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Examples of Bayesian and frequentist approach giving different answers

What are some practical examples where a Bayesian approach has an edge over frequentist statistics? Do you know of any success stories where a Bayesian model is superior to a traditional approach?

By more successful I mean "better predictions" or "lead to more valuable insights" or "made explaining it easier".

Examples should not discuss the philosophical nature of e.g. credible vs confidence intervals and its more "intuitive" interpretation.

Perhaps Bayesian spam filtering could be a starting point, but I don't know enough about it.

Similar? List of situations where a Bayesian approach is simpler, more practical, or more convenient

Might be linked to this question: Examples of Bayesian and frequentist approach giving different answers

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    $\begingroup$ simpler is not always better, sometimes the Bayesian approach is better because it is the direct answer to the question, but at the cost of having to solve tricky integrals (which often encourages people, including myself, to settle for a less direct frequentist alternative). $\endgroup$
    – Dikran Marsupial
    Commented Jan 22, 2013 at 17:01
  • $\begingroup$ This appears to be one-half of the linked question, Rico, and would thereby already be covered within its answers. Could you explain why we might need another thread to discuss this topic? $\endgroup$
    – whuber ♦
    Commented Jan 23, 2013 at 15:11

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Adrian Raftery examined a set of statistics about coal-dust explosions in 19th-century British mines. Frequentist techniques had shown the coal mining accident rates changed over time gradually. Our of curiosity, Raftery experimented with Bayes' Theorem, and discovered that accident rates had plummeted suddenly in the early 1890s. A historian suggested why: in 1889, the miners had formed a safety coalition.

from A history of Bayes Theorem. The original paper is here, though I would suggest the example in the PyMC literature for more clarity (i.e. less integrals haha)

Also, from another post, there is the Table Game from (the very accessible pdf) The Table Game. The two proposed solutions, Bayesian and frequentist, are very different (and only one is correct!).

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  • $\begingroup$ Raftery's paper is very nice, but the 'Table Game' in my view misses the point: (1) it's a contrived rather than a practical example; (2) there's "a physical mechanism for drawing a probability from a uniform prior", in which case anyone would use Bayes' rule - argument is about inference on parameters when there's no such mechanism; ... $\endgroup$
    – Scortchi - Reinstate Monica ♦
    Commented Jan 22, 2013 at 19:58
  • $\begingroup$ ... (3) the Bayesian approach is justified in terms of not inference, but prediction (which we can compare to actual outcomes because of (2)), & contrasted with a plug-in method that takes no account of the uncertainty in the parameter estimate - a straw man, as a better frequentist approach would be to use, say, profile likelihood. $\endgroup$
    – Scortchi - Reinstate Monica ♦
    Commented Jan 22, 2013 at 19:58
  • $\begingroup$ Could you elaborate on 3), I do not quite understand what you mean. $\endgroup$
    – Cam.Davidson.Pilon
    Commented Jan 22, 2013 at 20:12
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    $\begingroup$ The Bayesian predictive posterior is contrasted with an method where you take the maximum likelihood estimate of the binomial probability & use that to predict future observations as if it were the known value - i.e. you plug it into the probability mass function. It's not very good (for small samples) because it fails to take into account the uncertainty in the estimate. (If you were to use a more appropriate frequentist method it would be quite close to the Bayesian one using a uniform prior, I think.) $\endgroup$
    – Scortchi - Reinstate Monica ♦
    Commented Jan 22, 2013 at 20:44

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