Hostname: page-component-74656f9884-g96k6 Total loading time: 0 Render date: 2026-11-10T05:05:21.243Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical Methods for Absorbing Markov-Chain Models for Learning: Estimation and Identification

Published online by Cambridge University Press:  01 January 2025

Peter G. Polson  and
David Huizinga
Peter G. Polson
Affiliation:
University of Colorado
David Huizinga
Affiliation:
University of Colorado
Get access Rights & Permissions [Opens in a new window]

Abstract

General algorithms for computing the likelihood of any sequence generated by an absorbing Markov-chain are described. These algorithms enable an investigator to compute maximum likelihood estimates of parameters using unconstrained optimization techniques. The problem of parameter identifiability is reformulated into questions concerning the behavior of the likelihood function in the neighborhood of an extremum. An alternative characterization of the concept of identifiability is proposed. A procedure is developed for deciding whether or not this definition is satisfied.

Keywords

Statistical MethodPublic PolicyMaximum Likelihood EstimateLikelihood EstimateStatistical Theory

Information

Type
Original Paper
Information
Psychometrika , Volume 39 , Issue 1 , March 1974 , pp. 3 - 22
Copyright
Copyright © 1974 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

*

This research was undertaken within the Institute for the Study of Intellectual Behavior, University of Colorado, and is Publication No. 42 of the Institute. The work was supported by NSF Grant GB-34077X. The logic underlying Algorithm I was suggested by Clint Schumacher.

References

Bernbach, H. A. Derivation of learning process statistics for a general Markov model. Psychometrika, 1966, 31, 225–234.CrossRefGoogle ScholarPubMed
Chandler, J. P. (1965) Subroutine Stepit: An algorithm that finds the values of the parameters which minimize a given continuous function. A copyrighted program: J. P. Chandler, Copyright 1965. The program and writeup may be obtained from the Quantum Chemistry Program Exchange, Indiana University; request program QCPE 66.Google Scholar
Greeno, J. G. Paired associates learning with short term retention: Mathematical analysis and data regarding identification. Journal of Mathematical Psychology, 1967, 4, 430–472.CrossRefGoogle Scholar
Greeno, J. G. Identifiability and statistical properties of two-stage learning with no successes in the initial stage. Psychometrika, 1968, 33, 173–215.CrossRefGoogle ScholarPubMed
Greeno, J. G. How associations are memorized. In Norman, D. A. (Eds.), Models of human memory, 1970, New York: Academic Press.Google Scholar
Greeno, J. G., James, C. T., and DaPolito, F. A cognitive interpretation of negative transfer and forgetting of paired associates. Journal of Verbal Learning and Verbal Behavior, 1971, 10, 331–345.CrossRefGoogle Scholar
Greeno, J. G., Millward, R. B., and Merryman, C. T. Matrix analysis of identifiability of some finite Markov models. Psychometrika, 1971, 36, 389–408.CrossRefGoogle Scholar
Greeno, J. G. and Steiner, T. E. Markovian processes with identifiable states: General considerations and application to all-or-none learning. Psychometrika, 1964, 29, 309–333.CrossRefGoogle Scholar
Greeno, J. G. and Steiner, T. E. Comments on ‘Markovian processes with identifiable states: General considerations and applications to all-or-none learning’. Psychometrika, 1968, 33, 169–172.CrossRefGoogle Scholar
Hook, R. and Jeeves, J. Direct search solution of numerical and statistical problems. Journal of the Association for Computing Machinery, 1961, 8, 212–228.CrossRefGoogle Scholar
Kendall, M. G. and Stuart, A. The Advanced Theory of Statistics, Vol. II, 1963, London: C. Griffin.Google Scholar
Kemeny, J. G. and Snell, J. L. Finite Markov Chains, 1960, Princeton: Van Nostrand.Google Scholar
Millward, R. B. Derivations of learning statistics from absorbing Markov chains. Psychometrika, 1969, 34, 215–232.CrossRefGoogle Scholar
Polson, P. G. A quantitative study of the concept identification and paired-associates learning processes in the Hull paradigm. Unpublished doctoral dissertation. Indiana University, 1967..Google Scholar
Polson, P. G. Statistical methods for a general theory of all-or-none learning. Psychometrika, 1970, 35, 51–72.CrossRefGoogle Scholar
Polson, P. G. Presolution performance functions of Markov models. Psychometrika, 1972, 37, 453–459.CrossRefGoogle Scholar
Powell, M. J. D. A survey of numerical methods for unconstrained optimization. SIAM Review, 1970, 12, 79–97.CrossRefGoogle Scholar
Restle, F. and Greeno, J. G. Introduction to mathematical psychology, 1970, Reading, Mass.: Addison-Wesley.Google Scholar