Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T22:08:56.037Z Has data issue: false hasContentIssue false
  • English
  • Français

Derivations of Learning Statistics from Absorbing Markov Chains

Published online by Cambridge University Press:  01 January 2025

Richard B. Millward
Richard B. Millward*
Affiliation:
Brown University
Get accessRights & Permissions [Opens in a new window]

Abstract

Learning-process statistics for absorbing Markov-chain models are developed by using matrix methods exclusively. The paper extends earlier work by Bernbach by deriving the distribution of the total number of errors, u-tuples, autocorrelation of errors, sequential statistics, and the expectation and variance of all statistics presented. The technique is then extended to latency derivations including the latencies of sequential statistics. Suggestions are made for using the sequential-statistic algorithm in a maximum-likelihood estimation procedure. The technique is important because statistics for very large absorbing matrices can be easily computed without going through tedious theoretical calculations to find explicit mathematical expressions.

Information

Type
Original Paper
Information
Psychometrika , Volume 34 , Issue 2 , June 1969, pp. 215 - 232
Copyright
Copyright © 1969 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

*

The author is indebted to his students Thomas Wiekens and Richard Freund who were helpful in the development of this paper. Support was received for this work from Grant MH-11255 from the National Institutes of Mental Health.

References

Atkinson, R. C., & Estes, W. K. In Luce, R. D., Bush, R. R., & Galanter, E. (Eds.), Handbook of mathematical psychology. New York: Wiley. 1963, 121–268.Google Scholar
Bush, R. R. Sequential properties of linear models. In Bush, R. R. and Estes, W. K. (Eds.), Studies in mathematical learning theory. Stanford: Stanford Univ. Press. 1959, 215–227.Google Scholar
Bernbach, H. A. Derivation of learning process statistics for a general Markov model. Psychometrika, 1966, 31, 225–234.CrossRefGoogle ScholarPubMed
Bower, G. H. Application of a model to paired-associate learning. Psychometrika, 1961, 26, 255–280.CrossRefGoogle Scholar
Bower, G. H. & Trabasso, T. Concept-identification. In Atkinson, R. C. (Eds.), Studies in mathematical psychology. Stanford: Stanford Univ. Press. 1964, 32–94.Google Scholar
Kemeny, J. G., & Snell, L. J. Finite Markov chains, 1960, Princeton: D. Van Nostrand.Google Scholar
Millward, R. An all-or-none model for noncorrection routines with elimination of incorrect responses. Journal of Mathematical Psychology, 1964, 1, 392–404.CrossRefGoogle Scholar
Millward, R. Latency in a modified paired-associate learning experiment. Journal of Verbal Learning and Verbal Behavior, 1964, 3, 309–316.CrossRefGoogle Scholar
Nahinsky, I. D. Statistics-and-moment-parameter estimates for a duoprocess paired-associate learning model. Journal of Mathematical Psychology, 1967, 4, 140–150.CrossRefGoogle Scholar
Restle, F. Learning paired associates. In Atkinson, R. C. (Eds.), Studies in mathematical psychology. Stanford: Stanford Univ. Press. 1964, 116–172.Google Scholar
Suppes, P., Groen, G., & Schlag-Rey, M.. A model for response latency in paired-associate learning. Journal of Mathematical Psychology, 1966, 3, 99–128.CrossRefGoogle Scholar