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

By Itzhak Gilboa, Larry Samuelson, David Schmeidler

The e-book describes formal types of reasoning which are geared toward shooting the way in which that monetary brokers, and selection makers more often than not take into consideration their setting and make predictions according to their prior event. the focal point 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 review the dynamics of inductive reasoning by way of the mode of reasoning that's used to generate predictions.

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Additional info for Analogies and Theories: Formal Models of Reasoning

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In Gilboa and Schmeidler (2001) we call this process “second-order induction” and show that the additive formula cannot capture such a process. Combinations of inductive and deductive reasoning Another important class of examples in which the combination axiom is not very 8 Observe that if one were to use the maximum likelihood principle, one would have to specify a likelihood function. This exercise would highlight the fact that “the coin is biased” is not a fully specified theory. However, this does not imply that only theories that are given as conditional distributions are sufficiently specified to satisfy the combination axiom.

If some of the coefficients above are not zero, the vectors {ˆvxw , vˆ yw , vˆ zw } are linearly dependent, and this contradicts the diversity axiom. For instance, if vˆ xw is a non-negative linear combination of vˆ yw and vˆ zw , for no I will it be the case that y I z I w I x. We therefore obtain λ({x, w}, y) = λ({x, w}, z) for every y, z ∈ A\{x}. Hence for every x ∈ A there exists a unique λ({x, w}) > 0 such that, for every distinct x, y ∈ A vxy = λ({x, w})ˆvxw + λ({y, w})ˆvwy . Defining vxw = λ({x, w})ˆvxw completes the proof of the claim.

30 Inductive Inference: An Axiomatic Approach In conclusion, there are classes of counterexamples to our axioms that result from under-specification of cases, of eventualities, or of memories. There are others that are more fundamental. Among these, two seem to deserve special attention. First, there are situations where second-order induction is involved, and the similarity function itself is learned. Indeed, our model deals with accumulated evidence but does not capture the emergence of new insights.

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