Gibbs Random Field: Positive Probability Distribution Explained

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Discussion Overview

The discussion revolves around the relationship between Markov random fields (MRFs) and Gibbs measures, specifically focusing on the concept of positive probability distributions and the implications of negative probabilities in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion regarding the statement that a positive probability distribution is referred to as a Gibbs random field, questioning the existence and meaning of negative probability distributions.
  • Another participant references a source discussing negative probabilities, suggesting that it may provide clarity on the topic.
  • A third participant shares a link to a document that appears to explore interpretations of probability, indicating interest in the broader implications of the discussion.
  • One participant asserts that the focus should be on why a probability distribution might be zero rather than negative, emphasizing that the positive density statement does not imply negative probabilities exist.
  • This participant also suggests looking for clearer articles on Markov and Gibbs random fields, noting that alternative theories of probability are not relevant to the standard treatment of MRFs.

Areas of Agreement / Disagreement

Participants exhibit a mix of curiosity and confusion regarding the concept of negative probabilities, with some expressing interest in exploring the topic further while others emphasize the traditional understanding of probability distributions. No consensus is reached on the implications of negative probabilities in relation to Gibbs random fields.

Contextual Notes

Participants highlight the need for clearer definitions and understanding of probability distributions, particularly in the context of MRFs and Gibbs measures. There is an acknowledgment of alternative theories of probability, but these are noted as potentially irrelevant to the main discussion.

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.)
 

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