Bayesian Versus Frequentist Inference

1 Goals and Outline Throughout this book, the topic of order-restricted inference is dealt with almost exclusively from a Bayesian perspective. Some readers may wonder why the other main school for statistical inference – frequentist inference – has received so little attention here. Isn't it true that in the field of psychology, almost all inference is frequentist inference? The first goal of this chapter is to highlight why frequentist inference is a less-than-ideal method for statistical inference. The most fundamental limitation of standard frequentist inference is that it does not condition on the observed data. The resulting paradoxes have sparked a philosophical debate that statistical practitioners have conveniently ignored. What cannot be so easily ignored are the practical limitations of frequentist inference, such as its restriction to nested model comparisons. The second goal of this chapter is to highlight the theoretical and practical advantages of a Bayesian analysis. From a theoretical perspective, Bayesian inference is principled and prescriptive, and – in contrast to frequentist inference – a method that does condition on the observed data. From a practical perspective, Bayesian inference is becoming more and more attractive, mainly because of recent advances in computational methodology (e.g., Markov chain Monte Carlo and the WinBUGS program [95]). To illustrate, one of our fre-quentist colleagues had been working with the WinBUGS program and commented " I don't agree with the Bayesian philosophy, but the WinBUGS program does allow me to implement complicated models with surprisingly little effort ". This response to Bayesian inference is diametrically opposed to the one that was in vogue until the 1980's, when statisticians often sympathized with the Bayesian philosophy but lacked the computational tools to implement models with a moderate degree of complexity.