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Concerns About the Default Cauchy Are Often Exaggerated: A Demonstration with JASP 0.12

Contrary to most of the published literature, the impact of the Cauchy prior width on the t-test Bayes factor is seen to be surprisingly modest. Removing the most extreme 50% of the prior mass can at best double the Bayes factor against the null hypothesis, the same impact as conducting a one-sided instead of a two-sided test. We demonstrate this with the help of the “Equivalence T-Test” module, which was added in JASP 0.12.

We recently revised a comment on a scholarly article by Jorge Tendeiro and Henk Kiers (henceforth TK). Before getting to the main topic of this post, here is the abstract:

Tendeiro and Kiers (2019) provide a detailed and scholarly critique of Null Hypothesis Bayesian Testing (NHBT) and its central component –the Bayes factor– that allows researchers to update knowledge and quantify statistical evidence. Tendeiro and Kiers conclude that NHBT constitutes an improvement over frequentist p-values, but primarily elaborate on a list of eleven ‘issues’ of NHBT. We believe that several issues identified by Tendeiro and Kiers are of central importance for elucidating the complementary roles of hypothesis testing versus parameter estimation and for appreciating the virtue of statistical thinking over conducting statistical rituals. But although we agree with many of their thoughtful recommendations, we believe that Tendeiro and Kiers are overly pessimistic, and that several of their ‘issues’ with NHBT may in fact be conceived as pronounced advantages. We illustrate our arguments with simple, concrete examples and end with a critical discussion of one of the recommendations by Tendeiro and Kiers, which is that “estimation of the full posterior distribution offers a more complete picture” than a Bayes factor hypothesis test.

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Prediction is Easy, Especially About the Past: A Critique of Posterior Bayes Factors

The Misconception

Posterior Bayes factors are a good idea: they provide a measure of evidence but are relatively unaffected by the shape of the prior distribution.

The Correction

Posterior Bayes factors use the data twice, effectively biasing the outcome in favor of the more complex model.

The Explanation

The standard Bayes factor is the ratio of predictive performance between two rival models. For each model M_i, its predictive performance p(y | M_i) is computed as the likelihood for the observed data, averaged over the prior distribution for the model parameters \theta | M_i. Suppressing the dependence on the model, this yields p(y) = \int p(y | \theta) p(\theta) \, \text{d}\theta. Note that, as the words imply, “predictions” are generated from the “prior”. The consequence, of course, is that the shape of the prior distribution influences the predictions, and thereby the Bayes factor. Some consider this prior dependence to be a severe limitation; indeed, it would be more convenient if the observed data could be used to assist the models in making predictions — after all, it is easier to make “predictions” about the past than about the future.

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Einstein’s Riddle

Summary

Einstein confused his students with a riddle about probability – or was it Einstein himself who was confused?

Albert Einstein disliked the idea that the laws of nature were inherently probabilistic. ‘God does not play dice with the universe,’ he stated famously and repeatedly. Yet, physicists like Niels Bohr strongly advocated the idea –based on the ‘Copenhagen interpretation’ of quantum theory– that chance is an inalienable and inevitable aspect of nature itself.

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Cicero and the Greeks on Necessity and Fortune

Cicero eloquently summarized the philosophical position that the universe is deterministic – all events are preordained, either by nature or by divinity. Although “ignorance of causes” may create the illusion of Fortune, in reality there is only Necessity.

Cicero Citatus, Glans Inflatus?

The male academic who cites Cicero generally lacks the insight that, instead of imbuing his writing with gravitas, he inevitably conveys the impression of being a pompous dickhead (‘glans inflatus’). Particularly damaging to a writer’s reputation are Cicero quotations that occur at the start of an article; for, as Horace reminds us, “parturiunt montes, nascetur ridiculus mus”. Indeed, the only academics who seem to get away with citing Cicero are those who study Cicero’s work professionally.

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The Merovingian, or Why Probability Belongs Wholly to the Mind

Summary: When Bayesians speak of probability, they mean plausibility.

The famous Matrix trilogy is set in a dystopic future where most of mankind has been enslaved by a computer network, and the few rebels that remain find themselves on the brink of extinction. Just when the situation seems beyond salvation, a messiah –called Neo– is awakened and proceeds to free humanity from its silicon overlord. Rather than turn the other cheek, Neo’s main purpose seems to be the physical demolition of his digital foes (‘agents’), a task that he engages in with increasing gusto and efficiency. Aside from the jaw-dropping fight scenes, the Matrix movies also contain numerous references to religious themes and philosophical dilemma’s. One particularly prominent theme is the concept of free will and the nature of probability.

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