By way of background, I should disclose off the bat that I am a huge fan of the subjective approach to probability pioneered by Bruno de Finetti and Frank Ramsey, especially the idea that a person’s “priors” about the world are almost always derived from his own personal intuition and the related insight that one’s beliefs about the world don’t have to be “all or nothing” but can come in degrees or shades of grey. So this Yuletide I finally got around to reading Franz Huber’s 2009 survey of subjective probability, which is titled “Belief and Degrees of Belief“, and although this work is super-technical — filled with formal notation and a smattering of equations — I am glad I did for three reasons:

First off, Section 3 of Huber’s survey not only contains an excellent introduction to the standard betting model of subjective probability, i.e. the idea that a person’s probability estimate for a given proposition can be measured by “the highest price she is willing to pay for a bet that returns 1 Euro if [the proposition is true] and 0 otherwise” (Huber 2009, p. 6); Huber also explores two alternative theories of subjective probability, including one called the “transferable belief model” (pp. 13-15) and another based on “fuzzy set theory” (pp. 16-18).

Secondly, Huber identifies and discusses a deep and difficult foundational question in the probability literature, one that I had not given much thought to before. In a word, what is the relation between “belief” and “degrees of belief”? That is, what is the relation between one’s quantitative degree of belief about the truth value of a given proposition, which is supposed to be a real number from the interval [0, 1], and one’s qualitative belief about that proposition, which can only consist of one of three states: belief, disbelief, and suspension of judgement? At best, the betting model described above can only measure one’s degree of belief; it cannot, however, tell us what a degree of belief actually is; nor can it tell us when a degree of belief is strong or weak enough to be converted into belief or disbelief.

Third and last (and perhaps most importantly for me, given my own pro-subjective-probability priors!), Huber also explores two important logical problems in the literature: the so-called “lottery” and “preface” paradoxes. As it happens, both of these problems pose an existential challenge to the subjective approach to probability, so both deserve blog posts of their own. Stay tuned: I will address the the lottery paradox and preface paradox in a separate blog post, before proceeding to the works of Robert Darnton on the French Revolution and Paul Sagar on Adam Smith.