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

By Itzhak Gilboa, Larry Samuelson, David Schmeidler

The e-book describes formal types of reasoning which are aimed toward taking pictures the best way that fiscal brokers, and choice makers often take into consideration their atmosphere and make predictions in keeping with their previous event. 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 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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Analogies and Theories: Formal Models of Reasoning

The ebook describes formal types of reasoning which are geared toward taking pictures the way in which that monetary brokers, and selection makers quite often take into consideration their setting and make predictions according to their prior adventure. the point of interest 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.

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Define a M a if a has a higher value of the likelihood function given {(xi , yi )}i≤n than does a . This implies that M satisfies the combination axiom. Since the least squares estimator a is a maximum likelihood estimator of the parameter α (under the standard assumptions of regression analysis), choosing the estimate a is consistent with choosing a M -maximizer. Assume now that the units of analysis are the particular values of yp for a new value of xp . That is, rather than accepting the regression model y = α + βx + ε and asking what are the values of α and β, suppose that one is asked to predict (formulate M ) directly on potential values of yp .

In this case the proof relies on an auxiliary result that is of interest in its own right. Since the proof of this theorem applies to an infinite set of eventualities X, we do not restrict the cardinality of X in this case. Step 2 proceeds to deal with the case in which |T| is unrestricted, but X is finite. Lastly, Step 3 deals with the general case in which both |X| and |T| are unrestricted. 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.

Observe that “the coin generates random sequences” is a statement about sequences of cases. Similarly, statements such as “The weather always surprises” or “History repeats itself” are about sequences of cases, and are therefore likely to generate violations of the combination axiom. Second-order induction An important class of examples in which we should expect the combination axiom to be violated, for descriptive and normative purposes alike, involves learning of the similarity function. For instance, assume that one database contains but one case, in which Mary chose restaurant x over y.

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