danger zone

Prior to the ICMS meeting last year in Edinburgh, I spent two days in the Highlands, first in Glencoe climbing Bidean nam Bian, then on the Ben itself, with the classic Tower Ridge route. These were fantastic climbs in still wintry Scottish conditions and I enjoyed them tremendously without feeling any proximity with danger at any time (although I backed down from a snow corridor on Bidean nam Bian for missing an extra iceaxe…) On the previous weekends, there were alas two accidents on those same routes, first a group of four taken by an avalanche near Bidean nam Bian and second a lone climber on the Tower Ridge route… While climbing solo always involves some degree of objective danger, especially on exposed ridges like Tower Ridge, the casualties on Bidean nam Bian sounded more like “shit happens“. An unlikely and very rare accident cause by an accumulation of circumstances that were just too hard to predict. And to avoid (except by spending the day at the Clachaig Inn, which has its own risk!)

Posterior likelihood

At the Edinburgh mixture estimation workshop, Murray Aitkin presented his proposal to compare models via the posterior distribution of the likelihood ratio.

As already commented in a post last July, the positive aspect of looking at this quantity rather than at the Bayes factor is that the priors are then allowed to be improper if one simulates from the posteriors for each model, as in Aitkin et al. (2007). My overall feeling has not changed though, namely the ratio should be instead considered under the joint posterior of , which is [proportional to]

instead of the product of both posteriors. This of course makes a whole difference, as shown on the next R graph that compares the distribution of the likelihood ratio under the true posterior and under the product of posteriors (when comparing a Poisson model against a negative binomial with successes trials, when ). The joint simulation produces a much more supportive argument in favour of the negative binomial model, when compared with the product of the posteriors.

Obviously, this joint perspective also cancels the appeal of the approach under improper priors.