A new framework is explored for combining imprecise Bayesian methods with likelihood inference, and it is presented in the context of reliability growth models. The main idea of the framework is to divide a set of the model parameters of interest into two subsets related to fundamentally different aspects of the overall model, and to combine Walley's idea of imprecise Bayesian models related to one of the subsets of the model parameters with maximum likelihood estimation for the other subset. In accordance with the first subset and statistical data, the imprecise Bayesian model is constructed, which provides lower and upper predictive probability distributions depending on the second subset of parameters. These further parameters are then estimated by a maximum likelihood method, based on a novel proposition for maximum likelihood estimation over sets of distributions following from imprecise Bayesian models for the other subset of parameters. Use of this hybrid method is illustrated for reliability growth models and regression models, and some essential topics that need to be addressed in order to fully justify and further develop this framework are discussed.
Keywords. Bayesian inference, imprecise probabilities, linear regression, lower and upper probability distributions, maximum likelihood estimation, reliability growth models
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Authors addresses:
Lev Utkin
Institutski per. 5, 194021 St.Petersburg
Svetlana Zatenko
Department of Mathematics
St.Petersburg State Forest Technical Academy
Institutski per. 5
St.Petersburg, 194021
Frank Coolen
Department of Mathematical Sciences
Science Laboratories, South Road
Durham, DH1 3LE,
England
E-mail addresses:
| Lev Utkin | lev.utkin@mail.ru |
| Svetlana Zatenko | s_lana2004@mail.ru |
| Frank Coolen | Frank.Coolen@durham.ac.uk |