Archive for the Travel Category
“A frequent matter of debate in Bayesian inversion is the question, which of the two principle point-estimators, the maximum-a-posteriori (MAP) or the conditional mean (CM) estimate is to be preferred.”
An interesting topic for this arXived paper by Burger and Lucka that I (also) read in the plane to Montréal, even though I do not share the concern that we should pick between those two estimators (only or at all), since what matters is the posterior distribution and the use one makes of it. I thus disagree there is any kind of a “debate concerning the choice of point estimates”. If Bayesian inference reduces to producing a point estimate, this is a regularisation technique and the Bayesian interpretation is both incidental and superfluous.
Maybe the most interesting result in the paper is that the MAP is expressed as a proper Bayes estimator! I was under the opposite impression, mostly because the folklore (and even The Bayesian Core) have it that it corresponds to a 0-1 loss function does not hold for continuous parameter spaces and also because it seems to conflict with the results of Druihlet and Marin (BA, 2007), who point out that the MAP ultimately depends on the choice of the dominating measure. (Even though the Lebesgue measure is implicitly chosen as the default.) The authors of this arXived paper start with a distance based on the prior; called the Bregman distance. Which may be the quadratic or the entropy distance depending on the prior. Defining a loss function that is a mix of this Bregman distance and of the quadratic distance
produces the MAP as the Bayes estimator. So where did the dominating measure go? In fact, nowhere: both the loss function and the resulting estimator are clearly dependent on the choice of the dominating measure… (The loss depends on the prior but this is not a drawback per se!)
I started my stay in Banff with an interesting ice-climb on Cascade Mountain, just next to the Icefields Parkway exit to the town. (So we climbed the redundant Cascade Fall!) While the difficulty of the climb was much lower [grade III] than for my earlier ice-climb in Banff, it was incredibly cold (when we started, the temperature was -27⁰C… and rose to -19⁰C by the mid-afternoon, freezing the water in my thermos bottle) and I was a little worried at getting numb fingers, which would definitely not help with the climbing. And at the ice getting too brittle. As it happened, the cold temperature did not bother us at all during the climb which ended up being highly enjoyable. (The missing thumb did not bother me either. Except when clipping gear in and out, where I was rather clumsy.) The mountain guide who took us there was Joe McKay, who was hilarious and highly laid-back. He is also involved in filming climbing tricks and advices, so I may see him again at the Banff Centre this week… (In a video, Joe states that one should not be climbed if it’s 25 below!)