Analogies and Theories: Formal Models of Reasoning by Itzhak Gilboa, Larry Samuelson, David Schmeidler

By Itzhak Gilboa, Larry Samuelson, David Schmeidler

The publication describes formal versions of reasoning which are aimed toward taking pictures the best way that fiscal brokers, and choice makers quite often take into consideration their surroundings and make predictions in keeping with their earlier adventure. the point of interest is on analogies (case-based reasoning) and common theories (rule-based reasoning), and at the interplay among them, in addition to among them and Bayesian reasoning. A unified method permits one to check the dynamics of inductive reasoning when it comes to the mode of reasoning that's used to generate predictions.

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With constant columns), that respects case equivalence, such that u = λv + β. Observe that, by the richness assumption, C is infinite, and therefore the matrix v has infinitely many columns. Moreover, the theorem does not restrict the cardinality of X, and thus v may also have infinitely many rows. Given any real matrix of order |X| × |C|, one can define for every M ∈ M a weak order on X through (2). It is easy to see that it will satisfy A1 and A2. If the matrix also respects case equivalence, A3 will also be satisfied.

One need not consciously assign conditional probabilities p (c|x) for every case c given every theory x, and one need not know probability calculus in order to generate predictions in accordance with the likelihood criterion. Rather, whenever 4 We do not assume that the cases that have been observed (M) constitute an exhaustive state space. Correspondingly, there is no requirement that the sum of conditional probabilities c∈M p(c|x) be the same for all x. 26 Inductive Inference: An Axiomatic Approach one satisfies our axioms, one may be ascribed conditional probabilities p (c|x) such that one’s predictions are in accordance with the resulting likelihood functions.

Cases, the elements of C, may also be called observations or facts. A memory M in M represents the predictor’s knowledge and will be referred to also as a database. 2 Axioms We will use the four axioms stated below. In their formalization let and ≈M denote the asymmetric and symmetric parts of M , as usual. is complete if x M y or y M x for all x, y ∈ X. A1 Order: For every M ∈ M, M M M is complete and transitive on X. A2 Combination: For every disjoint M, N ∈ M and every x, y ∈ X, if x (x M y) and x N y, then x M∪N y(x M∪N y).

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