# Efficient Algorithms for Minimax Decisions Under Tree-Structured Incompleteness

## Abstract

When decisions must be based on incomplete (coarsened) observations and the coarsening mechanism is unknown, a minimax approach offers the best guarantees on the decision maker’s expected loss. Recent work has derived mathematical conditions characterizing minimax optimal decisions, but also found that computing such decisions is a difficult problem in general. This problem is equivalent to that of maximizing a certain conditional entropy expression. In this work, we present a highly efficient algorithm for the case where the coarsening mechanism can be represented by a tree, whose vertices are outcomes and whose edges are coarse observations.

## Keywords

Coarse data Incomplete observations Minimax decision making Maximum entropy## Notes

### Acknowledgments

This research was supported by Vici grant 639.073.04 and Veni grant 639.021.439 from the Netherlands Organization for Scientific Research (NWO).

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