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list1
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Re: negative probabilities

November 6th, 2016, 4:04 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.
No, it's just a hint where it was come from. Historically probability was associated with frequency. That is sufficient interpretation with fixed states observations. The simple illustrations are coin, dice trials. When we consider limit transition to continuous states frequency as a definition of probability fails and measure model is developed. Probability is a normalised, positive case of a general measure. In physics they study a charge which can be negative and positive additive function forms by electrons and protons. With such analogy in abstract measure theory they study math. charge as an additive function taking negative and positive values and admitting infinite values only of one sign. Of course the math charge as only additive function does not related to frequency or probability but if we deal with the probability properties which use only additivity then it might be an intention to replace charge by the word probability as far as we do not think about their distinctions. There is no sense talk about -.0.3 chance to observe an event in a dice trial.The problem with negative probability even not a definition but a simple discrete state example that illustrate negative chance. If simple nontrivial example are available then one can try present a formal definition.
 
frolloos
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Re: negative probabilities

November 6th, 2016, 8:37 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.
No, it's just a hint where it was come from. Historically probability was associated with frequency. That is sufficient interpretation with fixed states observations. The simple illustrations are coin, dice trials. When we consider limit transition to continuous states frequency as a definition of probability fails and measure model is developed. Probability is a normalised, positive case of a general measure. In physics they study a charge which can be negative and positive additive function forms by electrons and protons. With such analogy in abstract measure theory they study math. charge as an additive function taking negative and positive values and admitting infinite values only of one sign. Of course the math charge as only additive function does not related to frequency or probability but if we deal with the probability properties which use only additivity then it might be an intention to replace charge by the word probability as far as we do not think about their distinctions. There is no sense talk about -.0.3 chance to observe an event in a dice trial.The problem with negative probability even not a definition but a simple discrete state example that illustrate negative chance. If simple nontrivial example are available then one can try present a formal definition.
I dont know anything about measure theory. I am just saying it is not Hana-Jordana but Hahn-Jordan so that someone googling doesn't end up on a most popular girls names 2016 page.
 
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Cuchulainn
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Re: negative probabilities

November 6th, 2016, 9:07 pm

Just so that everyone is on one page: it's Hahn-Jordan decomposition.
No, it's just a hint where it was come from. Historically probability was associated with frequency. That is sufficient interpretation with fixed states observations. The simple illustrations are coin, dice trials. When we consider limit transition to continuous states frequency as a definition of probability fails and measure model is developed. Probability is a normalised, positive case of a general measure. In physics they study a charge which can be negative and positive additive function forms by electrons and protons. With such analogy in abstract measure theory they study math. charge as an additive function taking negative and positive values and admitting infinite values only of one sign. Of course the math charge as only additive function does not related to frequency or probability but if we deal with the probability properties which use only additivity then it might be an intention to replace charge by the word probability as far as we do not think about their distinctions. There is no sense talk about -.0.3 chance to observe an event in a dice trial.The problem with negative probability even not a definition but a simple discrete state example that illustrate negative chance. If simple nontrivial example are available then one can try present a formal definition.
I dont know anything about measure theory.  I am just saying it is not Hana-Jordana but Hahn-Jordan so that someone googling doesn't end up on a most popular girls names 2016 page.
Notwithstanding, it is list1 who is posing the right questions. A vital clue is probability measure.
http://mathworld.wolfram.com/ProbabilityMeasure.html
Last edited by Cuchulainn on November 6th, 2016, 9:10 pm, edited 1 time in total.
 
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Cuchulainn
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Re: negative probabilities

November 6th, 2016, 9:10 pm

I still don't know what it means if we say there's a -30% probability of Hillary winning the general
It probably means nothing., All over in 2 days time.
 
frolloos
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Re: negative probabilities

November 6th, 2016, 9:18 pm

I agree. And I think I have said this before - I do think list1 is strong in pure measure theory and probability. But for some abd mysterious reason Black-Scholes / finance is his 'angstgegner'.
 
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list1
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Re: negative probabilities

November 6th, 2016, 11:25 pm

Just so that everyone is on one page: it's Hahn-Jordan decomposition.
No, it's just a hint where it was come from. Historically probability was associated with frequency. That is sufficient interpretation with fixed states observations. The simple illustrations are coin, dice trials. When we consider limit transition to continuous states frequency as a definition of probability fails and measure model is developed. Probability is a normalised, positive case of a general measure. In physics they study a charge which can be negative and positive additive function forms by electrons and protons. With such analogy in abstract measure theory they study math. charge as an additive function taking negative and positive values and admitting infinite values only of one sign. Of course the math charge as only additive function does not related to frequency or probability but if we deal with the probability properties which use only additivity then it might be an intention to replace charge by the word probability as far as we do not think about their distinctions. There is no sense talk about -.0.3 chance to observe an event in a dice trial.The problem with negative probability even not a definition but a simple discrete state example that illustrate negative chance. If simple nontrivial example are available then one can try present a formal definition.
I dont know anything about measure theory.  I am just saying it is not Hana-Jordana but Hahn-Jordan so that someone googling doesn't end up on a most popular girls names 2016 page.
I referred on H-J decomposition to point on formal example in which measure with negative values comes and nothing more, ie measure with negative values does not something unusual. In electrostatic when the total charge is a sum of positive and negative charges is a illustration from physics. But the charge from measure theory is more general issue and generally speaking the reduction of the charge to probability measure does not look reasonable .
 
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Cuchulainn
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Re: negative probabilities

November 7th, 2016, 4:35 pm

Here a claim of an example:

"Let us consider the situation when an attentive person A with the high knowledge of English writes some text T. We may ask what the probability is for the word “texxt” or “wrod” to appear in his text T. Conventional probability theory gives 0 as the answer. However, we all know that there are usually misprints. So, due to such a misprint this word may appear but then it would be corrected. In terms of extended probability, a negative value (say, −0.1) of the probability for the word “texxt” to appear in his text T means that this word may appear due to a misprint but then it’ll be corrected and will not be present in the text T."

????
 
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outrun
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Re: negative probabilities

November 7th, 2016, 5:07 pm

People creating their own little garbage island in search for a throne...
 
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list1
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Re: negative probabilities

November 7th, 2016, 5:13 pm

Here a claim of an example:

"Let us consider the situation when an attentive person A with the high knowledge of English writes some text T. We may ask what the probability is for the word “texxt” or “wrod” to appear in his text T. Conventional probability theory gives 0 as the answer. However, we all know that there are usually misprints. So, due to such a misprint this word may appear but then it would be corrected. In terms of extended probability, a negative value (say, −0.1) of the probability for the word “texxt” to appear in his text T means that this word may appear due to a misprint but then it’ll be corrected and will not be present in the text T."

????
The term of probability in your example should be refined as if we can associate probability with frequency a priori chance zero does not match to its refining adjustment.
 
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Cuchulainn
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Re: negative probabilities

November 8th, 2016, 10:40 am

Here a claim of an example:

"Let us consider the situation when an attentive person A with the high knowledge of English writes some text T. We may ask what the probability is for the word “texxt” or “wrod” to appear in his text T. Conventional probability theory gives 0 as the answer. However, we all know that there are usually misprints. So, due to such a misprint this word may appear but then it would be corrected. In terms of extended probability, a negative value (say, −0.1) of the probability for the word “texxt” to appear in his text T means that this word may appear due to a misprint but then it’ll be corrected and will not be present in the text T."

????
The term of probability in your example should be refined as if we can associate probability with frequency a priori chance zero does not match to its refining adjustment.
I still don't get what [$]-w[$]  is 3rd line of page 61. It's not defined as far as I can see and we are getting various interpretations, in fact.
 
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list1
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Re: negative probabilities

November 8th, 2016, 5:09 pm

Here a claim of an example:

"Let us consider the situation when an attentive person A with the high knowledge of English writes some text T. We may ask what the probability is for the word “texxt” or “wrod” to appear in his text T. Conventional probability theory gives 0 as the answer. However, we all know that there are usually misprints. So, due to such a misprint this word may appear but then it would be corrected. In terms of extended probability, a negative value (say, −0.1) of the probability for the word “texxt” to appear in his text T means that this word may appear due to a misprint but then it’ll be corrected and will not be present in the text T."

????
The term of probability in your example should be refined as if we can associate probability with frequency a priori chance zero does not match to its refining adjustment.
I still don't get what [$]-w[$]  is 3rd line of page 61. It's not defined as far as I can see and we are getting various interpretations, in fact.
As a new object one should present simple examples to illustrate negative probability notions. In the paper there are a lot of abstract mathematics but not enough common sense examples. 
Financial applications in the paper make sense for the standard probability. If they try to apply negative probability then the question is whether does the negative probability presents other values. If the answer is no then with probability 1 it looks like wrong. If the answer is yes then it is a positive chance that it might make sense but the next question is why we should apply low understanding negative probabilities  when we can apply well known with clear sense standard probability notion. Only one case which will prove that negative probability has an advantage is the case when positive probability does not sufficient to present a real world problem while negative one successfully resolve the problem.  
 
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Cuchulainn
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Re: negative probabilities

May 2nd, 2018, 11:24 am

Physicists don't use Cauchy sequences it seems.

"As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases. ... Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua."
 
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Traden4Alpha
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Re: negative probabilities

May 2nd, 2018, 1:11 pm

Physicists don't use Cauchy sequences it seems.

"As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases. ... Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua."
Interesting!

Perhaps this is because the population total of a physical quantity (mass, energy, velocity, charge, etc.) is bounded and therefore the per-particle or per-unit average is bounded, too. (But if the universe is infinite then.....)

Does Cauchy ever occur in the real world or is it all the minds of mathematicians?
 
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Cuchulainn
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Re: negative probabilities

May 2nd, 2018, 4:22 pm

Cauchy sequences have amazing properties that can be used to understand the behavior of a system as time progresses. They are heavily used in fields like satellite design, manufacturing, construction, treatment plants, and so on.

Let’s say we want to analyze how a structure is going to respond to weather conditions in the next couple of decades. Based on historical data, if we know that this can be modeled as a Cauchy sequence, we can extract a lot of insights. The goal is to model the system and make predictions about how this structure is going to react under various conditions. If we know that a given physical process can be described using a Cauchy sequence, we know that the process will converge to a single point.

And squeeze
https://en.wikipedia.org/wiki/Squeeze_t ... th_example