ABSTRACT
The terms uncertainty quantification and uncertainty propagation have become so widely used as to almost have little meaning unless they are further explained. Here we focus primarily on two basic types of problems:
1. Modeling and inverse problems where one assumes that a precise mathematical model without modeling error is available. This is a standard assumption underlying a large segment of what is taught in many modern statistics courses with a frequentist philosophy. More precisely, a mathematical model is given by a dynamical system
dx
dt (t) = g(t,x(t), q) (1.1)
x(t0) = x0 (1.2)
with observation process
f (t; θ) = Cx(t; θ), (1.3) where θ = (q,x0). The mathematical model is an n-dimensional deterministic system and there is a corresponding “truth” parameter θ0 = (q0,x00) so that in the presence of no measurement error the data can be described exactly by the deterministic system at θ0. Thus, uncertainty is present entirely due to some statistical model of the form
Y j = f(tj ; θ0) + Ej , j = 1, . . . , N, (1.4)
where f (tj ; θ) = Cx(tj ; θ), j = 1, . . . , N , corresponds to the observed part of the solution of the mathematical model (1.1)–(1.2) at the jth covariate or observation time and Ej is some type of (possibly state dependent) measurement error. For example, we consider errors that
include those of the form Ej = f(tj ; θ0) γ ◦ E˜j where the operation γ◦
denotes component-wise exponentiation by γ followed by componentwise multiplication and γ ≥ 0.
