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Bayes Factors for Those Who Hate Bayes Factors

This post is inspired by Morey et al. (2016), Rouder and Morey (in press), and Wagenmakers et al. (2016a).

The Misconception

Bayes factors may be relevant for model selection, but are irrelevant for
parameter estimation.

The Correction

For a continuous parameter, Bayesian estimation involves the computation of an infinite number of Bayes factors against a continuous range of different point-null hypotheses.

The Explanation

Let H0 specify a general law, such that, for instance, the parameter θ has a fixed value θ0. Let H1 relax the general law and assign θ a prior distribution p(θ | H1). After acquiring new data one may update the plausibility for H1 versus H0 by applying Bayes’ rule (Wrinch and Jeffreys 1921, p. 387):

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Popular Misconceptions About Bayesian Inference: Introduction to a Series of Blog Posts

“By seeking and blundering we learn.”

– Johann Wolfgang von Goethe

Bayesian methods have never been more popular than they are today. In the field of statistics, Bayesian procedures are mainstream, and have been so for at least two decades. Applied fields such as psychology, medicine, economy, and biology are slow to catch up, but in general researchers now view Bayesian methods with sympathy rather than with suspicion (e.g., McGrayne 2011).

The ebb and flow of appreciation for Bayesian procedures can be explained by a single dominant factor: pragmatism. In the early days of statistics, the only Bayesian models that could be applied to data were necessarily simple – the more complex, more interesting, and more appropriate models escaped the mathematically demanding derivations that Bayes’ rule required. This meant that unwary researchers who accepted the Bayesian theoretical outlook effectively painted themselves into a corner as far as practical application was concerned. How convenient then that the Bayesian paradigm was “absolutely disproved” (Peirce 1901, as reprinted in Eisele 1985, p. 748); how reassuring that it would “break down at every point” (Venn 1888, p. 121); and how comforting that it was deemed “utterly unacceptable” (Popper 1959, p. 150).

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