Foundations: Indifference, Independence & MaxEnt
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Through completing an under specified probability model, Maximum Entropy (MaxEnt) supports non-monotonic inferences. Some major aspects of how this is done by MaxEnt can be understood from the background of two principles of rational decision: the concept of Indifference and the concept of Independence. In a formal specification MaxEnt can be viewed as (conservative) extension of these principles; so these principles shed light on the “magical” decisions of MaxEnt. But the other direction is true as well: Since MaxEnt is a “correct” representation of the set of models (Concentration Theorem), it elucidates these two principles (e.g. it can be shown, that the knowledge of independences can be of very different information-theoretic value). These principles and their calculi are not just arbitrary ideas: When extended to work with qualitative constraints which are modelled by probability intervals, each calculus can be successfully applied to V.Lifschitz’s Benchmarks of Non-Monotonic Reasoning and is able to infer some instances of them ([Lifschitz88]). Since MaxEnt is strictly stronger than the combination of the two principles, it yields a powerful tool for decisions in situations of incomplete knowledge. To give an example, a well-known problem of statistical inference (Simpson’s Paradox) will serve as an illustration throughout the paper.
KeywordsMaximum Entropy Undirected Graph Linear Constraint Propositional Logic Elementary Event
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