# 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
### A Premise. Bayes Estimation of the Frequency of Exceedances
The likelihood function of the sample, i.e., of data D, is the probability that \(M\) exceedances are observed in time interval (0,\(t\)), computed by mean of the above Poisson PDF,  
\[\mathrm{L}\big{(}\mathbf{D}\big{|}\varphi\big{)}=P[N(t)=M]=e^{-\phi t}\frac{( \phi t)^{M}}{M!} \tag{24}\]  
It is then straightforward to deduce that the product of the gamma prior PDF of (23) and the likelihood function of (24) yields a posterior PDF of \(\phi\), \(q(\phi|\mathbf{D})\), which is still a gamma PDF, i.e., a _Gampdf_ (\(\phi;p_{0}\),\(\theta_{0}\)) with updated parameters [26; 27]:  
\[\rho_{1}=\rho_{0}+M;\ \theta_{1}=\frac{\theta_{0}}{1+\theta_{0}} \tag{25}\]Therefore, the BE of \(\phi\) being expressed, the mean of its posterior PDF, is computed as follows (see also Appendix C for the mean of a gamma PDF):  
\[\varphi^{\circ}=\mathbb{E}\big{(}\phi\big{|}\mathbf{D}\big{)}=\
u_{1}\delta_{1} =\frac{\delta_{0}(\
u_{0}+n)}{1+U\delta_{0}} \tag{26}\]  
Of course, the same arguments apply to the Expected number of exceedances (ENE) RV, W = \(\phi\)t. It is easily proved [28] that, if \(\phi\) is a gamma RV with parameters \(\
u\) and \(\delta\), then the RV W = \(\phi\)t is still a gamma RV with parameters \(p^{\prime}\) and \(\theta^{\prime}\) simply related to \(p\) and \(\theta\) as follows: \(p^{\prime}\) = \(p\); \(\theta^{\prime}\) = \(\theta\)t.  
It is also straightforward to verify that the gamma conjugate PDF for \(\phi\) also implies a gamma conjugate PDF for W. Thus, in the paper, a gamma prior PDF (\(\phi\); \(p_{0}\), \(\theta_{0}\) _t_) for W is assumed with parameters deduced from the gamma prior PDF of \(\phi\). As above said, the posterior PDF of W is still a gamma prior PDF \(Gampdf(\phi;p_{1},\theta_{1})\),  
\[R=R(t,z)=1-e^{-WS} \tag{18}\]0  
and the BE of \(W\) is as follows:  
\[R=R(t,z)=1-e^{-WS} \tag{18}\]1  
A final remark seems opportune before concluding this subsection devoted to the estimation of \(\phi\). The assessment of the prior PDF on \(S\) is dealt with in the following subsections. Once the posterior PDF of \(S\), \(q_{S}(s)\), is also obtained, it can be used together with the posterior PDF of \(W\), \(q_{W}(w)\), for determining the joint posterior PDF of \(W\) and \(S\), which--assuming the statistical independence of \(W\) and \(S\)--will be given by the following:  
\[R=R(t,z)=1-e^{-WS} \tag{18}\]2  
Then, an alternative (but equivalent) formulation of the BE of \(R\), instead of (22), can be made in terms of joint posterior PDF of \(W\) and \(S\) as follows:  
\[R=R(t,z)=1-e^{-WS} \tag{18}\]3  
It is remarked that, in some cases, the above double integration can be avoided as well as the explicit evaluation of the single integration of (22). First, let us consider the prior PDF assessment. An opportune method which will be pursued in the following for a feasible estimation of the RI is the one assessing the prior PDF on the CCDF \(S\) of (18) in order for--at least in principle--easy evaluation of the implied prior PDF of the product, \(V=WS\), and consequently of the RI:  
\[\left\{ \begin{array}{l}R=1-e^{-V}\\ V=WS\end{array} \right. \tag{31}\]  
Therefore, an alternative way to evaluate the BE of \(R\) may be obtained by means of a single integrations, as follows: if the PDF of \(V\) is computed, then the BE of \(R\) can be computed in terms of such a posterior PDF of \(V\), \(q(v|\mathbf{D})\), with \(R=e^{-V}\):  
\[r^{\circ}=1-\mathbb{E}\Big{[}e^{-V}\big{|}\mathbf{D}\Big{]}=1-\int\limits_{0} ^{\infty}e^{-v}q(v|\mathbf{D})dv \tag{32}\]It is noticed that the above formulation establishes the BE of the RI in terms of a \"Laplace Transform\" of the posterior PDF of \(V\). A useful analytical tool, given that a gamma prior PDF is assumed for \(W\), is the assessment of the prior PDF for the probability \(S(z)=1-F(z)\) by mean of a model that possesses hopefully the two following properties:  
1. First, the obvious property that such a model is very manageable analytically and rather flexible for a RV with support in the interval (0,1). 2.