbl 0861.62023 |

[31] | Hannan, E. J.; Quinn, B. G.: The determination of the order of an autoregression. Journal of the royal statistical society, series B 41, 190-195 (1979) · Zbl 0408.62076 |

[32] | Hoeting, J.A., Raftery, A.E., Madigan, D., 1995. Simultaneous variable and transformation selection in linear regression. Technical Report 9506, Statistics Department, Colorado State University. · Zbl 0900.62352 |

[33] | Hoeting, J. A.; Raftery, A. E.; Madigan, D.: A method for simultaneous variable selection and outlier identification in linear regression. Computational statistics and data analysis 22, 251-271 (1996) · Zbl 0900.62352 |

[34] | Kass, R. E.; Raftery, A. E.: Bayes factors. Journal of the American statistical association 90, 773-795 (1995) · Zbl 0846.62028 |

[35] | Kass, R. E.; Wasserman, L.: A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. Journal of the American statistical association 90, 928-934 (1995) · Zbl 0851.62020 |

[36] | Laud, P. W.; Ibrahim, J. G.: Predictive model selection. Journal of the royal statistical society, series B 57, 247-262 (1995) · Zbl 0809.62024 |

[37] | Laud, P. W.; Ibrahim, J. G.: Predictive specification of prior model probabilities in variable selection. Biometrika 83, 267-274 (1996) · Zbl 0864.62012 |

[38] | Leamer, E. E.: Specification searches: ad hoc inference with nonexperimental data. (1978) · Zbl 0384.62089 |

[39] | Lee, H., 1996. Model selection for consumer loan application data, Mimeo, Technical Report 650, Statistics Department Carnegie Mellon University. |

[40] | Lindsey, J. K.: Some statistical heresies (with discussion). The statistician 48, 1-40 (1999) |

[41] | Madigan, D.; Gavrin, J.; Raftery, A. E.: Eliciting prior information to enhance the predictive performance of Bayesian graphical models. Communications in statistics, theory and methods 24, 2271-2292 (1995) · Zbl 0937.62576 |

[42] | Madigan, D.; Raftery, A. E.: Model selection and accounting for model uncertainty in graphical models using Occam’s window. Journal of the American statistical association 89, 1535-1546 (1994) · Zbl 0814.62030 |

[43] | Madigan, D.; York, J.: Bayesian graphical models for discrete data. International statistical review 63, 215-232 (1995) · Zbl 0834.62003 |

[44] | Min, C.; Zellner, A.: Bayesian and non-Bayesian methods for combining models and forecasts with applications to forecasting international growth rates. Journal of econometrics 56, 89-118 (1993) · Zbl 0800.62800 |

[45] | Mitchell, T. J.; Beauchamp, J. J.: Bayesian variable selection in linear regression (with discussion). Journal of the American statistical association 83, 1023-1036 (1988) · Zbl 0673.62051 |

[46] | O’hagan, A.: Fractional Bayes factors for model comparison (with discussion). Journal of the royal statistical society, series B 57, 99-138 (1995) · Zbl 0813.62026 |

[47] | Osiewalski, J.; Steel, M. F. J.: Regression models under competing covariance structures: a Bayesian perspective. Annales d’economie et de statistique 32, 65-79 (1993) |

[48] | Pericchi, L. R.: An alternative to the standard Bayesian procedure for discrimination between normal linear models. Biometrika 71, 575-586 (1984) · Zbl 0562.62028 |

[49] | Phillips, P. C. B.: Bayesian model selection and prediction with empirical applications (with discussion). Journal of econometrics 69, 289-365 (1995) · Zbl 0925.62528 |

[50] | Poirier, D.: Bayesian hypothesis testing in linear models with continuously induced conjugate priors across hypotheses. Bayesian statistics 2, 711-722 (1985) · Zbl 0671.62030 |

[51] | Poirier, D.: Frequentist and subjectivist perspectives on the problem of model building in economics (with discussion). Economic perspectives 2, 121-144 (1988) |

[52] | Poirier, D.: Prior beliefs about fit. Bayesian statistics 5, 731-738 (1996) |

[53] | Raftery, A. E.: Approximate Bayes factors and accounting for model uncertainty in generalised linear models. Biometrika 83, 251-266 (1996) · Zbl 0864.62049 |

[54] | Raftery, A. E.; Madigan, D.; Hoeting, J. A.: Bayesian model averaging for linear regression models. Journal of the American statistical association 92, 179-191 (1997) · Zbl 0888.62026 |

[55] | Raftery, A. E.; Madigan, D.; Volinsky, C. T.: Accounting for model uncertainty in survival analysis improves predictive performance (with discussion). Bayesian statistics 5, 323-349 (1996) |

[56] | Richard, J. F.: Posterior and predictive densities for simultaneous equation models. (1973) · Zbl 0271.62136 |

[57] | Richard, J. F.; Steel, M. F. J.: Bayesian analysis of systems of seemingly unrelated regression equations under a recursive extended natural conjugate prior density. Journal of econometrics 38, 7-37 (1988) |

[58] | Schwarz, G.: Estimating the dimension of a model. The annals of statistics 6, 461-464 (1978) · Zbl 0379.62005 |

[59] | Smith, M.; Kohn, R.: Nonparametric regression using Bayesian variable selection. Journal of econometrics 75, 317-343 (1996) · Zbl 0864.62025 |

[60] | Smith, A. F. M.; Spiegelhalter, D. J.: Bayes factors and choice criteria for linear models. Journal of the royal statistical society, series B 47, 213-220 (1980) · Zbl 0433.62045 |

[61] | Vandaele, W.: Participation in illegitimate activities; ehrlich revisited. Deterrence and incapacitation, 270-335 (1978) |

[62] | Volinsky, C. T.; Madigan, D.; Raftery, A. E.; Kronmal, R. A.: Bayesian model averaging in proportional hazard models: assessing the risk of a stroke. Applied statistics 46, 433-448 (1997) · Zbl 0903.62093 |

[63] | Zellner, A.: On assessing prior distributions and Bayesian regression analysis with g-prior distributions. Bayesian inference and decision techniques: essays in honour of bruno de Finetti, 233-243 (1986) · Zbl 0655.62071 |

[64] | Zellner, A.; Siow, A.: Posterior odds ratios for selected regression hypotheses (with discussion). Bayesian statistics, 585-603 (1980) · Zbl 0457.62004 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.