Bayesian credibility and the K constant

Named after the mathematician (and pastor!) Thomas Bayes, of course, who first articulated the concept in a paper published posthumously in 1764. The Wikipedia entry on him states:

"He is known to have published two works in his lifetime: Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures (1731), and An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst (published anonymously in 1736), in which he defended the logical foundation of Isaac Newton's calculus against the criticism of George Berkeley, author of The Analyst."

Ooooooh. Sounds so very geeky. In a virile (and godly!) sort of way, I mean. And yes, those two things *can* go together! In the right man.. :P

Real actuarial work for a change! :) Woot! With the boss off to Zurich this week, I have some time in my schedule to work on bigger projects, and this one has a little of the science in it that I spent so many years learning.

People have asked me why I keep all the technical papers, textbooks and binders of actuarial exam material (which takes up about 30 linear feet of bookshelf space) - could I possibly still use them? Oh, yeah, baby! Just this morning, in fact. And darned if I didn't know exactly which book to pull, and which formulae to look for. Mind like a steel trap, yes sir.

I'm glad I don't have to *derive* formulae anymore.. just apply them. :)

So, what I was looking for was a 1967 paper by Switzerland's Hans Buhlmann (the paper was already about 20 years old when I was studying it - which, by the way, is one of the reasons I missed what was going on musically and culturally in the 80's. No time!) That, and the later contributions of Stuart Klugman (department chair at Iowa State, and oh, so dry as a speaker) and Gary Venter (no slouch at wearing down an audience, either) were pretty much what I needed.

The mission? To find the elusive K.

As in Z = P/(P+K) where P is the number of (insurance) claims you have in your data, Z is the credibility value (the percent you can believe what your own data are telling you), and K is the constant that lets you asymptotically approach 1 (or full credibility) as your dataset gets larger, but not ever quite believe it fully. This is appropriate because your data are really only a subset of the universe of data, regardless of how much you have.

The assumption is that the emergence of your claims follow a Poisson distribution, and that the size of those claims is modeled by some relatively thick-tailed distribution, such as Lognormal, Weibull, generalized Pareto, or the ever-popular three-parameter Gamma.

I've always been partial to a thick-tailed distribution myself, when it comes to modeling. ;) And oh, I know it's old-fashioned of me, but.. her classic, pretty lines and that easy way she has of telling you everything you want to know (mmm.. those simple parameters.. so sweetly agreeable) with no need for a lot of extra "interpreting", leads me back to the Lognormal every time. (sigh) Lovely... and so exquisitely parsimonious.

(Who else but an actuary can work himself into a bit of a lather over the beauty and ease of handling of a certain type of loss distribution? Anyway, I digress.) With those assumptions K should represent the ratio of Process Variance to Parameter Variance, converted to the same base as the item you are modeling, in my case claim counts.

So, since minds brighter than mine (including old Stu from Iowa State) have done the experimenting, how about if I try some of their tried and true values of K? So I do, and woohoo! The results are so intuitively pleasing.. exactly (well, within +/- 5% of) the results I expected a priori, just from instinct and past experience.

Oooh, I love it when the data confirm my intuition! And wouldn't every Myers-Briggs INTJ feel exactly the same?

Gee, it's nice to have some fun at work for a change. :)

Must be the turn of the month. I always seem to cheer up about now..