Introduction to Econometrics - Basic principles of probabilities - Jeffreys Axiom System

I.I.2 Jeffreys' axiom system

In this section, we introduce an axiom system (according to Jeffreys) of probability theory. Then we will deduce important theorems and formalize in a general way some results of section I.I.1.

Jeffreys uses the convention that if probabilities are expressed by numbers, larger numbers correspond to more probable statements than lower numbers. Furthermore, he uses expression (I.I.1-6) conventionally to be true.

Axiom 1: comparability

Only one of these alternatives is (exclusively and exhaustively) true.

Axiom 2: transitivity

Axiom 3: deducibility

For all propositions A: for all propositions B_i for which A implies B_i, it follows that

for all i, j.

For all propositions A: for all propositions B_i for which A does NOT imply B_i, it follows that

for all i, j.

Axiom 4:

Axiom 5:

All relations

can be expressed by numbers (i.e., a set of real increasing numbers).

This axiom implies that there are "enough" numbers such that all probability preferences can be expressed. It is also implicitly assumed that a probability of 1 is equivalent with certainty.

Thus

if A implies B.

Axiom 6:

Axiom 7: the product rule

Theorem A:

If A implies "not B", then

.

Using axiom 3 it is easy to prove this theorem:

It is obvious from this theorem and the fact that "more probable propositions" have larger probability numbers than "less probable propositions", that all probabilities should be larger or equal to 0:

Above this, it is assumed that a probability of 1 represents certainty (see previous discussion).

Theorem B:

If B is true if and only if C is true (we say that B and C are equivalent) then it follows that

Proof:

It is obvious that

(*: given) and that

(axiom 7 refers to the product rule)

Theorem C:

(axiom 7 refers to the product rule)

Theorem D:

(see also axiom 4)

Theorem E:

Theorem F:

According to Jeffreys, this can be interpreted as "... given that a set of alternatives are equally probable, exclusive and exhaustive, the probability that some one of any subset is true is the ratio of the number in that subset to the whole number of possible cases" (GRILICHES, ZVI and INTRILIGATOR, MICHAEL, D.: editors, ZELLNER, A.: author, 1983)

Theorem G:

The derivative of the cpdf can be written as

(I.I.2-1)

Theorem H:

(see also theorem F, and axiom 7)

Theorem I:

(I.I.2-2)

or

(I.I.2-3)

(also known as Bayes' theorem).