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 aimed toward shooting the way in which that fiscal brokers, and selection makers quite often take into consideration their surroundings and make predictions in keeping with their prior adventure. the point of interest is on analogies (case-based reasoning) and normal 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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In all three steps, memories in M are represented by vectors of non-negative integers, counting how many cases of each type appear in memory. Formally, for every T ⊂ T define JT = ZT+ = {I | I : T → Z+ } where Z+ stands for the nonnegative integers. I ∈ JT is interpreted as a counter vector, where I(t) counts how many cases of type t appear in the memory represented by I. 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 .

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. To complete the proof of the lemma, we apply the claim consecutively. In case X is not countable, the induction is transfinite (and assumes that X can be well ordered). Note that Lemma 3, unlike Lemma 2, guarantees the possibility to rescale simultaneously all the vxy -s from Lemma 1 such that the Jacobi identity will hold on X.

Then there are vectors {vxy }x,y∈A∪{w},x=y , as in Lemma 1, and for any three distinct acts, x, y, z ∈ X, vxy + vyz = vxz holds. Proof: Choose distinct x, y, z ∈ A. Let vˆ xw ,ˆvyw , and vˆ zw be the vectors provided by Lemma 1 when applied to the pairs (x, w), (y, w), and (z, w), respectively. Consider the triple {x, y, w}. By Lemma 2 there are unique coefficients λ({x, w}, y), λ({y, w}, x) > 0 such that 40 Inductive Inference: An Axiomatic Approach vxy = λ({x, w}, y)ˆvxw + λ({y, w}, x)ˆvwy (8) Applying the same reasoning to the triple {x, z, w}, we find that there are unique coefficients λ({x, w}, z), λ({z, w}, x) > 0 such that vxz = λ({x, w}, z)ˆvxw + λ({z, w}, x)ˆvwz .

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