Epistemic Methods

This chapter covers theoretical aspects of methods for propagating epistemic uncertainty.

Dempster-Shafer theory of evidence (DSTE)

In Dempster-Shafer theory, the event space is defined by a triple \((\mathcal{S},\mathbb{S},m)\) which defines \(\mathcal{S}\) the universal set, \(\mathbb{S}\) a countable collection of subsets of \(\mathcal{S}\), and a notional measure \(m\). \(\mathcal{S}\) and \(\mathbb{S}\) have a similar meaning to that in classical probability theory; the main difference is that \(\mathbb{S}\), also known as the focal elements, does not have to be a \(\sigma\)-algebra over \(\mathcal{S}\). The operator \(m\) is defined to be

(149)\[\begin{split}\begin{aligned} m(\mathcal{U}) = \left\{ \begin{array}{rr} > 0 & \mathrm{if} \ \mathcal{U} \in \mathbb{S}\\ 0 & \mathrm{if} \ \mathcal{U} \subset \mathcal{S} \ \mathrm{and} \ \mathcal{U} \notin \mathbb{S} \end{array} \right.\end{aligned}\end{split}\]
(150)\[\begin{aligned} \displaystyle\sum_{\mathcal{U} \in \mathbb{S}} m(\mathcal{U}) = 1\end{aligned}\]

where \(m(\mathcal{U})\) is known as the basic probability assignment (BPA) of the set \(\mathcal{U}\). In the DSTE framework, belief and plausibility are defined as:

(151)\[\mathrm{Bel}(\mathcal{E}) = \displaystyle\sum_{\{ \mathcal{U} \ | \ \mathcal{U} \subset \mathcal{E}, \ \mathcal{U} \in \mathbb{S}\}} m(\mathcal{U})\]
(152)\[\mathrm{Pl}(\mathcal{E}) = \displaystyle\sum_{\{ \mathcal{U} \ | \ \mathcal{U} \cap \mathcal{E} \neq \emptyset, \ \mathcal{U} \in \mathbb{S}\}} m(\mathcal{U})\]

The belief Bel(\(\mathcal{E}\)) is interpreted to be the minimum likelihood that is associated with the event \(\mathcal{E}\). Similarly, the plausibility Pl(\(\mathcal{E}\)) is the maximum amount of likelihood that could be associated with \(\mathcal{E}\). This particular structure allows us to handle unconventional inputs, such as conflicting pieces of evidence (e.g. dissenting expert opinions), that would be otherwise discarded in an interval analysis or probabilistic framework. The ability to make use of this information results in a commensurately more informed output.

The procedure to compute belief structures involves four major steps:

  1. Determine the set of \(d\)-dimensional hypercubes that have a nonzero evidential measure

  2. Compute the composite evidential measure (BPA) of each hypercube

  3. Propagate each hypercube through the model and obtain the response bounds within each hypercube

  4. Aggregate the minimum and maximum values of the response per hypercube with the BPAs to obtain cumulative belief and plausibility functions on the response (e.g. calculate a belief structure on the response).

The first step involves identifying combinations of focal elements defined on the inputs that define a hypercube. The second step involves defining an aggregate BPA for that hypercube, which is the product of the BPAs of the individual focal elements defining the hypercube. The third step involves finding the maximum and minimum values of the response value in each hypercube, and this part can be very computationally expensive. Finally, the results over all hypercubes are aggregated to form belief structures on the response.