Conditional Independence,
Imprecise Probabilities, Null-Events and Graph-Theoretic Models
In this presentation I review two recent challenges to the use of
conditional probability as an account of probabilistic irrelevance –
challenges that threaten both the adequacy of the likelihood principle and that
threaten to make invalid the use of familiar graph-theoretic techniques for
computing conditional independencies.
1. The first challenge arises within IP
theory – Imprecise Probability theory – where a set of
probability distributions is used to depict uncertainty. That is, in IP
theory there are distinct lower and upper probabilities for events, rather than
what comes from using a single probability distribution. The
straightforward account of probabilistic irrelevance
in IP theory – that conditioning does not change the lower
and upper probabilities for an event – is not a symmetric relation!
2. The second challenge arises with
(real-valued) conditional probability given a null event, i.e. when the
conditioning event has 0 probability, as in the theory of full conditional
distributions due to work by de Finetti-Dubins-Krauss.
This, too, produces an asymmetric irrelevance relation. Of course,
the second problem can be embedded within IP theory, as when the
conditioning event is null for at least one of the probability distributions in
the IP set.
I offer responses to these challenges that
return us to a symmetric (“Bayesian”) analysis of probabilistic
irrelevance in each case.
This work on these two challenges reflects
collaborations, respectively, (1) with Mark Schervish
and Jay Kadane of CMU, and (2) with Fabio Cozman of U. Sao Paulo.