By William Feller
Significant adjustments during this variation contain the substitution of probabilistic arguments for combinatorial artifices, and the addition of recent sections on branching methods, Markov chains, and the De Moivre-Laplace theorem.
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Additional resources for An Introduction to Probability Theory and Its Applications, Volume 2
Sclove University of Illinois at Chicago ABSTRACT Statistical model selection criteria provide answers to the questions, "How much improvement in fit should be achieved to justify the inclusion of an additional parameter in a model. " Mathematically, statistical model selection criteria are defined as estimates of suitable functionals of the probability distributions corresponding to alternative models. This paper discusses different approaches to model-selection criteria, with a view toward illuminating their similarities and differences.
L. Sclove log (prior probability x likelihood) Const. +log (posterior probability), giving a link between the minimum-bit and Bayesian approaches (Cheeseman 1993). A. (1974) "Simultaneous Inference and the Choice of Variable Subsets," Technometrics 16, 221-227. [Akaike 1969] Akaike, H. (1969) "Fitting Autoregressive Models for Prediction," Ann. Inst. Statist. Math. 21,243-247. [Akaike 1973] Akaike, H. (1973) "Information Theory and an Extension of the Maximum Likelihood Principle," Proc. N. Petrov and F.
Define W" = Pie - P"-l' Then for k ~ ko, we can write C(k,A) = EtE - (Etp"E - Ak( 2 ) = EtE - L", zl n ), i=l where zIn) = EtWiE - AU 2 . Minimizing C(k,A) is thus equivalent to maximizing Sln) ~7=1 Zi(n), k = ko, ko + 1, ... , K. Observe that the random vector (Z~n), ... , Z};')) converges in distribution to (Zl' ... , ZK), where the Zi'S are independent identically distributed random variables with distribution U2(X~ - A). Hence asymptotically, the problem of model selection is equivalent to the problem of finding the global maximizer of the random walk sequence S" = ~7=1 Zi, k = ko, ko + 1, ...
An Introduction to Probability Theory and Its Applications, Volume 2 by William Feller