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ppauper
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negative probabilities

November 4th, 2016, 9:33 pm

There's an article in wilmott magazine
Negative Probabilities in Financial Modeling: Wilmott Magazine Article – Mark Burgin and Gunter Meissner
this is new to me
I take it probabilities still must sum to 100%, so if there's a -30% probability of Hillary winning the general, that means there's a 130% probability of Trump winning
But what does that mean? If there's a 0% probability of Hillary winning, it means she won't win, period.
But what does a -30% probability convey that 0% does not?
And a 100% probability of Trump winning means that he is guaranteed to win.
So what does a 130% probability convey that 100% does not?
 
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outrun
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Re: negative probabilities

November 4th, 2016, 9:40 pm

 
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ppauper
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Re: negative probabilities

November 4th, 2016, 9:43 pm

ah, but that's not a probability
A negative profit is a loss
 
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outrun
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Re: negative probabilities

November 4th, 2016, 10:07 pm

true.

I looked at the intro of the paper, and this is what I think: it looks like it's a side effect of using a wrong model, a sign of internal inconsistency. In quantum mechanics you have effects like entanglement that can't be explained with classical models. I you still want to try to describe what you see with classical mechanics (maybe something like the interference patterns you see with the 2 slit experiment) then you have to resort to negative probability weirdness in intermediate steps to get your desired outcome (the interference pattern). It's implicitly saying that you can explain it with classical mechanics. 
 
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outrun
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Re: negative probabilities

November 4th, 2016, 10:12 pm

fromWikipedia:

A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Although quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, the ability to yield expectation values with respect to the weights of the distribution, they all violate the third probability axiom, because regions integrated under them do NOT represent probabilities of mutually exclusive states.
 
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outrun
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Re: negative probabilities

November 4th, 2016, 10:18 pm

If trackstar and I ever set foot in the same room, and if you talk to one of us, then..
* there is a 50% chance that you talk to a Dutch person
* a 50% chance that you're talking to a female
* a 50% chance that the first name start with a T

.. that's 150%! but sums of probabilities can't get above 100%... 

we see this violation because the probabilities of these states are not mutually exclusive.
 
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Cuchulainn
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Re: negative probabilities

November 5th, 2016, 8:53 pm

 We had a discussion in the past with the usual suspects here on this topic. Can't find the link but a précis is;

1. Group I; NP is OK ("Counsel call Paul Dirac to the stand", NP is fine, we have them in physics; even Kolmogorov say it is OK in parts..).
2. Group II; NP are  caused by defects, error and faults in numerical procedures (e.g. CRR for bad parameters) and/or model.

A. I read the paper. Section 2 is a list 2 pages of axioms that are probably trying to justify NP? What is NOT defined as fast as I can see is that antievent -w (the unary minus sticks in my mathematical kraw). What does it mean and give an example of an antievent. And maybe some other toy examples just for motivation.
B. The relation negative:NP is quite tenuous. It would have preferred a discussion on the possibility that the model is wrong when r < 0.
C. Do we have any evidence that the mathematical community accept NP? Intuition is not enough IMO.
D. Some more convincing use cases to show why and when things break down in the old regime.

I'm probably missing something but it all sounds fuzzy buzzy.
 
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ppauper
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Re: negative probabilities

November 6th, 2016, 9:18 am

I still don't know what it means if we say there's a -30% probability of Hillary winning the general
 
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Cuchulainn
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Re: negative probabilities

November 6th, 2016, 9:48 am

I still don't know what it means if we say there's a -30% probability of Hillary winning the general
Is that not an antievent 'value', I think that is what needs to be defined.
Last edited by Cuchulainn on November 6th, 2016, 9:53 am, edited 1 time in total.
 
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ppauper
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Re: negative probabilities

November 6th, 2016, 9:52 am

I still don't know what it means if we say there's a -30% probability of Hillary winning the general
Is that not an antievent 'value', I think that is what is defined.
then what would the "antievent" be for hillary winning the general?
 
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Cuchulainn
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Re: negative probabilities

November 6th, 2016, 12:35 pm

I still don't know what it means if we say there's a -30% probability of Hillary winning the general
Is that not an antievent 'value', I think that is what is defined.
then what would the "antievent" be for hillary winning the general?
Look, I need a definition.
 
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list1
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Re: negative probabilities

November 6th, 2016, 12:58 pm

Measure Theory studies Hana-Jordana decomposition $$\nu = \nu^{+} - \nu^{-}$$
where [$]\nu^{+} , \nu^{-}[$] are positive measures but not probabilities. It might be one decided when [$] \nu^{-}( \Omega) = 1[$] to call the negative measure [$]- \nu^{-}[$] by negative probability
 
frolloos
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Re: negative probabilities

November 6th, 2016, 1:27 pm

Measure Theory studies Hana-Jordana decomposition $$\nu = \nu^{+} - \nu^{-}$$
where [$]\nu^{+} , \nu^{-}[$] are positive measures but not probabilities. It might be one decided when [$] \nu^{-}( \Omega) = 1[$] to call the negative measure [$]- \nu^{-}[$] by negative probability
Just so that everyone is on one page: it's Hahn-Jordan decomposition.
 
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ppauper
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Re: negative probabilities

November 6th, 2016, 1:46 pm

Is that not an antievent 'value', I think that is what is defined.
then what would the "antievent" be for hillary winning the general?
Look, I need a definition.
I'm asking a simple question: 
If there's a -30% probability of Hillary winning the general, what does that mean?
 
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outrun
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Re: negative probabilities

November 6th, 2016, 2:40 pm

It means 
1) that that must come from a "quasiprobability distribution" instead of an actual probability distribution because actual probability distribution don't allow probabilities to be negative.
2) in an actual pdf the individual states to which the pdf assigns probabilities are disjoint, uncorrelated, non-negative and sum up to 1. Either Hilary, Donald or Gary wins the election. With quasiprobability the states are coupled.

Probability of winning the election: Hilary 48%, Donald 45%, Gary 7%

Pseudo probability of winning the election: Hilary or Donald 93%, Donald 45%, other -38%

In this Pseudo probability  they don't care to correct for the dependencies of states, they don't care that values go negatieve, they just decided they want them to add up to 1, ..or may be not even that! I think in QM they have different pseudo probabilities: don't make the values add up to one, but make the sum of squares add up to one. That means that you're no longer talking about a proper pdf

It should be called "there's a -30% pseudo probability for Hillary winning the general", but people don't care about using clear definitions and helping people understand, they much rather write things that look cool. 

I think it's disgusting. Just look at how much time get's wasted here to try and understand that paper. The confusion is deliberate.