Is conditional probability axiomatic?

Question

Is the conditional probability $P(A|B) = P(A \cap B)/P(B), P(B) \ne 0$ an axiom in the Kolmogorov scheme or is it derived from the other axioms?

Answer 1

Alfréd Rényi provided a foundation of Probability Theory in which conditional probabilities were taken as primitives and axioms given for those primitives.

Rényi has a book on the subject (Rényi, A. (1970).Foundations of Probability. Holden-Day) but I don't think it is easy to find. The paper "Conditional probability in Renyi spaces" by Gunnar Taraldsen is available as a PDF, and provides a starting point.

Answer 2

To give a more comprehensive answer, as GEdgar commented above in the original Kolmogorov formulation conditional probability $\mathbb{P}(\bullet\,|\,\bullet)$ is not included as an axiom, but is defined as an extra structure (see his Foundations of the Theory of Probability, 2e, Chelsea Publishing Company, p.6).

However at the same time there is a competing axiomatization of probability that takes a conditional probability $\mathbb{P}(\bullet\,|\, \bullet)$ as the primitive notion in a way compatible with the definition you cited. As Roger Purves cited in his answer, Rényi's book Foundations of Probability is based on this axiomatization. See also his earlier paper "On a new axiomatic theory of probability". According to this paper there were many others before him that detected that the axioms for a probability theory ought to be in terms of conditional probability (including Kolmogorov himself).

There are two reasons Rényi gives for the need for this conditional probability axiomatization:

1. The mathematical reason is that Kolmogorov takes the measure of a subset to be at most one, however in applications one encounters subsets of infinite measure; what is at most one ought to be conditional probabilities (this is along the lines of probability being asymptotic frequency of occurence; think laws of large numbers/ ergodic theorems).
2. The philosophical reason is that all probability is conditional probability (I find this declaration very insightful personally; I believe this not being clear is one of the things that makes naive probability confusing).

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For the sake of completeness, here is Rényi's axiomatization of probability theory, using a conditional probability as a primitive (I'm following the version in the aforementioned paper; in the book the formulation is slightly different):

A conditional probability space is a quadruple $(S,\mathcal{E},\mathcal{C},\mathbb{P})$, where

• $S$ is a set whose elements are called elementary events,
• $\mathcal{E}\leq \mathcal{P}(S)$ is a sub-$\sigma$-algebra of the $\sigma$-algebra of all subsets of $S$, whose elements are called events,
• $\mathcal{C}\subseteq \mathcal{E}$ is a nonempty collection of events whose elements are called conditions,
• $\mathbb{P}(\bullet\,|\, \bullet): \mathcal{E}\times\mathcal{C}\to[0,\infty]$ is a function satisfying

1. $\forall C\in \mathcal{C}: \mathbb{P}(C\,|\, C)=1$,
2. $\forall C\in\mathcal{C}: \mathbb{P}(\bullet\,|\, C):\mathcal{E}\to[0,\infty]$ is a measure,
3. $\forall E_1,E_2\in\mathcal{E},\forall C\in\mathcal{C}: E_2\cap C\in\mathcal{C} \implies \mathbb{P}( E_1 \,|\, E_2\cap C )\, \mathbb{P}( E_2 \,|\, C ) = \mathbb{P}( E_1\cap E_2 \,|\, C )$.