Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T09:53:41.387Z Has data issue: false hasContentIssue false

The Curve Fitting Problem: A Bayesian Rejoinder

Published online by Cambridge University Press:  01 April 2022

Prasanta S. Bandyopadhyay
Affiliation:
Montana State University
Robert J. Boik*
Affiliation:
Montana State University
*
Bandyopadhyay: Department of Philosophy, Montana State University, Bozeman, MT 59717; Boik: Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717.

Abstract

In the curve fitting problem two conflicting desiderata, simplicity and goodness-of-fit pull in opposite directions. To solve this problem, two proposals, the first one based on Bayes's theorem criterion (BTC) and the second one advocated by Forster and Sober based on Akaike's Information Criterion (AIC) are discussed. We show that AIC, which is frequentist in spirit, is logically equivalent to BTC, provided that a suitable choice of priors is made. We evaluate the charges against Bayesianism and contend that AIC approach has shortcomings. We also discuss the relationship between Schwarz's Bayesian Information Criterion and BTC.

Type
Probability and Statistical Inference
Copyright
Copyright © 1999 by the Philosophy of Science Association

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.)

Footnotes

Portions of this paper were presented at the APA, Central Division, 1998. We wish to thank James Allard, Scott DeVito, Malcolm Forster, William Harper, Henry Kyburg, Jr., Eric MacIntyre, Brian Skyrms, Elliot Sober and Greg Wheeler for discussion and encouragement. John G. Bennett and Gordon Brittan, Jr. also deserve special thanks for numerous discussions regarding the content of the paper.

References

Aitchison, J. (1975), “Goodness of Prediction Fit”, Biometrika 62: 547554.CrossRefGoogle Scholar
Akaike, H. (1973), “Information Theory as an Extension of the Maximum Likelihood Principle”, in Petrov, B. N. and Csaki, C. (eds.), Second International Symposium on Information Theory. Budapest: Akademiai Kiado, 267281.Google Scholar
Akaike, H. (1978). “A Bayesian Analysis of the Minimum AIC Procedure”, Annals of the Institute of Statistical Mathematics 30: 914.10.1007/BF02480194CrossRefGoogle Scholar
Bandyopadhyay, P., Boik, R., and Basu, P. (1996). “The Curve Fitting Problem: A Bayesian Approach”, Philosophy of Science 63 (Supplement): S264S272.10.1086/289960CrossRefGoogle Scholar
Berger, J. O. (1985), Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag.10.1007/978-1-4757-4286-2CrossRefGoogle Scholar
Forster, M. (1999), “The New Science of Simplicity”, forthcoming in Keuzenkamp, H. A., McAleer, M., and Zellner, A. (eds.), Simplicity, Inference and Economic Modelling. Cambridge: Cambridge University Press.Google Scholar
Forster, M. and Sober, E. (1994), “How to Tell when Simple, More Unified, or Less Ad Hoc Theories will Provide More Accurate Predictions”, The British Journal for the Philosophy of Science 45: 135.10.1093/bjps/45.1.1CrossRefGoogle Scholar
Jeffreys, H. (1961), Theory of Probability, 3rd ed. London: Oxford University Press.Google Scholar
Kieseppa, I. (1997), “Akaike Information Criterion, Curve Fitting, and the Philosophical Problem of SimplicityThe British Journal for the Philosophy of Science 48: 2148.10.1093/bjps/48.1.21CrossRefGoogle Scholar
Linhart, H. and Zucchini, W. (1986), Model Selection. New York: John Wiley & Sons.Google Scholar
Niiniluoto, I. (1994), “Descriptive and Inductive Simplicity,” in Salmon, W. and Wolters, G. (eds.), Logic, Language, and the Structure of Scientific Theories. Pittsburgh PA: University of Pittsburgh Press, 147170.Google Scholar
Reschenhofer, E. (1996), “Prediction with Vague Knowledge”, Communications in Statistics—Theory and Methods 25: 601608.CrossRefGoogle Scholar
San Martini, A. and Spezzaferri, F. (1984), “A Predictive Model Selection Criterion”, Journal of the Royal Statistical Stociety B 46: 296303.Google Scholar
Sawa, T. (1978), “Information Criteria for Discriminating Among Alternative Regression Models”, Econometrica 46: 12731291.10.2307/1913828CrossRefGoogle Scholar
Schwarz, G. (1978), “Estimating the Dimension of a Model”, The Annals of Statistics 6: 461464.CrossRefGoogle Scholar
Weinberg, S. (1992), Dreams of a Final Theory. New York; Pantheon Books.Google Scholar