density()

Following my earlier posts on the revision of Lack of confidence, here is an interesting outcome from the derivation of the exact marginal likelihood in the Laplace case. Computing the posterior probability of a normal model versus a Laplace model in the normal (gold) and the Laplace (chocolate) settings leads to the above histogram(s), which show(s) that the Bayesian solution is discriminating (in a frequentist sense), even for 21 observations. If instead I use R density() over the posterior probabilities, I get this weird and unmotivated flat density in the Laplace case. It looked as if the (frequentist) density of the posterior probability under the alternative was uniform, although there is no reason for this phenomenon!

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