LOCAL SENSITIVITY analysis FOR BAYESIAN MIXTURE MODELS

M. J. Rufo*, C. J. Pérez, J. Martín
Universidad de Extremadura, Departamento de Matemáticas, Spain
mrufo@unex.es

The analysis of finite mixture models is recently playing an important role in theoretical as well as in applied statistics (see, for example, Mclachlan and Peel [1] and Böhning and Seidel [2]). Mixture models are excellent tools that can be used to describe complex systems over a wide range of applications in many fields of knowledge. Since the advent of Markov Chain Monte Carlo (MCMC) methods, many advances have been achieved for the Bayesian analysis of finite mixture models. 

Inference of a probability distribution for lifetime data is necessary in any reliability analysis. Often this inference is complicated by the presence of multiple failure modes and variations in manufacturing that may create a heterogeneous component population. In these cases, standard distributions are inadequate. Mixture models provide additional modeling flexibility and have the physical interpretation of representing failures from a heterogeneous population. Titterington et al [3] give detailed references about mixtures for lifetime data.  

In spite of the modeling advantages, the practical implementation of finite mixture models sometimes introduces additional complexity because they often have many parameters and the calculations become difficult. Rufo et al [4] provided a general approach to address Bayesian analysis of finite mixture models of distributions from natural exponential families with quadratic variance function (NEF-QVF). Note that the families in this class are normal, gamma, hyperbolic-secant, Poisson, binomial and negative-binomial. This general approach solves the prior distribution choice and the unidentifiability problems in this kind of mixtures. However, a sensitivity analysis on the choice of the parameters in the prior distribution is needed. Most sensitivity analysis are informal ones and are based on changing the values in the prior parameters and observe how the output changes. This needs re-running the sampling algorithm for several parameter values with the corresponding computational cost. In this context, formal sensitivity analysis is a difficult task demanded by several authors. Pérez et al [5] proposed a computationally low-cost method to estimate local sensitivities in Bayesian models. This method is based on importance sampling and it requires to compute prior derivatives. It can be applied (but not only) to complex Bayesian models that need to be solved by MCMC methods, and it allows to estimate the sensitivity measures and their errors with no additional random sampling. 

In this work, a formal sensitivity analysis for the parameter values of the prior distribution in finite mixtures of distributions from NEF-QVF is described and applied. The derivatives that are needed to apply the method are obtained for all the distributions in this family class. This fact allows a direct implementation, and, therefore, the applicability to mixtures of lifetime distributions. The sensitivities are evaluated and give us information about a proper choice of the prior parameter values. This method can be applied in many applications arising in reliability contexts. 

Acknowledgement: This research has been partially supported by Ministerio de Educación y Ciencia, Spain (Project TSI2004-06801-C04-03).

References

[1] Mclachlan G, Peel McLachlan D: Finite Mixture Models. Wiley, 2000. 

[2] Böhning D, Seidel W: Editorial: recent developments in mixture models. Computational Statistics and Data Analysis 41, 349-357 (2003)
 
[3] Titterington DM, Smith, AFM, Makov UE: Statistical analysis of finite mixture distributions. Wiley, 1985.

[4] Rufo MJ, Martín J, Pérez CJ: Bayesian analysis of finite mixture models of distributions from exponential families. Computational Statistics 21, 621-637 (2006)

[5] Pérez CJ, Martín J, Rufo MJ: Sensitivity estimations for Bayesian inference models solved by MCMC methods. Reliability Engineering and System Safety 91, 1310-1314 (2006)