An Introduction to Differentiable Manifolds and Riemannian by W. Boothby PDF

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Multi-state static models such as Bayesian Belief Networks (BBN) (cf. 17), insofar as they link the undesired event e to multi-state indicator variables ein and esy representing the state of initiator events or conditional processes: e ¼ G ðein ; esy ; dÞ ð1:22Þ Accordingly, the computation of the risk measure f e involves discrete probability distributions and conditional dependence structures generalising Bernoulli distributions and common modes. ). Closed-form expressions still result in general, albeit of large dimension because of the multi-state features.

Some authors refer to it as the epistemic uncertainty or simply uncertainty about the risk level. Technicallyspeaking, it will be called ‘level-2’ uncertainty as it will materialise in uncertain parameters affecting the level-1 random variables, not directly tied to the physical states. Conversely, it may not be considered legitimate or practical to work within a probabilistic approach for step (i). This is either because samples may not be significant enough, or because of a lingering epistemological controversy about the quantification of return frequencies for very rare catastrophic APPLICATIONS AND PRACTICES OF MODELLING, RISK AND UNCERTAINTY 7 events.

According to the pdf of X (Zj)j¼1. N: Sample (of size N) of random outputs of interest generated by an uncertainty propagation algorithm (typically Monte-Carlo Sampling or alternative designs of experiment) h X, hU: Vectors of parameters (of dimension np and nu respectively) of the measure of uncertainty of X or U: in the probabilistic setting, this comprises the parameters of the joint pdf. In simplified notations yX yU kn h X , h Xun: Vectors representing the known (kn) or unknown (un) components of vector h X in an inverse probabilistic or model calibration approach p(h j f): Joint density of random vector Y, modelling epistemic uncertainty in the parameters h X, h U describing the distribution of (aleatory) uncertainty in X, U.

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An Introduction to Differentiable Manifolds and Riemannian Geom. by W. Boothby

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