110 Metastatistics for the Non-Bayesian Regression Runner
opening is a consequence of his dialing the right numbers on the combination.
It might be expressed thus: simply having the belief, and doing nothing further,
has in general no consequences, while simply performing the action, and doing
nothing further does have consequences.While the connections between Bayesian probability and Bayesian decision
theory are a matter of debate as well, the connections seem tighter.^19 More impor-
tantly, an example from “decision theory” will, I think, highlight an important
difference between Bayesians and non-Bayesians.
A useful case study comes from L.J. Savage, an important figure in the devel-
opment of Bayesian ideas, who argued that the role of a mathematical theory of
probability “is to enable the person using it to detect inconsistencies in his own
real or envisaged behavior. It is also understood that, having detected an incon-
sistency, he will remove it” (Savage, 1972, p. 57). Indeed, the first seven chapters
of Savage (1972) are an introduction to the “personalistic” tradition in probability
and utility.
3.3.1 What’s utility got to do with it?
To me, the idea of probability as primarily a tool for detecting inconsistencies
sounds strange; nonetheless, it appears to be a view held by many. Savage him-
self provides an interesting example of “detecting an inconsistency” and then
removing it. This case study was the result of a “French” complaint about crazy
“American” ideas in economics. The Frenchman issuing the complaint, Allais
(1953), wrote a hotly contested article arguing against the “American” School’s
view of a “rational man.”^20
Savage, like some Bayesians, argued that maximizing expected utility is good
normativeadvice. Although the ideas will probably be familiar as the “Allais para-
dox,” it may be a good idea to sketch the main idea. If we considerx 1 ,x 2 ,...xk
mutually exclusive acts that occur with probabilityp 1 ,p 2 ,...pk, respectively, where
∑k
i= 1 pi=1, and we can define utility over these acts with a single utility func-
tionU(x)with the “usual” properties (increasing inx, and so on), we can define
expected utility as:
E[U]=∑ki= 1U(xi)pi.If utility is, say, increasing in money, then a “rational” person “should” prefer
the gamble that yields the highest expected utility. (Note we postpone a discussion
of what probability is until the next section.)
One of the gambles Allais devised to demonstrate that maximization of Expected
Utility (what Allais referred to as the “Principle of Bernoulli”) wasn’t necessarily a
good idea went as follows:
Imagine 100 well-shuffled cards, numbered from 1 to 100, and consider the two
following pairs of bets and determine which you prefer.