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Cross-validation and selection of priors

It is not unusual for statisticians to check their model with cross-validation. Last week we have seen that several replications of cross-validation reduce the standard error of mean error estimate, and that cross-validation can also be used to obtain pseudo-posteriors alike those using Bayesian statistics and bootstrap. This posting will demonstrate that the choice of prior depends on the parameters of cross-validation: a “wrong” prior may result in overfitting or underfitting according to the CV criterion. Furthermore, the number of folds in cross-validation affects the choice of the prior.

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It has been recognized that the results of logistic regression and of the naive Bayes classifier are better if the weights are shrunk towards zero. Naive Bayes is a simple form of logistic regression where the weights for a multivariate logistic regression model are actually estimated in a univariate fashion: if all the variables were conditionally independent given the class, the results would be the same. The meaning of the shrinkage parameter is that we inject m cases into the data that have the same distribution in each variable as the original data, but have no association between any pair of variables – this corresponds to a variant of the usual conjugate Dirichlet prior. The usual recommendation in the context of machine learning is to use cross-validation to determine the “right” value of m, while a Bayesian would assume a prior distribution over m.

The trouble is that cross-validation itself has a parameter: the number of folds. As the graph above illustrates, the optimal value of m is 83 with predictive bootstrap (the cases that were not selected in a particular resample are used to asses the out-of-sample error), 41 with 2-fold CV, 54 with 3-fold CV, 59 with 5-fold CV, 64 with 10-fold CV, and 69 with leave-one-out. Moreover, should we use a different error measure, such as KL-divergence or classification error, the ideal amount of shrinkage would again be affected. A different data set would again imply a different amount of shrinkage.

One cannot escape assumptions. If this is so, what are reasonable assumptions? Data splitting is not an unreasonable assumption (analogous to a prior), and I find the above parameterization of shrinkage quite reasonable. But I personally feel uneasy expressing my prior as some sort of a distribution on weights. Secondly, having a more sophisticated model flunk the cross-validation test might just imply that there is a mismatch between the prior used and the prior that would be preferred by cross-validation.

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