# 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  
The Bayes estimation of the IRD model implies to estimate the parameter \(\alpha\) in the expression of the IRD CDF:  
\[F(x|\alpha)=e^{-(\frac{\alpha}{x^{2}})} \tag{34}\]  
For this estimation, two methods for are considered in the paper in order to establish appropriate prior distributions:  
1. A gamma prior distribution on \(\alpha\), which is a conjugate distribution (see Appendix F and [26; 27]). 2. A negative exponential beta distribution (NEBD) on \(\alpha\), (see Appendix D) which is not a conjugate distribution, but has the advantage of implying a beta distribution on \(S\). The choice of a gamma prior distribution is already reported in the literature and will not be further discussed here but will directly be used in the course of the numerical applications. Before discussing such method, it is recalled that, in the Bayesian setting, also the CDF and other parameter (e.g., moments and quantiles), being functions of \(\alpha\), are themselves RV, described by appropriate distributions. The same also holds for the CCDF:  
\[S(z)=1-F(z|\alpha) \tag{35}\]for which estimation, as outlined above, can directly allow for the estimation of the RI. The assessment of the prior information in terms of the parameter \(\alpha\), a quantile, the CDF, or any other parameter is a matter of convenience, depending on which information is available. Generally, it is advisable to base the estimation upon a parameter which possesses a clear physical meaning, which is not the case for \(\alpha\), which appears to be difficult to interpret. The information on a given quantile should be more easily available and also interpretable, so it is chosen here first how to use this kind of information. In particular, for sake of illustration, the median is chosen as the input RV for assessing the estimation, with no loss of generality, since any quantile \(X_{p}\) is proportional to the median through the following relations, which are simply deducible (see Appendix B):  
\[X_{p}=m\bigg{[}\frac{\psi}{K_{p}}\bigg{]}^{\frac{1}{2}};\ \psi=\log(2)=0.6931;\ \text{K}_{p}=-\log(p)>0 \tag{36}\]  
Therefore, an assessment of the prior information in terms of any quantile is essentially the same as an assessment of the prior information in terms of the median, which in turn implies a prior information in terms of \(\alpha\), since  
\[\alpha=\psi(m)^{2} \tag{37}\]  
The above relations also imply prior information in terms of IRD CCDF, expressible in terms of the median m as well as in terms of the parameter \(\alpha\):  
\[S=S(z|m)=1-F(z|m)=1-e^{-\psi(\frac{m}{2})^{2}}=1-F(z|\alpha)=1-e^{-(\frac{ \alpha}{2})} \tag{38}\]  
It is remarked that the function \(S(z)\) reported here, being relevant to a fixed threshold value, \(z\), of EWS has to be regarded as a constant, i.e., a parameter, in view of the estimation process. In the above Equation (34), instead, the CDF \(F(x)\) reported there denotes a function of the generic EWS value as a variable argument. Far from what is above discussed, the peculiar method suggested in this paper is that \(S\), a RV with support in (0,1), is assumed as a beta RV, which implies that the parameter \(\alpha\) has an NEBD PDF, as can be deduced from Appendix D, since  
\[\alpha=[-\log(1-S)]z^{2} \tag{39}\]  
In practice, assuming an NEBD prior PDF for \(\alpha\) implies that the negative exponential function of \(\alpha\) in (38) has a beta distribution. This assumption has the advantages already discussed in the previous section and may be put in practice, starting from prior information in terms of the median as follows:  
Let a prior guess on the median be given in terms of a \"credible interval\" (\(m_{1}\), \(m_{2}\)), i.e., a Bayesian (prior) confidence interval on the median, i.e., an \"interval\" (\(m_{1}\), \(m_{2}\)) such that, for a given probability value \(p\),  
\[P(m_{1}<m<m_{2})=p \tag{40}\]  
1.