# 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
### Bayes Estimation of the Parameters of the IRD Model
Since \(\alpha=\psi\) (\(m\))\({}^{2}\), the corresponding quantiles (\(\alpha_{1}\), \(\alpha\)) on \(\alpha\) are readily be evaluated as follows: \[\alpha_{1}=\psi\) (\(m_{1}\))\({}^{2}\); \(\alpha_{2}=\psi\) (\(m_{2}\))\({}^{2}\) so that \[P(\alpha_{1}<\alpha<\alpha_{2})=p\] (41)
2. On the basis of such quantiles of \(\alpha\), the corresponding quantiles of S are readily computed by (38) with S being a one-to-one increasing function of \(\alpha\), as follows: \[S_{1}=1-e^{-(\frac{\alpha_{1}}{2})};\ S_{2}=1-e^{-(\frac{\alpha_{2}}{2})}\] (42)This information is sufficient (see [27]) for assessing the beta PDF on \(S\). In any case, the posterior PDF on \(S\) must be evaluated numerically. In principle, this also holds for the posterior PDF on R, which is the basic tool for assessing the BE of R, i.e., its posterior mean. Some simplifications are however possible, which will be shown to be numerically acceptable. It was already discussed that, in cases in which \(S\) has a beta PDF, V has a computable PDF, the BGP PDF. Apart from such particular cases, the independence of \(W\) and \(S\) can be used as a basis for evaluating the posterior mean of \(V\):  
\[\mathrm{V}^{\circ}=\mathrm{E}[\mathrm{V}|\mathbf{D}]=\mathrm{E}[\mathrm{W}| \mathbf{D}]\mathrm{E}[\mathrm{S}|\mathbf{D}]=\rho_{1}\theta_{1}\mathrm{E}[ \mathrm{S}|\mathbf{D}] \tag{43}\]  
where \(p_{1}\) and \(\theta_{1}\) are the posterior gamma parameters computed in the previous subsection. This shows that only the posterior mean of \(S\) is needed in order to evaluate the posterior mean, i.e., the BE, of \(\mathrm{V}^{\circ}\). This does not imply a direct estimation of the RI, which theoretically requires the integration of (30) or (32). However, at least as a first-order approximation, with \(\mathrm{R}=e^{-\mathrm{V}}\), the following approximation of the BE of R can be evaluated:  
\[S(z)=1-F(z|\alpha) \tag{35}\]0  
Finally, if V can be described by a gamma distribution, then R has a \"negative log-gamma\" distribution (summarized in Appendix D and recalled in the previous subsection) and its posterior mean is evaluable analytically. A Framework for the Numerical Applications of the Bayes Method of Inference for the Risk Index with Reference to Practical Applications  
Before illustrating (in the next sections) extensive numerical applications of the Bayes method aimed at evaluating its performances, a framework for such numerical applications is established here in order to better highlight the rationale behind the computations which will be shown afterwards. For the sake of brevity, only the case of a gamma prior PDF on the IRD parameter a will be illustrated here, leaving the case of a of a beta prior PDF on the IRD CCDF \(S(\alpha)\) to be illustrated directly in the course of the numerical applications. Section 5.1 here is devoted to the prior assessment analysis, and the following Section 5.2 is devoted to the posterior analysis, still in the case of a gamma prior PDF on the IRD parameter \(\alpha\). the assumption of a gamma prior PDF on the Poisson parameter \(\phi\) is always maintained instead. It is remarked that the two subsections both referred to practical applications, being based upon typical wind-speed data as those reported in the Appendix B.