Dempster-Shafer Belief function

This last week, my bioinformatics class was introduced to this Bayesian function (see http://www.blutner.de/uncert/DSTh.pdf for more).  What got me wondering was this measure:

Given an event frame A, you can measure the probabilities not just of each individual event, but each possible set of events as follows:

For every possible subset of A, $\latex Bel(A) = \sum_{B \subseteq A} = m(B)$  

Now this looks like the beginnings of a Hopf algebra to me.  I ran it by the professor who was presenting, and he was intrigued but didn’t know the answer, although he mentioned one of his colleagues used posets in one of his papers on the mathematics of the Dempster-Shafer function.  Does it look like a possible Hopf algebra to you?  If so, do you think it would be useful to the probability behind the function?    

 

 

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3 comments

  1. Very interesting!
    You probably realized that Proposition 4.2.2 in that link is precisely the inclusion-exclusion formula applied to Definition 4.2.2. There is also a geometric interpretation of these formulas, which I can explain to you. You might also be able to express Definition 4.3.1 as a convolution product.
    This connection might be useful; a little bit of additional algebraic structure never hurt anybody! But I’d need to learn more about the applications. I’ve only browsed through this quickly. Perhaps we can find a time to discuss this with your bioinformatics professor. (Who is it?)

  2. Lisa Clayton

    Fabulous Singh. I wonder if I could talk Myriam into shifting gears.

  3. Lisa Clayton

    Rahul Singh, who is fabulous, but half a sentence disappeared here.

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