I'm working with functions (namely, representing incoherent degrees of belief) which resemble probabilities, but are actually, say, quasi-probabilites:
- their values on atomic events (here: atomic propositions) are in $[0;1]$, but
- they need not to sum up to $1$.
For example, if we have belief space $B=\{\phi_1$, $\phi_2\}$ and some credence function $c$, then it may be the case that $c(\{\phi_1\})=0.5$ and $c(\{\phi_2\})=0.7$, so obviously $c(B)\neq1$. Nevertheless, it's always $c(\{\phi_i\})\in[0;1]$.
This violation of probability laws creates many theoretical problems, so I'm in need of some proper theoretical framework. But I don't want to reinvent the wheel.
I wonder if there was any attempt to formulate alternative probability theory without the axiom of unit measure, so that not necessarily $P(\Omega)=1$?
Edit: In particular, I need something like conditional probability and Bayes' theorem.
Probability theory absent unit measure is developed by R. Christensen and T. Reichert: "Unit measure violations in pattern recognition: ambiguity and irrelevancy" Pattern Recognition, 8, No. 4 1976.