6.10 Plausibility Systems

The discussion in Sections 6.2–6.9 was carried out in terms of probability. However, probability did not play a particularly special role in the discussion; everything I said carries over without change to other representations of uncertainty. I briefly touch on some of the issues in the context of plausibility; all the other representations are just special cases.

As expected, a plausibility system is a tuple (, ₁, …, _n), where is a system and ₁,…, _n are plausibility assignments. CONS makes sense without change for plausibility systems; SDP, UNIF, PRIOR, and CP have obvious analogues (just replacing "probability" with "plausibility" everywhere in the definitions). However, for PRIOR and CP to make sense, there must be a conditional plausibility measure on the set of runs, so that conditioning can be applied. Furthermore, the sets (_i(r, m)) must be in ′, so that it makes sense to condition on them. (Recall that in Section 6.4 I assumed that μ_r,i((_i(r, m))) > 0 to ensure that conditioning on ((r, m)) is always possible in the case of probability.) I assume that in the case of plausibility, these assumptions are part of PRIOR.

It also makes sense to talk about a Markovian conditional plausibility measure, defined by the obvious analogue of Definition 6.5.1. However, Markovian probability measures are mainly of interest because of Proposition 6.5.2. For the analogue of this proposition to hold for plausibility measures, there must be analogues of + and . Thus, Markovian plausibility measures are mainly of interest in algebraic cps's (see Definition 3.9.1).