Book contents
- Frontmatter
- Contents
- Preface
- Notation and convention
- Part I Loss models
- Part II Risk and ruin
- Part III Credibility
- 6 Classical credibility
- 7 Bühlmann credibility
- 8 Bayesian approach
- 9 Empirical implementation of credibility
- Part IV Model construction and evaluation
- Appendix: Review of statistics
- Answers to exercises
- References
- Index
8 - Bayesian approach
from Part III - Credibility
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Notation and convention
- Part I Loss models
- Part II Risk and ruin
- Part III Credibility
- 6 Classical credibility
- 7 Bühlmann credibility
- 8 Bayesian approach
- 9 Empirical implementation of credibility
- Part IV Model construction and evaluation
- Appendix: Review of statistics
- Answers to exercises
- References
- Index
Summary
In this chapter we consider the Bayesian approach in updating the prediction for future losses. We consider the derivation of the posterior distribution of the risk parameters based on the prior distribution of the risk parameters and the likelihood function of the data. The Bayesian estimate of the risk parameter under the squared-error loss function is the mean of the posterior distribution. Likewise, the Bayesian estimate of the mean of the random loss is the posterior mean of the loss conditional on the data.
In general, the Bayesian estimates are difficult to compute, as the posterior distribution may be quite complicated and intractable. There are, however, situations where the computation may be straightforward, as in the case of conjugate distributions. We define conjugate distributions and provide some examples for cases that are of relevance in analyzing loss measures. Under specific classes of conjugate distributions, the Bayesian predictor is the same as the Bühlmann predictor. Specifically, when the likelihood belongs to the linear exponential family and the prior distribution is the natural conjugate, the Bühlmann credibility estimate is equal to the Bayesian estimate. This result provides additional justification for the use of the Bühlmann approach.
Learning objectives
Bayesian inference and estimation
Prior and posterior pdf
Bayesian credibility
Conjugate prior distribution
Linear exponential distribution
Bühlmann credibility versus Bayesian credibility
Bayesian inference and estimation
The classical and Bühlmann credibility models update the prediction for future losses based on recent claim experience and existing prior information.
- Type
- Chapter
- Information
- Nonlife Actuarial ModelsTheory, Methods and Evaluation, pp. 223 - 252Publisher: Cambridge University PressPrint publication year: 2009