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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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The Butler, The Maid, And The Bayes Factor

This post is based on the example discussed in Wagenmakers et al. (in press).

The Misconception

Bayes factors are a measure of absolute goodness-of-fit or absolute pre-
dictive performance.

The Correction

Bayes factors are a measure of relative goodness-of-fit or relative predictive performance. Model A may outpredict model B by a large margin, but this does not imply that model A is good, appropriate, or useful in absolute terms. In fact, model A may be absolutely terrible, just less abysmal than model B.

The Explanation

Statistical inference rarely deals in absolutes. This is widely recognized: many feel the key objective of statistical modeling is to quantify the uncertainty about parameters of interest through confidence or credible intervals. What is easily forgotten is that there is additional uncertainty, namely that which concerns the choice of the statistical model.

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