Probability distribution, entropy-sampling

A spectrum or chromatogram can be considered as a probability distribution. If the data are sampled at 1000 different points, then the intensity at each datapoint is a probability. For a flat spectrum, the intensity at each point in the spectrum equals 0.001, so die entropy is given by... 

Andricioaei and Straub have recently employed a similar acceptance probability where the trial step is sampled from a distribution function of a form proposed by Tsallis. In Tsallis statistics , the standard Gibbs entropy S = —k / dx p(x) In p(x) is modified to the form Sq = k J dx(l — Pq xf)j q — 1) which is equal to the Gibbs entropy formula in the limit that q =. The equilibrium probability distribution functions take the form... 

The required derivatives canbe obtained forthe FORM approach very efficiently as reported in Most and Knabe (2009). Based on this Taylor series approximation the failure probability can be calculated quite accurate for each specific sample of the stochastic parameters close to the mean value vector po- This calculation is performed here for all 10000 bootstrap samples. The resulting histogram of the reliability index is shown in Figure 6 using log-normally distributed

maximum entropy distributions. 

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