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

By Itzhak Gilboa, Larry Samuelson, David Schmeidler

ISBN-10: 0198738021

ISBN-13: 9780198738022

The booklet describes formal versions of reasoning which are aimed toward taking pictures the best way that monetary brokers, and determination makers ordinarily take into consideration their surroundings and make predictions in keeping with their prior event. the focal point is on analogies (case-based reasoning) and basic theories (rule-based reasoning), and at the interplay among them, in addition to among them and Bayesian reasoning. A unified technique permits one to check the dynamics of inductive reasoning by way of the mode of reasoning that's used to generate predictions.

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Extra info for Analogies and Theories: Formal Models of Reasoning

Example text

Section 2 presents the formal model and the main results. Section 3 discusses the relationship to kernel methods and to maximum likelihood rankings. Section 4 contains a critical discussion of the axioms, attempting to outline their scope of application. Finally, Section 5 briefly discusses alternative interpretations of the model, and, in particular, relates it to case-based decision theory. Proofs are relegated to the appendix. 1 Framework The primitives of our model consist of two non-empty sets X and C.

For I ∈ JT , if {t | I(t) > 0} is finite, define I ⊂ X × X as follows. Choose M ∈ M such that M ⊂ ∪t∈T t (recall that t ⊂ C is an equivalence class of cases) and I(t) = #(M ∩ t) for all t ∈ T, and define I = M . Such a set M exists since, by the richness assumption, |t| ≥ ℵ0 for all t ∈ T. For this reason, such a set M is not unique. However, if both M1 , M2 ∈ M satisfy these properties, then M1 ∼ M2 and M1 = M2 . Hence I is well-defined. Moreover, this definition implies the following property, which will prove useful in the sequel: if I ∈ JT and I ∈ JT where T ⊂ T , I (t) = I(t) for t ∈ T and I (t) = 0 for t ∈ T \T, then I = I .

Thus, one may argue, the combination axiom does not seem to be very compelling. Obviously, this example assumes that all of Lucifer’s deaths are equivalent. While this may be a reasonable assumption of a naive observer, the cat connoisseur will be careful enough to distinguish “first death” from “second death”, and so forth. Thus, this example suggests that one has to be careful in the definition of a “case” (and of case equivalence) before applying the combination axiom. Mis-specified theories Suppose that one wishes to determine whether a coin is biased.

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