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Finding a minimally informative Dirichlet prior distribution using least squares

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  • Kelly, Dana
  • Atwood, Corwin

Abstract

In a Bayesian framework, the Dirichlet distribution is the conjugate distribution to the multinomial likelihood function, and so the analyst is required to develop a Dirichlet prior that incorporates available information. However, as it is a multiparameter distribution, choosing the Dirichlet parameters is less straightforward than choosing a prior distribution for a single parameter, such as p in the binomial distribution. In particular, one may wish to incorporate limited information into the prior, resulting in a minimally informative prior distribution that is responsive to updates with sparse data. In the case of binomial p or Poisson λ, the principle of maximum entropy can be employed to obtain a so-called constrained noninformative prior. However, even in the case of p, such a distribution cannot be written down in the form of a standard distribution (e.g., beta, gamma), and so a beta distribution is used as an approximation in the case of p. In the case of the multinomial model with parametric constraints, the approach of maximum entropy does not appear tractable. This paper presents an alternative approach, based on constrained minimization of a least-squares objective function, which leads to a minimally informative Dirichlet prior distribution. The alpha-factor model for common-cause failure, which is widely used in the United States, is the motivation for this approach, and is used to illustrate the method. In this approach to modeling common-cause failure, the alpha-factors, which are the parameters in the underlying multinomial model for common-cause failure, must be estimated from data that are often quite sparse, because common-cause failures tend to be rare, especially failures of more than two or three components, and so a prior distribution that is responsive to updates with sparse data is needed.

Suggested Citation

  • Kelly, Dana & Atwood, Corwin, 2011. "Finding a minimally informative Dirichlet prior distribution using least squares," Reliability Engineering and System Safety, Elsevier, vol. 96(3), pages 398-402.
  • Handle: RePEc:eee:reensy:v:96:y:2011:i:3:p:398-402
    DOI: 10.1016/j.ress.2010.11.008
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    Cited by:

    1. Qin, Hao & Stewart, Mark G., 2020. "Construction defects and wind fragility assessment for metal roof failure: A Bayesian approach," Reliability Engineering and System Safety, Elsevier, vol. 197(C).
    2. Utkin, Lev V. & Coolen, Frank P.A. & Gurov, Sergey V., 2015. "Imprecise inference for warranty contract analysis," Reliability Engineering and System Safety, Elsevier, vol. 138(C), pages 31-39.
    3. Coolen, Frank P.A. & Coolen-Maturi, Tahani, 2015. "Predictive inference for system reliability after common-cause component failures," Reliability Engineering and System Safety, Elsevier, vol. 135(C), pages 27-33.
    4. Le Duy, Tu Duong & Vasseur, Dominique, 2018. "A practical methodology for modeling and estimation of common cause failure parameters in multi-unit nuclear PSA model," Reliability Engineering and System Safety, Elsevier, vol. 170(C), pages 159-174.
    5. Mi, Jinhua & Beer, Michael & Li, Yan-Feng & Broggi, Matteo & Cheng, Yuhua, 2020. "Reliability and importance analysis of uncertain system with common cause failures based on survival signature," Reliability Engineering and System Safety, Elsevier, vol. 201(C).
    6. Zheng, Xiaoyu & Yamaguchi, Akira & Takata, Takashi, 2013. "α-Decomposition for estimating parameters in common cause failure modeling based on causal inference," Reliability Engineering and System Safety, Elsevier, vol. 116(C), pages 20-27.
    7. Xiang, W. & Zhou, W., 2021. "Bayesian network model for predicting probability of third-party damage to underground pipelines and learning model parameters from incomplete datasets," Reliability Engineering and System Safety, Elsevier, vol. 205(C).
    8. Troffaes, Matthias C.M. & Walter, Gero & Kelly, Dana, 2014. "A robust Bayesian approach to modeling epistemic uncertainty in common-cause failure models," Reliability Engineering and System Safety, Elsevier, vol. 125(C), pages 13-21.

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