# Evidence Theory

This section discusses Dempster-Shafer evidence theory. In this approach, one does not assign a probability distribution to each uncertain input variable. Rather, one divides each uncertain input variable into one or more intervals. The input parameters are only known to occur within intervals: nothing more is assumed.

Each interval is defined by its upper and lower bounds, and a Basic Probability Assignment (BPA) associated with that interval. The BPA represents a probability of that uncertain variable being located within that interval.

The intervals and BPAs are used to construct uncertainty measures on the outputs called “belief” and “plausibility.” Belief represents the smallest possible probability that is consistent with the evidence, while plausibility represents the largest possible probability that is consistent with the evidence. For more information about the Dempster-Shafer theory of evidence, see [OH03] and [HO04].

Similar to the interval approaches, one may use global or local methods to determine plausbility and belief measures for the outputs.

*Usage Notes*

Note that to calculate the plausibility and belief cumulative distribution functions, one has to look at all combinations of intervals for the uncertain variables. Within each interval cell combination, the minimum and maximum value of the objective function determine the belief and plausibility, respectively. In terms of implementation, global methods use LHS sampling or global optimization to calculate the minimum and maximum values of the objective function within each interval cell, while local methods use gradient-based optimization methods to calculate these minima and maxima.

Finally, note that many non-deterministic keywords apply to
the evidence methods,
but one needs to be careful about the interpretation
and translate probabilistic measures to epistemic ones. For example,
if the user specifies distribution of type complementary, a
complementary plausibility and belief function will be generated
for the evidence methods (as
opposed to a complementary distribution function in the
`sampling`

case). If the user specifies a set of responses levels,
both the belief and plausibility will be calculated for each response
level. Likewise, if the user specifies a probability level, the
probability level will be interpreted both as a belief and
plausibility, and response levels corresponding to the belief and
plausibility levels will be calculated. Finally, if generalized
reliability levels are specified, either as inputs (
`gen_reliability_levels`

) or outputs ( `response_levels`

with `compute`

`gen_reliabilities`

), then these are directly converted to/from
probability levels and the same probability-based mappings described
above are performed.