Representation of Random Variables

As previously observed, random variables are a comfortable tool to model variability, whenever statistical data is available. Nevertheless, the complete knowledge of their distribution is rarely available, since only samples may be obtained. In some situations, even the variable itself cannot be observed – it is a hidden variable – and only its effects may be observed: variability of the response of the system is observed, but the cause of this variability remains unidentified. To use random variables in models, it is necessary to have information such as its distribution (for instance, its cumulative distribution or its density) and some of its statistical properties (for instance, mean, variance, mode). Classical approaches start by assumptions on the distribution – for instance, assuming that the variable under consideration is Gaussian or a particular transformation of a Gaussian variable. In the classical approaches, severe errors of model are difficulty to correct, even in the Bayesian approaches, since the conjugate distributions are predetermined, so that the choice of the prior distribution seriously constrains the result for the posterior distribution. Statistical Learning approaches are more flexible but request a large amount of data and face difficulties when the cause of the heterogeneity is unobserved. Uncertainty Quantification (UQ) proposes an alternative approach tending to introduce both more flexibility and economic use of data: on the one hand, the connection of the basic variable chosen for the representation (which may be interpreted as a “prior”) and the variable to be represented (which may be interpreted as a “posterior”) is relaxed, so that severe errors may be corrected. In addition, variability generated by hidden variables may be represented by explicit ones. On the other hand, reasonable quantities of data are enough to the determination of the representations.