In earlier posts the relationship of the
material conditional to conditional probability and the
role of Leibniz in the early philosophy of probability where discussed. In both posts the case for taking conditional probability as fundamental was made or implied. How far this will resolve the difficulties in combining aspects of propositional logic with probability theory remains to be seen but it is worth taking time to explain a full axiomisation with conditional probability as fundamental. A further consideration is the clarification of distinct role of conditional probability in the
epistemic and the
objective (ontological) interpretations.
In his
Foundations of Probability Renyi provided an alternative axiomisation to that of Kolmogorov that takes conditional probability as the fundamental notion, otherwise he stays as close as possible to Kolmogorov. Renyi has provided a direct axiomatisation of quantitative conditional probability. In brief, Renyi's conditional probability space $(\Omega, \mathcal{F} (, \mathcal{G}, P(F | G))$ is defined as follows. The set $\Omega$ is the sample space of elementary events and $\mathcal{F}$ is a $\sigma$-field of subsets of $\Omega$ (so far as with Kolmogorov) and $\mathcal{G}$, a subset of $\mathcal{F}$ (called the
set of admissible conditions) having the properties:
(a) $ G_1, G_2 \in \mathcal{G} \Rightarrow G_1 \cup G_2 \in \mathcal{G}$,
(b) $\exists \{G_n\}, \cup_{n=1}^{\infty} G_n = \Omega,$
(c) $\emptyset \notin \mathcal{G}$,
$P$ is the conditional probability function satisfying the following four axioms.
R0. $ P : \mathcal{F} \times \mathcal{G} \rightarrow [0, 1]$,
R1. $ (\forall G \in \mathcal{G} ) , P(G | G) = 1.$
R2. $(\forall G \in \mathcal{G}) , P(\centerdot | G)$ , is a countably additive measure on $\mathcal{F}$.
R3. $(\forall G_1, G_2 \in \mathcal{G}) G_2 \subseteq G_1 \Rightarrow P(G_2 | G_1) > 0$, $$(\forall F \in \mathcal{F}) P(F|G_2 ) = { \frac{P(F \cap G_2 | G_1)}{P(G_2 | G_1)}}.$$
What has this won over the more well known Kolmogorov formulation?
A number of examples have been highlighted by Stephen Mumford, Rani Lill Anjum and Johan Arnt Myrstad in
What Tends to Be, Chapter 6. These have been analysed by them using absolute probabilities as fundamental, so a Kolmogorov type framework, and these examples will be revisited here using Renyi's formulation. The critique of Mumford
et all is based on a development of the development of an ontological point of view that has the potential to clarify physical propensities as a degree of causal disposition. The explicit clarification of the example within Renyi's axiomisation shows that by adopting it the path is open to mathematically modelling physical propensities as causal dispositions.
Here is the first example that is thought to indicate a problem with absolute probability (absolute probability will be denoted by $\mu$ below to avoid confusion with Renyi's function $P$).
P1. Let $\mu(A) = 1$ then $\mu(A | B) =1$, $\mu$ is Kolmogorov's absolute probability
We can calculate this result from Kolmogorov's conditional probability postulate as follows: since $\mu(A \cap B) = \mu(B)$, $\mu(A|B) = \mu(A \cap B)/\mu(B) = \mu(B)/\mu(B)=1$. Why is this problematic? Not at all if you stay inside the formal mathematics but is if $\mu(A|B)$ is to be understood as a degree of implication. Is it not reasonable that there must exist a a condition under which the probability of $A$ decreases? A consequence of Renyi's theory is that these Kolmogorov absolute probabilities can be obtained by conditioning on the entire sample space
$$ \mu(A) \equiv P(A|\Omega).$$
Then $\mu(A)=1$ means $P(A|\Omega)=1$ and again (by
R3.)
$$P(A|B) = { \frac{P(F \cap B | \Omega)}{P(B |\Omega)}}=1 $$
independently of choice of $B$ which must be a subset of $\Omega$. Thus, giving the same result. However if we are not working within a global conditioning on the entire $\Omega$ but on a proper subset of $\Omega$ called $G$, say, then $\mu(A)=1$ has no consequence for $P(A|G)$ and in a addition it is now possible to pick another conditioning subset of $\Omega$, $G'$, such that $G' \not\subseteq G$ then
R3. does not apply and therefore the value of $P(A|G)$ and $P(A|G')$ have to be evaluated separately. That is, it is a modelling choice. How they are evaluated depends on whether an epistemic or an objective interpretation of $P$ is being used.
A further problematic consequence of Kolmogorov's conditional probability is when $A$ and $B$ are probabilistically independent
P2. $\mu (A\cap B)=\mu(A )\mu(B)$ implies $\mu(A|B)=\mu(A)$⋅
In general Renyi's formulation does not allow this analysis to be carried out. This is because the Kolmogorov conditional probability formula only holds under special conditions, see
R3. Independence, in Renyi's scheme, is only defined with reference to some conditioning set, $C$ say. In which case probabilistic independence is defined by the condition
$$ P(A \cap B |C) = P(A|C)P(B|C)$$
and as a consequence it is only if $B \subseteq C$ that
$$ P(A|B ) = { \frac{P(A \cap B | C)}{P(B | C)}} = P(A|C)$$
that is, only if $C$ includes $B$. Therefore, $P(A|C)$ being large only implies $P(A|B)$ is equally large when $C$ inudes $B$, using the mapping to the material implication in propositional logic as shown in the earlier
post.
The third example,
P3., is that regardless of the probabilistic relation between $A$ and $B$, a third consequence of the Kolmogorov conditional probability definition is that whenever the probability of $A$ and $B$ is high $\mu(A|B)$ is high and so is $\mu(B|A)$:
P3. $(\mu(A \cap B) \sim 1) \Rightarrow((\mu(A|B) \sim 1) \land \mu(B|A) \sim 1))$
As above this carries over into Renyi's axiomisation only for the case of conditionalisation on the whole sample space. If another conditioning set is used, call it $C$ again, then
P3. does not hold in general. It does hold, or its equivalent does, when both $A$ and $B$ are subsets of $C$ but that is then a reasonable conclusion for the special case of both $A$ and $B$ included in $C$.
