By Itzhak Gilboa, Larry Samuelson, David Schmeidler
The booklet describes formal versions of reasoning which are geared toward taking pictures the way in which that financial brokers, and choice makers generally take into consideration their setting and make predictions in line 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 process 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
Assume further that this database does contain many choices between other pairs of restaurants, and it turns out that John and Mary consistently choose different restaurants. When combining the two databases, it makes sense to predict that John would choose y over x. This is an instance in which the similarity function is learned from cases. Linear aggregation of cases by fixed weights embodies learning by a similarity function. But it does not describe how this function itself is learned. In Gilboa and Schmeidler (2001) we call this process “second-order induction” and show that the additive formula cannot capture such a process.
Proof of Theorem 2: We present the proof for the case |X| ≥ 4. The proofs for the cases |X| = 2 and |X| = 3 will be described as by-products along the way. Part 1: (i) implies (ii). 1 For every I ∈ ZT + and every k ∈ N, I= kI . Proof: Follows from consecutive application of the combination axiom. In view of this claim, we extend the definition of I to functions I whose T values are non-negative rationals. Given I ∈ QT + , let k ∈ N be such that kI ∈ Z+ and define I = . I is well-defined in view of Claim 1.
If one were to specify the theories more fully, the combination axiom would hold. 8 Theories about patterns A related class of examples deal with concepts that describe, or are defined by patterns, sequences, or sets of cases. Assume that a single case consists of 100 tosses of a coin. A complex sequence of 100 tosses may lend support to the hypothesis that the coin generates random sequences. But many repetitions of the very same sequence would undermine this hypothesis. Observe that “the coin generates random sequences” is a statement about sequences of cases.
Analogies and Theories: Formal Models of Reasoning by Itzhak Gilboa, Larry Samuelson, David Schmeidler