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Abstract:
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The optimal design of a sequence of experiments needs to account for feedback between experiments and the future consequences of each design decision. Common practices such as batch design (no feedback) and myopic design (not forward-looking) are thus suboptimal. We formulate the optimal sequential design problem using dynamic programming (DP), and employ a Bayesian framework that seeks to maximize expected information gain for continuous parameters. The DP problem is solved using approximate value iteration, and employs transport maps to represent non-Gaussian posteriors and to enable fast approximate Bayesian inference. This is achieved by constructing a Knothe-Rosenblatt map that couples a standard Gaussian to the joint distribution of designs, observations, and parameters. The maps are constructed using sample trajectories from exploration and exploitation, favoring accuracy over state regions that are more likely to be visited. Since the measure induced by the optimal policy is unknown, we devise a method to iteratively update it as better policies become available. The overall method is demonstrated on a sensor placement problem for source inversion.
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