Palgrave Handbook of Econometrics: Applied Econometrics

114 Metastatistics for the Non-Bayesian Regression Runner

The same is true for pairwise mutually exclusive events so, for example, we can

write:

P(A 1

⋃

A 2

⋃

...

⋃

Ak) =

∑k

j= 1

P(Aj)

= 1. (3.1)

If we were intending a proper introduction to probability, even here complications
arise. Iskfinite, for example? To address that issue properly one would introduce
a set of measure theoretic considerations, but there is no need to cavil about such
issues at present.^24
The most important observation to make is that these are, so far, simplyaxioms.
At this level, they are mere statements of mathematics. Indeed, we don’t even have
to consider them to be “probabilities.” They may or may not be readily associated
with anything “real” in the world. As Feller (1950, p. 1) explains:

Axiomatically, mathematics is concerned solely with relations among undefined

things.This propertyis well illustrated by the game of chess. It is impossible to

“define” chess otherwise than by stating a set of rules...The essential thing

is to know how the pieces move and act. It is meaningless to talk about the

“definition” or the “true nature” of a pawn or a king. Similarly, geometry does

not care what a point and a straight line “really are.” They remain undefined

notions, and the axioms of geometry specify the relations among them: two

points determine a line, etc. These are rules, and there is nothing sacred about

them. We change the axioms to study different forms of geometry, and the

logical structure of the several non-Euclidean geometries is independent of their

relation to reality. Physicists have studied the motion of bodies under laws of

attraction different from Newton’s, and such studies are meaningful if Newton’s

law of attraction is accepted as true in nature.^25

In sum, these axioms don’t commit you to believing anything in particular. One
reason you might adopt such axioms (and the reason I do) is because they seem
convenient and useful if you are interested in the properties of chance set-ups or
things that resemble chance set-ups.
I belabor this obvious point because I think it useful to consider that wecould
begin with different axioms. A nice example comes from Hacking (2001). Consider
representing the probability of a certain event,A,asP(A)=∞, and ifAwere
impossible,P(A)=−∞. In such a system:

• If the eventAand ̃A(the event “notA”) have the same probability, thenP(A)=
  P( ̃A)= 0


• If the eventAis more probable than ̃A, thenP(A)> 0


• If the event ̃Ais more probable thanA, thenP(A)<0.