Non-informative prior pdf

In this study, a non-informative prior PDF for the uncertain parameters is used and the prior PDF is absorbed into the normalizing constant. Then, the updated PDF is proportional to the likelihood function ... 

In what follows the procedure is shown only for the use of a non-informative prior pdf. Details on the use of other prior pdfs are found in [25, 33]. 

In case of a Poisson likelihood function the non-informative prior pdf is proportional to X [25], i.e. 

Example 9.16 Calculation of the failure for refrigerating units using Bayes rule with a non-informative prior pdf... 

A lifetime observation of centrifugal pumps in a process plant resulted in k = 36 leaks from casings to the outside during an accumulated time of observation of t = Tc = 934,984 h. What are the expected value, the median, and the 5th and 95th percentiles, if a Bayesian evaluation with a non-informative prior pdf is made The lifetimes are supposed to be exponentially distributed. 

In the case where a non-informative prior is used, the first term can be simply neglected. Furthermore, the updated PDF of the parameter vector 0 can be well approximated by a Gaussian distribution G(0 0, Ti(0 ) ) with mean 0 and covariance matrix H(0 ), where H 0 ) denotes the Hessian matrix of the objective function J calculated at = ... 

To simplify the problem, a non-informative prior distribution for U was used so the updated PDF for V was ... 

In the Bayesian procedure the failure rate X is considered as a random variable, which is described by a probability distribution. The calculation of sums and products of random variables are well-defined operations. Thus the propagation of uncertainties through the fault tree represents no difficulty. For this reason the formula of Bayes is applied as well to cases where no appropriate prior information is available, no matter how many failures were observed. In such a case the non-informative pdf is used in order to arrive at a probability distribution for the result. 

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