# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## 4 A Bayes Method of Inference for the Risk Index Based upon the IRD Model
### The Case of a Gamma Prior PDF on the IRD Parameter \(\alpha\)--Posterior Analysis
Numerical Application for Performances of the Bayes Approach in the Case of a Beta Prior PDF on the IR Parameter S(\(\alpha\))  
In this section, the topic of the previous section is continued, assuming a beta prior PDF on the IR parameter S(\(\alpha\)) while maintaining the customary assumption of a gamma prior PDF for the random parameter \(\phi\). Also, in this case, interest is focused on the evaluation of the performances of the Bayes approach by numerical methods since, as highlighted in Section 4, also in this case, generally an analytical solution is not available or is very cumbersome [37; 38]. In practice, as illustrated in Section 4, a prior guess on the parameter S (\(\alpha\) can, as in the case of a prior guess on \(\alpha\), be originated by a prior guess on the median or may be directly expressed on the basis of \"credible interval\" (S\({}_{1}\), S\({}_{2}\)), i.e., a Bayesian (prior) confidence interval on S). As already discussed (see the discussion after Equation (38)), it is convenient that S, a RV with support in (0,1), is assumed as a beta RV (already introduced in Section 4 and with more details in Appendix D), since this is a very flexible distribution in the interval (0,1), which implies that the parameter \(\alpha\) has an NEBD PDF (Appendix D), since:  
\[\alpha=[-\log(1-S)]z^{2} \tag{66}\]  
For brevity, here, only three cases are examined. In detail, we start from the same cases examined in previous section, here recalled only in the \"most unfavorable\" case of both CV = 0.10:  
Case A22 (\"low risk\" values: E[R] \(\approx\) 0.0735): E[\(\phi\)] = 2.5, E[\(\alpha\)] = 15, CV = 0.10 for both \(\phi\) and \(\alpha\);  
Case B22 (\"medium risk\" values: E[R] \(\approx\) 0.3585): E[\(\phi\)] = 5, E[\(\alpha\)] = 45, CV = 0.10 for both \(\phi\) and \(\alpha\);  
corresponding to an approximate expected value of the RI: E[R] \(\approx\) 0.3585;  
Case C22 (\"high risk\" values: E[R] \(\approx\) 0.8167: E[\(\phi\)] = 10, E[\(\alpha\)] = 90, CV = 0.10 for both \(\phi\) and \(\alpha\). For easier comparison with the cases examined above, we assume that the beta(r,s) prior PDF for \(S\) has the mean value corresponding to the above mean values of \(\alpha\), with the value CV = 0.10 assigned to \(S\), so to obtain the following three cases, denoted with apexes A'22, B'22, and A'22 corresponding respectively to A22, B22, and C22. **Case A'22**: E[S] = 0.0305 and CV[S] = 0.10 (implying a beta(r,s) PDF with: r = 96.92, s = 3081) and E[\(\phi\)] = 2.5, CV[\(\phi\)] = 0.10;
**Case B'22**: E[S] = 0.0888 and CV[S] = 0.10 (implying a beta(r,s) PDF with: r = 91.03, s = 934.1) and E[\(\phi\)] = 5, CV[\(\phi\)] = 0.10;
**Case C'22**: E[S] = 0.1697 and CV[S] = 0.10 (implying a beta(r,s) PDF with: r = 82.86, s = 405.4) and E[\(\phi\)] = 10, CV[\(\phi\)] = 0.10. It is apparent that the results are quite comparable and indeed very similar to those of the previous section, confirming the validity of the approach with respect to different methods of assessing the researcher's prior information. Finally, it is remarked that an analysis of the robustness of the Bayesian Estimation with respect to the assumed prior hypotheses has been successfully carried out following the same methodology illustrated in [27]. In particular, many different prior PDFs were chosen instead of the assumed ones, e.g., lognormal or uniform prior PDF with the same mean value and CV of the assumed gamma and beta PDF. For reasons of length, these results are not reported here but are available on request from the authors.