Chapter 2: Bayesian Statistics

15 Oct

Bayesian statistics is one of the few areas in the actuarial syllabus I’ve seen before, but when I first encountered it as a beginning statistics major, it made no sense, both from the point of view of how to do it, and from the point of view of what for.

To understand why Bayesian statistics might be important to an actuary, well the best thing to do is to read the rest of the CT6 (or C/4) notes. To understand why and how it is interesting to a statistician you could read the Scholarpedia article -> http://www.scholarpedia.org/article/Bayesian_statistics

This article has been written and reviewed by some of the biggest names currently working in the field!

The scholarpedia version is, despite being relatively short, magisterial and comprehensive. As one might expect from those involved, some of the biggest names in Bayesian statistics academic practice.

For another short and sweet view of the topic, one could also read the introduction at the bottom of the page (a small amount of scrolling may be required before you get to it as of today, 15/10/2012, due to election notices) at bayesian.org under the heading ‘What is Bayesian Analysis?’

For me, an obvious omission from the Acted treatment of this topic is that after emphasising the use of conjugate priors, there was no discussion on how to find the damn things. Also, not much discussion of diffuse priors. With respect to the first point, the notes make it seem like conjugate priors are usually available, whereas they are very rare outside the exponential family (although the exponential family does, of course, contain some of the most used probability distributions). The strangeness is compounded given that it is essential to understand exponential families of distributions in order to understand generalised linear models, and hence this family of distributions is taught later in the subject.

With respect to diffuse priors, it should be noted that they are also difficult critters, and it is hard to find truly non-informative priors. James Berger, one of the heavyweights of Bayesian decision theory apparently only admits the existence of four (4) (the word apparently appears in the preceeding sentence because my only reference is my third year Bayesian Statistics lecture notes, and the quote is not referenced.Most likely it appears in this paper http://www.stat.duke.edu/~berger/papers/catalog.html(1985), but I can’t be certain because I only found the paper one second before rewriting this parenthesis. Reading it will have to wait), although I think at least one of his four is a set of priors, rather than a single specific distribution.

To give weight to my rant about how easy it is to find conjugate priors, I give below the steps to finding the proposed by Raiffa and Schlaifer (not quite the originators of the term and the concept, but they appear to have it given natural conjugates a lot of momentum), as written in S. James Press Applied Multivariate Analysis: Using Bayesian and Frequentist Methods of Inference, Second Edition (Dover Books on Mathematics)
) , a text available in a Dover reprint for only slightly more than a nominal amount.

“…write the density or likelihood function for the observable random variables and then interchange the roles of the observable random variables and the parameters, assuming the latter to be random and the former to be fixed and known. Modifying the proportionlity constant appropriately so that the new ‘density’ integrates to unity and letting the fixed parameters be abitrary provides a density that is in this sense is ‘conjugate’ to the original.”

Not terrifying difficult, but maybe not trivial enough to not be a distraction if you’re not specifically after testing it?

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