Back
 OJS  Vol.1 No.3 , October 2011
A Kullback-Leibler Divergence for Bayesian Model Diagnostics
Abstract: This paper considers a Kullback-Leibler distance (KLD) which is asymptotically equivalent to the KLD by Goutis and Robert [1] when the reference model (in comparison to a competing fitted model) is correctly specified and that certain regularity conditions hold true (ref. Akaike [2]). We derive the asymptotic property of this Goutis-Robert-Akaike KLD under certain regularity conditions. We also examine the impact of this asymptotic property when the regularity conditions are partially satisfied. Furthermore, the connection between the Goutis-Robert-Akaike KLD and a weighted posterior predictive p-value (WPPP) is established. Finally, both the Goutis-Robert-Akaike KLD and WPPP are applied to compare models using various simulated examples as well as two cohort studies of diabetes.
Cite this paper: nullC. Wang and M. Ghosh, "A Kullback-Leibler Divergence for Bayesian Model Diagnostics," Open Journal of Statistics, Vol. 1 No. 3, 2011, pp. 172-184. doi: 10.4236/ojs.2011.13021.
References

[1]   C. Goutis and C. P. Robert, “Model Choice in Generalised Linear Models: A Bayesian Approach via Kullback- Leibler Projections,” Biometrika, Vol. 85, No. 1, 1998, pp. 29-37. doi:10.1093/biomet/85.1.29

[2]   H. Akiaike, “A New Look at the Statistical Identification Model,” IEEE Transactions on Automatic Control, Vol. 19, No. 6, 1974, pp. 716-723. doi:10.1109/TAC.1974.1100705

[3]   C. E. Shannon, “A Mathematical Theory of Communication,” Bell System Technical Journal, Vol. 27, 1948, pp. 379-423 and pp. 623-656.

[4]   S. Kullback and R. A. Leibler, “On Information and Sufficiency,” The Annals of Mathematical Statistics, Vol. 22, No. 1, 1951, pp. 79-86. doi:10.1214/aoms/1177729694

[5]   D. V. Lindley, “On a Measure of the Information Provided by an Experiment,” The Annals of Mathematical Statistics, Vol. 27, No. 4, 1956, pp. 986-1005. doi:10.1214/aoms/1177728069

[6]   J. M. Bernardo, “Expected Information as Expected Utility,” The Annals of Statistics, Vol. 7, No. 3, 1979, pp. 686-690. doi:10.1214/aos/1176344689

[7]   G. Schwarz, “Estimating the Dimension of a Model,” The Annals of Statistics, Vol. 6, No. 2, 1978, pp. 461-464. doi:10.1214/aos/1176344136

[8]   I. Guttman, “The Use of the Concept of a Future Observation in Goodness-of-Fit Problems,” Journal of the Ro- yal Statistical Society B, Vol. 29, No. 1, 1967, pp. 83-100.

[9]   D. B. Rubin, “Bayesianly Justifiable and Relevant Frequency Calculations for the Applies Statistician,” Annals of Statistics, Vol. 12, No. 4, 1984, pp. 1151-1172. doi:10.1214/aos/1176346785

[10]   A. Gelman, J. Carlin, H. S. Stern and D. Rubin, “Bayesian Data Analysis,” Chapman and Hall, London, 1996.

[11]   H. P. Hazuda, S. M. Haffner, M. P. Stern and C. W. Eifler, “Effects of Acculturation and Socioeconomic Status on Obesity and Diabetes in Mexican Americans: The San Antonio Heart Study,” American Journal of Epidemiology, Vol. 128, No. 6, 1988, pp. 1289-1301.

[12]   S. Ghosal and T. Samanta, “Expansion of Bayes Risk for Entropy Loss and Reference Prior in Nonregular Cases,” Statistics and Decisions, Vol. 15, 1997, pp. 129-140.

[13]   I. Ibragimov and R. Hasminskii, “Statistical Estimation: Asymptotic Theory,” Springler-Verlag, New York, 1980.

 
 
Top