By Royal Skousen (auth.)
Analogy and Structure offers the required starting place for figuring out the character of analogical and structuralist (or rule-based) methods to describing habit. within the first a part of this ebook, the mathematical houses of rule ways are constructed; within the moment half, the analogical replacement to ideas is built. This ebook serves because the mathematical foundation for AnalogicalModeling of Language (Kluwer, 1989). good points comprise:
A common degree of Uncertainty: The war of words among randomly selected occurences avids the problems of utilizing entropy because the degree of uncertainty.
Optimal Descriptions: The implicit assumption of structuralist descriptions (namely, that descriptions of habit will be corrected and minimum) may be derived from extra primary statements concerning the uncertainty of rule structures.
Problems with Rule Approaches: the proper description of nondeterministic habit ends up in an atomistic, analog substitute to structuralist (or rule-based) descriptions.
Natural Statistics: conventional statistical exams are eradicated in prefer of statistically similar determination principles that contain very little mathematical calculation.
Psycholinguistic Factors: Analogical types, in contrast to, neural networks, without delay account for probabilistic studying in addition to response occasions in world-recognition experiments.
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Extra resources for Analogy and Structure
Consider the case of J = 3. One efficient way to ask questions is: The average number of questions for this strategy is 1%: outcome number of questions probability of outcome (,)1 1 1/3 (,)2 2 1/3 (,)3 2 1/3 1'1/3 + 2'1/3 + 2'1/3 = 5/3 However, the previous examples (when J was a power of two) suggest log23 ( .. 5850) as a measure of the average number of questions. Can this be motivated? The answer is yes. For a single occurrence of a probabilistic rule with three equally likely outcomes, 1% is the average number of questions.
Thus the entropy H = -LP)Og2Pj represents the average number of questions needed to determine the correct outcome, even when the probabilities are not all equally likely. This demonstration assumes of course that all the probabilities are rational numbers. But the irrational numbers can be approximated by rational numbers to any degree of accuracy and H = - LPj log2Pj itself is a continuous function of the Pi" Therefore, this formulation for H also works when the probabilities are irrational. And of course it also applies when the probabilities are all equal, in which case Pj = 1jJ for allj: H = -L(ljJ)log2(ljJ) = L(ljJ)log21 = log21.
S) In particular, we have the following special cases: H = II = -log C1; Q = U2 = 1 - C2 = 1 - z. For both H and Q, the uncertainty of the system is the average uncertainty of the rules in the system: H(S) = LPl+· H(Rj); i Q(S) = LPl+· Q(R j ). j And since Z is a linear function of Q, we also have 44 ANALOGY AND STRUCTURE Z(S) =E Pj+· Z(Rj). j But Cl is an exponential function of H (since H C1(S) ;II E Pj+· C1(R = -logCl ). Therefore, j ). j Instead, we have the following: C1(S) = II [C1(R j )f'+.