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# INTRODUCTION

Agents choose a plan of action by comparing its possible outcomes against the outcomes of other plans. Bayesian theory suggests that the basis for such comparisons is expected loss with a single probability distribution. Quasi-Bayesian theory, as axiomatized by Giron and Rios [Giron and Rios1980], also relies on expected loss, but uses a convex set of probability distributions to represent the agent's beliefs. Many scholars agree that assuming an agent uses a single probability distribution is too restrictive [Breese and Fertig1991, Levi1980, Shafer1987]. But there has been little agreement on how to make decisions with several distributions; many seem to think that theories with several distributions will always lead to intractable decision making problems.

Recently, great attention has been given to Robust Bayesian Statistics, which uses Quasi-Bayesian sets of distributions to represent imprecision of subjective probability assessments [Berger1985, Walley1991]. A robust decision is one that can be safely taken despite the imprecision in the probability assessments; a non-robust decision is one that may produce wildly different results depending on the adopted distribution.

In this paper, we explore a Quasi-Bayesian approach to plan generation. Generating a plan means enumerating the actions to be taken and providing information about the robustness of the actions. Our approach puts less emphasis on the search for unique ``best'' decisions than the usual Bayesian approach. Essentially, the agent is required to choose admissible decisions and to monitor and report robustness of these decisions. We clarify the terms involved in this requirement in sections 2 and 3.

The central point of this work can be expressed by a short fable. Suppose two archers try to hit a target (Figure 1).

Figure: The Archers Fable

The first archer, a Bayesian, considers that hitting the center of the target is the only satisfactory result and orders a new, expensive bow. But the judge only detects if an archer hits the hatched region. If both archers hit the hatched region, the judge considers them tied and calls other procedures to solve the dispute. That does not prevent the Bayesian archer from trying to hit the center. The second archer, a New Quasi-Bayesian archer, tries simply to reach the hatched circle with a cheap bow. The New Quasi-Bayesian strategy seems wiser given the lack of precision of the target.

Our main contribution is to show that the Archers Fable can be formalized for the planning to observe problem. The analogy here is that a point in the target corresponds to a distribution: the Bayesian agent has one, the Quasi-Bayesian agent has many. The Bayesian seeks an answer to the question, how to create an optimal sequence of actions? Such a question is very demading computationally. The Quasi-Bayesian is attentive to the limitations of real probability assessments and seeks an answer to the question, how to create a sequence of admissible actions and quantify the robustness of such actions? The surprising result is that we can answer the latter question without examining the full solution for the former question. To illustrate the details of our solution, we apply it to the planning to observe problem for material classification with a robotic probe.

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