Inference and Learning for Active Sensing, Experimental Design and Control

In this paper we argue that maximum expected utility is a suitable framework for modeling a broad range of decision problems arising in pattern recognition and related fields. Examples include, among others, gaze planning and other active vision problems, active learning, sensor and actuator placement and coordination, intelligent humancomputer interfaces, and optimal control. Following this remark, we present a common inference and learning framework for attacking these problems. We demonstrate this approach on three examples: (i) active sensing with nonlinear, non-Gaussian, continuous models, (ii) optimal experimental design to discriminate among competing scientific models, and (iii) nonlinear optimal control. 1 The Principle of Maximum Expected Utility Broadly speaking, utility reflects the preferences of an agent. That is, if outcome o1 is preferred to o2 (i.e. o1 o2), we say that o1 has higher utility than o2. More formally, let o1 o2 denote weak preference, o1 o2 denote strong preference and o1 ∼ o2 denote indifference. Define a lottery to be a random set of outcomes with corresponding probabilities: l = [(o1, p1), (o2, p2), . . . , (ok, pk)], where the probabilities satisfy pi ≥ 0 and ∑k i pi = 1 as usual. Now consider the following axioms: 1. Completeness: ∀o1, o2, we have o1 o2, o2 o1 or o1 ∼ o2. 2. Transitivity: If o1 o2 and o2 o3, then o1 o3. 3. Substitutability: If o1 ∼ o2, then for all sequences of outcomes o3, . . . , ok and sets of probabilities p, p3, . . . , pk for which p + ∑k i=3 pi = 1, we have [(o1, p), (o3, p3), . . . , (ok, pk)] ∼ [(o2, p), (o3, p), . . . , (ok, pk)]. 4. Decomposability: Let Pl(oi) be the probability that outcome oi is selected by lottery l. If for all oi: Pl1(oi) = Pl2(oi), then l1 ∼ l2. 5. Monotonicity: If o1 o2 and p > q, then [(o1, p), (o2, 1−p)] [(o1, q), (o2, 1− q)]. 6. Continuity: If o1 o2 and o2 o3 then ∃p ∈ [0, 1] such that o2 ∼ [(o1, p), (o3, 1− p)]. Using these axioms, von Neumann and Morgenstern [16] proved the following fundamental result showing the existence of utility: