Eliezer,
Ben, is the indefinite probability approach compatible with local propagation in graphical models?
Hmmm... I haven't thought about this before, but on first blush, I don't see any reason why you couldn't locally propagate indefinite probabilities through a Bayes net...
We have code that carries out Bayes Rule and other probabilistic rules using indefinite probabilities, so implementing an "indefinite Bayes net" would basically be a matter of taking Bayes net code and replacing all the applications of probability theory operations, with the appropriate C++ function implementing that operation using indefinite probabilities.
[I note that the current indefinite probabilities code makes some distributional assumptions that are best views as heuristic, but it seems to work reasonably in the practical cases where we've tried it. The current code is also too slow to use for large-scale applications; we know how to speed it up though, and this has to be done for Novamente-internal reasons anyway.]
Making an Indefinite Bayes Net might be a fun thing to do, actually.... (if there is no flaw in the above thinking...)
We don't use Bayes Nets in Novamente because Novamente's knowledge network is loopy. And the peculiarities that allow standard Bayes net belief propagation to work in standard loopy Bayes nets, don't hold up in Novamente, because of the way you have to update probabilities when you're managing a very large network in interaction with a changing world, so that different parts of which get different amounts of focus. So we use different mechanisms to avoid "repeated evidence counting" whereas in loopy Bayes nets they rely on the fact that in the standard loopy Bayes net configuration, extra evidence counting occurs in a fairly constant way across the network.... However, when you have a set of interrelated knowledge items that you know are going to be static for a while, and you want to be able to query them probabilistically, then building a Bayes Net (i.e. "freezing" part of Novamente's knowledge network and mapping it into a Bayes Net) may be useful. For this reason we've discussed incorporating Bayes Nets as an extra tool within NM, but this hasn't been prioritized since there is a lot of more basic NM stuff still unimplemented...
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