## ASC 2012 (#1)

**T**his morning I attende**d Alan Gelfand talk on directional data, i.e. on the torus (0,2π), and found his modeling via wrapped normals (i.e. normal reprojected onto the unit sphere) quite interesting and raising lots of probabilistic questions. For instance, usual moments like mean and variance had no meaning in this space. The variance matrix of the underlying normal, as well of its mean, obviously matter. One thing I am wondering about is how restrictive the normal assumption is. Because of the projection, any random change to the scale of the normal vector does not impact this wrapped normal distribution but there are certainly features that are not covered by this family. For instance, I suspect it can only offer at most two modes over the range (0,2π). And that it cannot be explosive at any point.**

**T**he keynote lecture this afternoon was delivered by Roderick Little in a highly entertaining way, about calibrated Bayesian inference in official statistics. For instance, he mentioned the inferential “schizophrenia” in this field due to the between design-based and model-based inferences. Although he did not define what he meant by “calibrated Bayesian” in the most explicit manner, he had this nice list of eight good reasons to be Bayesian (that came close to my own list at the end of the Bayesian Choice):

- conceptual simplicity (Bayes is prescriptive, frequentism is not), “
having a model is an advantage!”- avoiding ancillarity angst (Bayes conditions on everything)
- avoiding confidence cons (confidence is not probability)
- nails nuisance parameters (frequentists are either wrong or have a really hard time)
- escapes from asymptotia
- incorporates prior information and if not weak priors work fine
- Bayes is useful (25 of the top 30 cited are statisticians out of which … are Bayesians)
~~Bayesians go to Valencia!~~[joke! Actually it should have been Bayesian go MCMskiing!]- Calibrated Bayes gets better frequentists answers

He however insisted that frequentists should be Bayesians *and also* that Bayesians should be frequentists, hence the *calibration* qualification.

**A**fter an interesting session on Bayesian statistics, with (adaptive or not) mixtures and variational Bayes tools, I actually joined the “young statistician dinner” (without any pretense at being a young statistician, obviously) and had interesting exchanges on a whole variety of topics, esp. as Kerrie Mengersen adopted (reinvented) my dinner table switch strategy (w/o my R simulated annealing code). Until jetlag caught up with me.

July 11, 2012 at 12:03 pm

I don’t understand why “usual moments like mean and variance had no meaning in this space”. I can understand if it doesn’t have any moments E(|x|^k) is infinite for all k>=1, but the concept still means the same thing doesn’t it? I mean, a torus is a nice manifold…

July 11, 2012 at 8:54 pm

Dan: moments are defined just like for every regular random variable. But take X with mass at 1 and at 2π-1, its mean would be π… Zero would be better. In other words, E[X] depends on the origin chosen for the torus (0,2π), which makes it a poor representative of the random variable. (Alan made those points during the talk, mind.)

July 11, 2012 at 11:47 am

Of course the wrapped normal distribution is limited in terms of the features it can capture. There is a *lot* of work on this area. A recently published flexible wrapped distribution can be found here

http://onlinelibrary.wiley.com/doi/10.1111/j.1541-0420.2011.01651.x/abstract

Multimodality is typically preferred to be modelled using finite mixtures.