BUILDING A NEW APPROACH TO DECISION-MAKING

 

A Bayesian agent can always say that a plan is better than, worse than, or equal to another plan. A Quasi-Bayesian agent may be in a different situation. Consider two plans, $a$₁ and $a$₂, and two distributions $p$₁ and $p$₂ in the credal set. Suppose plan $a$₁ has smaller expected loss than plan $a$₂ with respect to a probability distribution $p$₁, but $a$₂ has smaller expected loss with respect to another probability distribution $p$₂. In this case, $a$₁ and $a$₂ are not comparable by expected loss; both are admissible. What should be done?

Reactions to this question vary. Fertig and Breese, in their work with interval probabilities, simply report all admissible plans [Breese and Fertig1991, Fertig and Breese1990]. This leaves the actual actions unspecified. Levi argues that plans should not only be admissible, but also be optimal with respect to some distribution in the credal set. He calls such a plan E-admissible [Levi1980]. Since there may be several E-admissible plans, Levi suggests secondary guidelines that enforce ``security''. Others have suggested the agent should minimize the maximum possible value of expected loss, an approach common in Robust Bayesian Statistics under the name of $\Gamma$-minimax [Berger1985].

We suggest that Quasi-Bayesian strategies should specify the admissible decisions and allow the agent to monitor the robustness of such decisions. These are the two requirements on a plan. There should be no artificially enforced preference among admissible plans: any admissible plan provide useful guidance if an action must be chosen. Robustness should always be monitored; what use is a ``best'' plan if it is based on a skewed set of assumptions? As long as a plan provides a method for the detection of non-robust situations, the agent can pick the first admissible decision that admits convenient manipulation in the time available for decision-making. We call this the New Quasi-Bayesian strategy.

The strategy above contains important elements of decision-making as it must be exercised by finite, bounded agents. The agent is required to produce an admissible answer as quickly as possible, and have that as a default solution, as usually required in anytime planning. Further, the agent is required to detect the situations that require additional computation and refinement: those are the non-robust situations.

Compared to the Bayesian strategy, the New Quasi-Bayesian strategy has some remarkable differences. The Bayesian strategy will always be appropriate if there is total confidence on the precision of probability assessments. If that is not the case, the Bayesian strategy calls for a decision analysis of the value of further computation and/or introspection [Heckerman and Jimison1989, Horvitz1989, Matheson1968, Russell and Wefald1991]. Such meta-analysis requires probabilities over probabilities, which may be harder to elicit than a simple set of bounds on distributions.

So far we have specified the New Quasi-Bayesian strategy, but it is still unclear how we can use this strategy in any decision problem. In order to do so, we must be able to quickly generate actions and monitor robustness. Ideally, we must be able to do so faster than the usual Bayesian solution, which involves generating actions and either checking the sensitivity of such actions or checking the meta-analysis for those actions. In the remainder of the paper, we show that these goals are met for the planning to observe problem with Gaussian measurements. This is a classic Markov decision problem; although we describe the solution for univariate data, the ideas readily extend to multivariate data.