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Derivation of Learning Process Statistics for a General Markov Model

Published online by Cambridge University Press:  01 January 2025

Harley A. Bernbach
Harley A. Bernbach*
Affiliation:
University of Michigan
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Abstract

A Markov learning model may be stated in the form of a transition matrix, starting vector, and response probability vector. Utilizing these and some general properties of absorbing Markov chains, general expressions are derived for several statistics of the learning process which can be applied to any model of this form. Included are derivations for the mean learning curve, number of total errors, trial numbers of the first success and the last error, and the number of error runs. As an illustration, all derivations are worked out for the simple two-state one-element model.

Type
Original Paper
Information
Psychometrika , Volume 31 , Issue 2 , June 1966 , pp. 225 - 234
Copyright
Copyright © 1966 Psychometric Society

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Footnotes

*

This work was partially supported by Public Health Service Grant NIH 1T1 GM 1231-01.

†

Appreciation is expressed to David Birch for his continuing advice and support during the course of this work, and to Frank Goode for his helpful criticism of the manuscript and notation. The author is now at Cornell University.

References

Atkinson, R. C. and Crothers, E. J. A comparison of paired-associate learning models having different acquisition and retention axioms. J. math. Psychol., 1964, 1, 285–315.CrossRefGoogle Scholar
Bernbach, H. A. A forgetting model for paired-associate learning. J. math. Psychol., 1965, 2, 128–144.CrossRefGoogle Scholar
Bower, G. H. Application of a model to paired-associate learning. Psychometrika, 1961, 26, 255–280.CrossRefGoogle Scholar
Bower, G. H. and Trabasso, T. Concept identification. In Atkinson, R. C. (Eds.), Studies in mathematical psychology, Stanford: Stanford Univ. Press, 1964.Google Scholar
Estes, W. K. Component and pattern models with Markovian interpretations. In Bush, R. R. and Estes, W. K. (Eds.), Studies in mathematical learning theory, Stanford: Stanford Univ. Press, 1959.Google Scholar
Kemeny, J. G. and Snell, J. L. Finite Markov chains, Princeton, N. J.: D. Van Nostrand, 1959.Google Scholar
McGuire, W. J. A multiprocess model for paired-associate learning. J. exp. Psychol., 1961, 62, 335–347.CrossRefGoogle Scholar
Theios, J. Simple conditioning as two-stage all-or-none learning. Psychol. Rev., 1963, 70, 403–417.CrossRefGoogle ScholarPubMed