Gibbs Random Field: Positive Probability Distribution Explained

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pamparana
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Hello everyone,

I am trying to understand markov random fields and how it is related to the Gibbs measure and basically trying to understand the Gibbs-MRF equivalancy.

Anyway, while browsing Wikipedia documents, I was looking at the page on MRFs and when I came across the following line;

When the probability distribution is positive, it is also referred to as a Gibbs random field

I got confused with this. Aren't probabilities supposed to be positive. Why would be a probability distribution be negative? What does a negative probability distribution even mean? Would one use it in any possible case? So, are not ALL probability distributions gibbs random field?

Thanks,
/L
 
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Hey pamparana.

I do recall hearing about this once before in the context of Dirac, but I never really gave it much thought, however the wiki page is probably a good place to start on learning this:

http://en.wikipedia.org/wiki/Negative_probability

The above says that a guy named M.S. Bartlett did the mathematical and logical consistency analysis of these kinds of distributions, so that would be a good place to start if you can't get something immediate on google.
 
You've got me interesting in this, and a quick search came up with the following:

http://cs5824.userapi.com/u11728334/docs/8db4cf52c20c/Khrennikov_Interpretations_of_probability_34766.pdf
 
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Wow, thanks! Crazy stuff!

Would need a lot of time to process this. In any case, it seems most of the probability distributions we encounter most of the time are Gibbs fields.

Many thanks for your replies.

Luc
 
pamparana said:
Why would be a probability distribution be negative?

Instead of that, you shoud ask "Why would a probability distribution be zero?". (It could be zero at certain values.)

The statement in that article that the "the probability density is positive" doesn't imply that probability distributions can be negative. Look up clearer articles about Markov and Gibbs random fields. (There are various alternative theories of probability, but they are irrelevant to the usual treatment of Markov random fields.)