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A Variant of AIC Based on the Bayesian Marginal Likelihood

Author

Listed:
  • Yuki Kawakubo

    (Chiba University)

  • Tatsuya Kubokawa

    (University of Tokyo)

  • Muni S. Srivastava

    (University of Toronto)

Abstract

We propose information criteria that measure the prediction risk of a predictive density based on the Bayesian marginal likelihood from a frequentist point of view. We derive criteria for selecting variables in linear regression models, assuming a prior distribution of the regression coefficients. Then, we discuss the relationship between the proposed criteria and related criteria. There are three advantages of our method. First, this is a compromise between the frequentist and Bayesian standpoints because it evaluates the frequentist’s risk of the Bayesian model. Thus, it is less influenced by a prior misspecification. Second, the criteria exhibits consistency when selecting the true model. Third, when a uniform prior is assumed for the regression coefficients, the resulting criterion is equivalent to the residual information criterion (RIC) of Shi and Tsai (J. R. Stat. Soc. Ser. B 64, 237–252 2002).

Suggested Citation

  • Yuki Kawakubo & Tatsuya Kubokawa & Muni S. Srivastava, 2018. "A Variant of AIC Based on the Bayesian Marginal Likelihood," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 60-84, May.
    • Handle: RePEc:spr:sankhb:v:80:y:2018:i:1:d:10.1007_s13571-018-0152-7
    DOI: 10.1007/s13571-018-0152-7
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    References listed on IDEAS

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    1. Peide Shi & Chih‐Ling Tsai, 2002. "Regression model selection—a residual likelihood approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 237-252, May.
    2. Tomohiro Ando, 2007. "Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models," Biometrika, Biometrika Trust, vol. 94(2), pages 443-458.
    3. Srivastava, Muni S. & Kubokawa, Tatsuya, 2010. "Conditional information criteria for selecting variables in linear mixed models," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 1970-1980, October.
    4. Hamparsum Bozdogan, 1987. "Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions," Psychometrika, Springer;The Psychometric Society, vol. 52(3), pages 345-370, September.
    5. Florin Vaida & Suzette Blanchard, 2005. "Conditional Akaike information for mixed-effects models," Biometrika, Biometrika Trust, vol. 92(2), pages 351-370, June.
    6. David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika Van Der Linde, 2002. "Bayesian measures of model complexity and fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 583-639, October.
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