Archive for density

miXed distributions

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , on November 3, 2015 by xi'an

A couple of questions on X validated showed the difficulty students have with mixed measures and their density. Actually, my students always react with incredulity to the likelihood of a censored normal sample or to the derivation of a Bayes factor associated with the null (and atomic) hypothesis μ=0…

I attribute this difficulty to a poor understanding of the notion of density and hence to a deficiency in the training in measure theory, since the density f of the distribution F is always relative to a reference measure dμ, i.e.

f(x) = dF/dμ(x)

(Hence Lebesgue’s moustache on the attached poster!) To handle atoms in the distribution requires introducing a dominating measure dμ with atomic components, i.e., usually a sum of the Lebesgue measure and of the counting measure on the appropriate set. Which is not so absolutely obvious: while the first question had {0,1} as atoms, the second question introduced atoms on {-θ,θ}and required a change of variable to consider a counting measure on {-1,1}. I found this second question actually of genuine interest and a great toy example for class and exams.


Posted in R, Statistics with tags , , , , on June 28, 2011 by xi'an

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!