## Risk Management Collaboratory - Imprecise Probability

### Research Objectives

- Develop extended probabilistic graphical models based on theories of imprecise probability. In particular we will develop influence diagrams based on more qualitative probabilities and utilities, and on interval-valued probabilities and utilities.
- Develop deterministic and probabilistic graphical models based on multi-attribute utility functions to support multi-criteria decision making under certainty and uncertainty.
- Devise and implement efficient computational algorithms for the extended graphical models, based on variable elimination and search-based approaches.

### Activities

We will start by refining a system of order of magnitude probability and utilities [Wilson, 95] for reasoning with imprecise information. These kinds of probabilities and utilities can be elicited from the decision maker in an ad-hoc or more structured manner and incorporated into a probabilistic decision model such as an influence diagram. Therefore, the new model called an order of magnitude influence diagram can be used to support decision making under uncertainty in situations when the probabilistic information as well as the decision maker's preferences are rather difficult to quantify exactly.

In many situations, a decision maker has more than one objective, and mapping several objectives to a single utility scale can be problematic, since the decision maker may be unwilling or unable to provide precise tradeoffs between objectives. We therefore consider multi-objective influence diagrams models where the utility values are represented by multi-dimensional vectors. We also define a simple formalism for imprecise tradeoffs; this allows the decision maker, during the elicitation stage, to specify a preference for one multi-objective utility vector over another, and uses such inputs to infer other preferences.

For decision problems under certainty and multiple objectives, we extend the framework of multi-objective constraint optimization to include additional imprecise tradeoffs between the objectives. Furthermore, we show that our tradeoffs approach which is based on a preference inference technique can also be given an alternative semantics based on the well known Multi-Attribute Utility Theory (MAUT).

Influence diagrams models for sequential decision making under uncertainty assume that the decision maker remembers all past decisions when he or she is supposed to make the current decision. This may incurr a huge computational overhead for large problems. Therefore, we are investigating a class of limited memory influence diagrams using our proposed formalisms for representing and reasoning with imprecise probabilistic and deterministic information.