# 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 \(\alpha\): Prior Analysis
As for the choice of the prior PDF \(p(\phi\), \(\alpha)\), i.e., the gamma PDF (on both \(\phi\) and \(\alpha\)) and/or beta PDF on \(S(\phi\), \(\alpha)\), their parameters are chosen in order to obtain the expected values of the cases A, B, and C above; moreover, in each of the three cases, different values of the prior coefficient of variation (CV)--i.e., the ratio between standard deviation and mean values of \(\phi\) and a--are chosen in order to show different degrees of prior uncertainty (a larger prior CV value implies a higher degree of prior uncertainty about the random parameter values). Therefore, denoting for a given RV Q, its CV is as follows in terms of its mean value E[Q] and standard deviation STD[Q]:  
\[\text{CV}[\text{Q}]=\frac{\text{STD}[\text{Q}]}{\text{E}[\text{Q}]} \tag{47}\]  
The following values of CV for both \(\phi\) and \(\alpha\) have been chosen: 0.05 and 0.10, giving rise to four different subcases of the three cases A, B, and C above, each labeled with (i,j) indexes, \"i\" standing for 1 or 2 according to \(\text{CV}[\phi]\) = 0.05 or \(\text{CV}[\phi]\) = 0.10 and \"j\" having the same meaning for \(\text{CV}[\alpha]\). In detail, \(12=3\times 4\) sets of combinations of prior values (corresponding to different simulation cases) among the many more performed are shown in the following sections with respect to the mean value and CV of the input RV(\(\phi\), \(\alpha\)), as listed with more details in the next section, while here, they are only exemplified in Table 2:  
Details on the results of numerous simulations performed for each one of the above cases will be shown in the following sections, while here, some highlights are reported which give some insight on the process of the prior and posterior PDF assessments and on the BE of the RI. For the sake of exemplification, case C11 is examined. Therefore, it is assumed that, concerning the prior PDF assessment,  
- \(\alpha\) has a prior gamma PDF with E[\(\alpha\)] = 90, CV[\(\alpha\)] = 0.05;  
- \(\phi\) has a prior gamma PDF with E[\(\phi\)] = 10, CV[\(\phi\)] = 0.05. Let us examine first the prior gamma PDF for \(\alpha\): as already reported (see also Appendices C and F), the mean and variance of such prior PDF with parameters \(\
u_{0}\) and \(\delta_{0}\) are  
\[\text{E}[\alpha]=\
u_{0}\delta_{0};\text{ VAR}[\alpha]=\
u_{0}\delta_{0}^{2} \tag{48}\]  
which imply  
\[\text{STD}[\alpha]=\
u_{0}^{1/2}\delta_{0}\rightarrow\text{ CV}[\alpha]=\frac{\text{STD}[\alpha]}{\text{E}[\alpha]}=\frac{1}{\
u_{0}^{1/2}} \tag{49}\]  
Inverting the relations between the desired values of mean (90) and CV (0.05) of \(\alpha\) and the above prior parameters (\(\
u_{0}\) and \(\delta_{0}\)), the following values are obtained for \(\
u_{0}\) and \(\delta_{0}\): \(\
u_{0}\) = 400; \(\delta_{0}\) = 0.2250. The corresponding prior gamma PDF with such parameters (\(\
u_{0}\) and \(\delta_{0}\)), i.e., with mean E[\(\alpha\)] = 90 and CV: CV[\(\alpha\)] = 0.05, is depicted in Figure 3. Due to the high value of the shape parameter n\({}_{0}\) = 400, such a PDF closely resembles a Gaussian one [28]. Analogous computations for RV \(\phi\)--having a prior gamma PDF with E[f] = 10 and CV[\(\phi\)] = 0.05--show that its prior gamma parameters are \(p_{0}\) = 400 and \(\theta_{0}\) = 0.02250 (the shape parameter \(p_{0}\) = 400 is equal to the one, \(\
u_{0}\), of the PDF of \(\alpha\), since it depends only upon the CV, which is the same for both RV). With \(t\) = 1 year, also \(W\) = \(\phi\)t has the same prior gamma PDF. Such a PDF has exactly the same shape as the one of \(\alpha\) in Figure 4 but centered on E[\(W\)] = 10.