This work (in two parts) will present a novel
predictive modeling methodology aimed at obtaining “best-estimate results with
reduced uncertainties” for the first four moments (mean values, covariance,
skewness and kurtosis) of the optimally predicted distribution of model results
and calibrated model parameters, by combining fourth-order experimental and
computational information, including fourth (and higher) order sensitivities of
computed model responses to model
parameters. Underlying the construction of this fourth-order predictive
modeling methodology is the “maximum entropy principle” which is
initially used to obtain a novel closed-form expression of the (moments-constrained) fourth-order Maximum
Entropy (MaxEnt) probability distribution
constructed from the first four moments (means, covariances, skewness,
kurtosis), which are assumed to be known, of an otherwise unknown distribution
of a high-dimensional multivariate uncertain quantity of interest. This
fourth-order MaxEnt distribution provides optimal compatibility of the available information while simultaneously
ensuring minimal spurious information content, yielding an estimate of a
probability density with the highest uncertainty among all densities satisfying
the known moment constraints. Since this novel generic fourth-order MaxEnt
distribution is of interest in its own right for applications in addition to
predictive modeling, its construction is presented separately, in this first
part of a two-part work. The fourth-order predictive modeling methodology that
will be constructed by particularizing this generic fourth-order MaxEnt
distribution will be presented in the accompanying work (Part-2).
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