Parameterizations make different model selections: Empirical findings from factor analysis

Abstract

How parameterizations affect model selection performance is an issue that has been ignored or seldom studied since traditional model selection criteria, such as Akaike’s information criterion (AIC), Schwarz’s Bayesian information criterion (BIC), difference of negative log-likelihood (DNLL), etc., perform equivalently on different parameterizations that have equivalent likelihood functions. For factor analysis (FA), in addition to one traditional model (shortly denoted by FA-a), it was previously found that there is another parameterization (shortly denoted by FA-b) and the Bayesian Ying-Yang (BYY) harmony learning gets different model selection performances on FA-a and FA-b. This paper investigates a family of FA parameterizations that have equivalent likelihood functions, where each one (shortly denoted by FA-r) is featured by an integer r, with FA-a as one end that r = 0 and FA-b as the other end that r reaches its upper-bound. In addition to the BYY learning in comparison with AIC, BIC, and DNLL, we also implement variational Bayes (VB). Several empirical finds have been obtained via extensive experiments. First, both BYY and VB perform obviously better on FA-b than on FA-a, and this superiority of FA-b is reliable and robust. Second, both BYY and VB outperform AIC, BIC, and DNLL, while BYY further outperforms VB considerably, especially on FA-b. Moreover, with FA-a replaced by FA-b, the gain obtained by BYY is obviously higher than the one by VB, while the gain by VB is better than no gain by AIC, BIC, and DNLL. Third, this paper also demonstrates how each part of priors incrementally and jointly improves the performances, and further shows that using VB to optimize the hyperparameters of priors deteriorates the performances while using BYY for this purpose can further improve the performances.

Key words： model selection; factor analysis; parameterizations; maximum likelihood; variational Bayes; Bayesian Ying-Yang learning

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