Parameters

Define a scalar-valued quantity of interest (qoi) as \(f = f\left(\boldsymbol{x} \right)\) where \(\boldsymbol{x} = (x^{(1)}, \ldots, x^{(d)} )\) is a \(d\)-dimensional vector of mutually independent variables. Thus the function \(f\) is a map from \(\mathbb{R}^{d} \rightarrow \mathbb{R}\). Here, each parameter \(x^{(i)}\) belongs to a domain \(\mathcal{X}^{(i)} \in \mathbb{R}\), that can be an infinite interval \(\left(-\infty, \infty \right)\), a semi-infinite interval \(\left(-\infty, b \right]\) or \(\left[a, \infty \right)\), or a closed (finite) interval \(\left[a, b\right]\). This interval represents the support of each parameter. We generally consider the input domain to be a non-compact hypercube decomposed as a Cartesian product of the form \(\mathcal{X}=\mathcal{X}^{(1)} \times \dots \times \mathcal{X}^{(d)}\).

Along each interval, consider a positive weight function \(\rho_{i} \left( x^{(i)} \right)>0\) over the domain \(\mathcal{X}^{(i)}\) such that

\begin{equation} \int_{\mathcal{X}^{i}}\left(x^{(i)}\right)^{k}\rho_{i}\left(x^{\left(i\right)}\right)dx^{\left(i\right)}<\infty, \; \; \; \text{where} \; \; \; \int_{\mathcal{X}^{i}}\rho_{i}\left(x^{\left(i\right)}\right)dx^{\left(i\right)}=1, \label{eq:pdf}{\tag{1}} \end{equation}

for \(k=1, 2, \ldots\), and where \(i=1, \ldots, d\). The second expression in \(\eqref{eq:pdf}\) naturally leads to the interpretation that \(\rho_{i} \left( x^{(i)} \right)\) is a probability density function over the domain \(\mathcal{X}^{(i)}\). In making the assumption that \(\boldsymbol{x}\) is a vector of independent variables, the joint density \(\boldsymbol{\rho}\) of all the probability distributions associated with \(\boldsymbol{x}\) is given by \begin{equation} \boldsymbol{\rho}(\boldsymbol{x})=\prod_{i=1}^d \rho_i\left(x^{(i)}\right), \label{eq:marginals}{\tag{2}} \end{equation} defined on \(\mathbb{R}^d\); should the variables be correlated then \(\eqref{eq:marginals}\) no longer holds.

Defining a parameter in equadratures can be found in the tutorial here.