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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.

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  • $\begingroup$ I have not touch these topics before, just curious if you can "normalize" by dividing each of them by the sum of them. $\endgroup$ – BGM Mar 12 '16 at 2:33
  • $\begingroup$ You may wish to look into the theory of measures and measure spaces. I don't see whats wrong about defining conditional quasiprobability in exactly the same way as before: $QPr(E|F)=\frac{QPr(E\cap F)}{QPr(F)}$. Rearranging will give you baye's theorem. It is worth noting that conditional quasiprobabilities with this definition will always be less than or equal to one. How useful they'll be, I'm not sure. $\endgroup$ – JMoravitz Mar 12 '16 at 3:54
  • $\begingroup$ Defining independent events will be tricky as well as a result, but it could again take exactly the same definition as before that $E$ and $F$ are independent iff $QPr(E|F)=QPr(E)$, which implies that if either $F$ or $E$ have quasi-probabilities greater than one, that it is impossible for them to be independent. $\endgroup$ – JMoravitz Mar 12 '16 at 3:58
  • $\begingroup$ @BGM, I can't normalize, since I need to distinguish incoherent functions from coherent ones. $\endgroup$ – machaerus Mar 15 '16 at 16:00
  • $\begingroup$ @JMoravitz, thanks for your comments. Yes, I think that I can define these concepts just like in the standard theory. I'm not sure though if it won't have some unwanted consequences that I may overlook. That's why I'm curious if anyone already took the time to formulate and examine such axiomatic theory so that I don't have to. $\endgroup$ – machaerus Mar 15 '16 at 16:05
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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.

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