Negative Probabilities: Exploring the Weird MWI Branches

[Dmitry67]

I wonder why negative probabilities should be avoided. Except the obvious “it just not possible!”. What happens if we assume that it is possible and just try to go ahead with it?

I suggest looking at it on the MWI platform (even if you don’t like MWI). Why? Because MWI is deterministic, so there are no probabilities at all. We don’t know how probability is defined in MWI. So may be it is bad, but for the discussion of the subject it is good.

So we have a number. We don’t call it a probability. It is >1 or <0? That’s fine! We know that in MWI the feeling or ‘being real’ does not depend on the value of what we call a ‘probability’.

So my question is, except for the fact that it is ‘weird’, are any observable inconsistencies on the microscopic level for the ‘frogs’ in the ‘weird’ branches with ‘probabilities’ outside of 0..1 range?

 

[Cthugha]

In some branches of physics there actually are quasi-probability distributions, which take values smaller than 0.

For example the Wigner quasi-probability distribution in quantum optics behaves in this manner. It is a phase-space distribution and takes negative values for states, which have no classical counterpart. However the negative values usually occur only in small regions and disappear if one tries to take the classical limit. The Glauber-Sudarshan P-representation behaves similarly.

 

[Dmitry67]

In some branches of physics there actually are quasi-probability distributions, which take values smaller than 0.

For example the Wigner quasi-probability distribution in quantum optics behaves in this manner. It is a phase-space distribution and takes negative values for states, which have no classical counterpart. However the negative values usually occur only in small regions and disappear if one tries to take the classical limit. The Glauber-Sudarshan P-representation behaves similarly.

Interesting.
So may be Standard Model does not 'fall apart' at High Energies at all?
There are some negative probabilities predicted by SM - but they also disappear on the classical limt and low energies...

 

[kote]

I wonder why negative probabilities should be avoided. Except the obvious “it just not possible!”. What happens if we assume that it is possible and just try to go ahead with it?

I suggest looking at it on the MWI platform (even if you don’t like MWI). Why? Because MWI is deterministic, so there are no probabilities at all. We don’t know how probability is defined in MWI. So may be it is bad, but for the discussion of the subject it is good.

So we have a number. We don’t call it a probability. It is >1 or <0? That’s fine! We know that in MWI the feeling or ‘being real’ does not depend on the value of what we call a ‘probability’.

So my question is, except for the fact that it is ‘weird’, are any observable inconsistencies on the microscopic level for the ‘frogs’ in the ‘weird’ branches with ‘probabilities’ outside of 0..1 range?

Dmitry... probability has a specific mathematical definition. Saying probability doesn't have to be bound between 0 and 1 is like saying 1 + 1 doesn't have to equal 2. It's a logical contradiction. Of course, literally any conclusion can be derived from a contradictory premise. This last point is why it can be so appealing (knowingly or unknowingly) to slip an obfuscated contradiction into a theory or explanation. Unfortunately, it just doesn't work.

 

[gel]

Dmitry... probability has a specific mathematical definition. Saying probability doesn't have to be bound between 0 and 1 is like saying 1 + 1 doesn't have to equal 2. It's a logical contradiction. Of course, literally any conclusion can be derived from a contradictory premise. This last point is why it can be so appealing (knowingly or unknowingly) to slip an obfuscated contradiction into a theory or explanation. Unfortunately, it just doesn't work.

I agree completely.

You can have negative numbers, but they aren't probabilities. Even if someone calls them that.

 

[haushofer]

Dirac wrote about this in his "interpretations of QM", in which he stated

"Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money."

In physics negative probabilities are linked to negative norm states, which means that you need ghostfields to account for that. In a certain way, these negative norm states mean that you have to many degrees of freedom, and adding these ghost fields means basically cutting down the degrees of freedom of your theory.

 

[Cthugha]

Dmitry... probability has a specific mathematical definition. Saying probability doesn't have to be bound between 0 and 1 is like saying 1 + 1 doesn't have to equal 2. It's a logical contradiction. Of course, literally any conclusion can be derived from a contradictory premise. This last point is why it can be so appealing (knowingly or unknowingly) to slip an obfuscated contradiction into a theory or explanation. Unfortunately, it just doesn't work.

This is wrong. You are jumping to conclusions. The mathematical and logical consistency of negative valued probabilities has been shown as early as 1945 by M. S. Bartlett ("Negative Probability". Math Proc Camb Phil Soc 41: 71–73). Feynman and Dirac also proposed the usage of negative probabilities as mathematical tools.

However, the interpretation of these probabilities is of course more complicated. If negative probabilities occur somewhere in QM, this is usually a hint at a state, which does not show up alone, but only as a part of a segregated system or in combination with another state. So in terms of a physical interpretation, you will need to have a look at the total system and combine a negative and a positive probability to some ordinary probability. Negative probabilities can be a sensible mathematical tool and it is just wrong to call them a logical contradiction.

 

[Dmitry67]

Also, it is possible that SOMETHING we got used to call a probability in QM becomes a probability in a low energy limit. If it is true, there will be nothing 'wrong' found in Standard Model at high energies, where negative probabilites start to appear.

 

[Naty1]

I wonder why negative probabilities should be avoided

You have to qualify what you are trying to accomplish...what's the context. They should clearly be avoided in many contexts, perhaps not in others as above posts suggest.

Kote posted

probability has a specific mathematical definition.

which is a fine classical answer. But maybe another superior definition will someday be uncovered. In the everyday world we observe, negative probabilities don't exist. You don't see a negative probability in the rolling of dice, for example.

Negative probabilities are an example of many, many mathematical results that do not appear to apply to our everyday world. We have more math than we can use in THIS universe; but if there really are an infinite number of universes, then maybe all our math falls real short...and describes only a small portion of the infinite alternatives.

Galilean transforms don't generally apply because we have relativity and Lorentz Tranforms which match experimental observations; String theory has so far many more particles than we observe; There are many possible formulations of the Einstein Tensor, but the one he picked is supported by experimental observation.

What ARE we to make of all the other math?? I am not sure.

 

[kote]

This is wrong. You are jumping to conclusions. The mathematical and logical consistency of negative valued probabilities has been shown as early as 1945 by M. S. Bartlett ("Negative Probability". Math Proc Camb Phil Soc 41: 71–73). Feynman and Dirac also proposed the usage of negative probabilities as mathematical tools.

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

Modern definition: The modern definition starts with a set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by $$\Omega=\left \{ x_1,x_2,\dots\right \}$$. It is then assumed that for each element $$x \in \Omega\,$$, an intrinsic "probability" value $$f(x)\,$$ is attached, which satisfies the following properties:

1. $$f(x)\in[0,1]\mbox{ for all }x\in \Omega\,;$$
2. $$\sum_{x\in \Omega} f(x) = 1\,.$$

That is, the probability function $$f(x)$$ lies between zero and one for every value of x in the sample space Ω, and the sum of $$f(x)$$ over all values x in the sample space Ω is equal to 1. An event is defined as any subset $$E\,$$ of the sample space $$\Omega\,$$. The probability of the event $$E\,$$ is defined as

$$P(E)=\sum_{x\in E} f(x)\,.$$​

If you'd like to redefine probability I suggest you instead just use a different word. You are free, of course, to make up whatever meaning you want for a word. However, using the standard definition, negative probabilities are a logical contradiction.

 

[kote]

You have to qualify what you are trying to accomplish...what's the context. They should clearly be avoided in many contexts, perhaps not in others as above posts suggest.

But maybe another superior definition will someday be uncovered.

There is no such thing as a superior definition of a purely analytic concept. Any difference in meaning necessarily makes a new concept. The fact that we can use the same word to represent varying concepts is an artifact of our language. Are you suggesting that maybe someday someone will come up with a better definition for the word "two"?

Also, as a purely analytic concept, probability is in no way dependent on any conception of universes or any external truths. The concept of probability, as with all mathematical concepts, has been defined a priori. Its entire meaning is contained and represented in its definition. It is not representative of some external object that we can learn more about.

 

[Dmitry67]

You have to qualify what you are trying to accomplish...what's the context.

Context - Negative probabilities predicted by Standard Model at high energies (LHC energies, not Planck/quantum gravity level)

Options:
1. Accept them and try to deal with them. Negative probabilities are an indication that (in a simplest case of the Born rule) square of a probability dencity function is a probability only at the lower energy limit
2. (Widely believed) It means that SM falls apart at high energies and frequire corrections.

 

[Cthugha]

http://en.wikipedia.org/wiki/Probability_theory
[...]
If you'd like to redefine probability I suggest you instead just use a different word. You are free, of course, to make up whatever meaning you want for a word. However, using the standard definition, negative probabilities are a logical contradiction.

There is no need to redefine it. What you hide in your quote is that you did not quote the definition of probability, but the definition of probability in terms of discrete probability distributions. You will also find another definition of probability in terms of continuous on the very page you quoted. And - most importantly - you will also find the definition of probability in the generalizing case of measure-theoretic probability theory, which also includes strange cases like the Cantor distribution and already differs significantly from the definition in terms of discrete probability distributions. From there, the generalization to negative probabilities is not a huge step. So there is no redefinition, just a generalization. The more general definition includes the one you posted as a limiting case, just as relativity includes classical mechanics as a limiting case.

 

[kote]

There is no need to redefine it. What you hide in your quote is that you did not quote the definition of probability, but the definition of probability in terms of discrete probability distributions.

I'm pretty sure that what I quoted was very plainly the "modern definition." Anything inconsistent with what I quoted, $$p$$ requires a different definition, $$p'$$. Let $$q$$ be the statement "negative probabilities are impossible."

$$p \rightarrow q$$
$$p' \rightarrow\neg q$$
$$p$$
$$\therefore\neg p'$$

You are arguing that $$p \equiv p'$$, which gives you $$p \land\neg p$$. This is the definition of a contradiction.

 

[Cthugha]

I'm pretty sure that what I quoted was very plainly the "modern definition." Anything inconsistent with what I quoted requires a different definition,

I hate repeating, but the page you quote already has two paragraphs labeled "modern definition". One for discrete probability distributions and one for continuous probability distributions, which are different definitions. Is one more modern and standard than the other or why do you choose one of them? The page also hints at one further definition by telling "The modern approach to probability theory solves these problems using measure theory to define the probability space".

It is clear that a definition of what a probability is can not make any sense without also defining what kind of probability distributions you have in mind. The definition you quoted is fine and the standard definition for probability in terms of discrete distributions, but not for probability in general. This is a huge difference. Discrete distributions are a special case for a special choice of which sample space, algebra, definition of set and measure you use. However, these choices must anyway be adapted to the kind of problem you work on. That many problems lead to discrete probability distributions does not mean that this is the standard definition for all kinds of probability distributions.

Whatever, I think we are already moving far from the original topic.

 

[kote]

Discrete distributions are a special case for a special choice of which sample space, algebra, definition of set and measure you use.

You're right. I should have been more specific. You'll notice the continuous definition of probability is also bound by 0 and 1. I chose the discrete definition because it deals with all possible outcomes and events. If we're talking about something being possible or not, we're necessarily not concerned with impossible outcomes. Continuous probability will tell you that there is a pretty good chance you will have between 2.1 and 2.9 children. This is logically impossible, a priori. Discrete probability deals with the complete set of logically consistent outcomes to a situation.

 

[Cthugha]

Continuous probability will tell you that there is a pretty good chance you will have between 2.1 and 2.9 children. This is logically impossible, a priori. Discrete probability deals with the complete set of logically consistent outcomes to a situation.

Right, I agree. In this (and almost all) cases, the given probability distribution deals with all logically consistent outcomes. However there are also situations, where your knowledge of the logically consistent outcomes is necessarily limited and you want to say something about situations, where the logically consistent outcomes are not well defined.

The easiest example is some arbitrary two-dimensional phase space, for example with position and momentum as dimensions. Looking at a classical particles under well defined circumstances, you can easily give the joint probability to find this particle at position x having momentum p. If you now perform a measurement of x alone, you will find, that p(x=x_i) will just be the integral over all possible momentum values p for a certain x_i.

For a quantum particle, the joint probability density does not have a well defined meaning because momentum is not well defined if the position is well known and you are in a position eigenstate. Nevertheless you can find an unambiguous (quasi)probability distribution, which gives you correct results for both marginal distributions in momentum and position (Wigner function). So you get a probability distribution, which gives you correct results for p(x_i) if you integrate over all momentum values at a given x_i and it gives you correct results for p(p_i) if you integrate over all position values at a given p_i. Unfortunately this gives you negative probability values at single points in phase space, but does that matter, if the joint probability density can not be measured (and is not really well defined) anyway?

 

[kote]

For a quantum particle, the joint probability density does not have a well defined meaning because momentum is not well defined if the position is well known and you are in a position eigenstate. Nevertheless you can find an unambiguous (quasi)probability distribution, which gives you correct results for both marginal distributions in momentum and position (Wigner function). So you get a probability distribution, which gives you correct results for p(x_i) if you integrate over all momentum values at a given x_i and it gives you correct results for p(p_i) if you integrate over all position values at a given p_i. Unfortunately this gives you negative probability values at single points in phase space, but does that matter, if the joint probability density can not be measured (and is not really well defined) anyway?

I would argue that QM has falsified the idea that position and momentum can both be persistent basic properties of particles. From what I understand this agrees with the Copenhagen and Bohmian interpretations. A particle with simultaneous, classical, position and momentum, is as possible as 2.6 children (assuming QM). The negative probability result is a reductio ad absurdum proof of this. Assume a definite position and momentum (garbage in) and you get negative probabilities (garbage out). It's the same as when I used to screw up on my math homework and end with the equation 1=-1. I didn't disprove logic, I just screwed up in one of my assumptions.

QM absolutely forces us give up a lot about our naive classical views. Sometimes I think the weirdness and obfuscation of the math and theory can cause us to lose sight and go too far. But yeah, as far as I can tell I agree with everything you just posted.

 

[alexepascual]

When two state vectors are orthogonal, we can find the total probability by adding their squared moduli, which in turn represent the sum of the individual probabilities. If they are not othogonal, there will be an interference term which can be negative for destructive interferenct. I guess we could interpret this term as negative probability. But it looks that we can only see negative probability when subtracted from a positive probability and the result of this subtraction can't be less than 0.

 

[Tac-Tics]

I saw this thread hoping there'd be discussion about negative probabilities. Instead, all I found was arguments over nomenclature and the usual QM nonsense.

There is no standard nomenclature for "negative probabilities" because it's a recent idea and because it's not an important area of research (scientists get by with sticking to complex numbers). They aren't the probabilities of classical probability theory, but they are a natural extension of them, just as pseudoriemannian manifolds are a natural extension to riemannian manifolds.

If anyone has any solid information on this study from a mathematical viewpoint (with or without physical applications), I'd be very interested to hear about it. The only reference I've ever seen to this idea comes from Sigfpe's blog:

http://blog.sigfpe.com/2008/04/negative-probabilities.html

 

[Dmitry67]

Very good link, thank you.
So what do you think about a question I posted earlier: negative probabilities predicted by the Standard Model - reality we need to deal with or an end of Standard model?
If it is reality that very likely we will not see any new particles (except Higgs may be) in LHC but instead we will see a new sort of quantum weirdeness with the particles we already have.

 

[kote]

They aren't the probabilities of classical probability theory, but they are a natural extension of them...

http://blog.sigfpe.com/2008/04/negative-probabilities.html

You realize the link you posted contradicts what you just said, right?

We start with tweaking probability theory a bit. One of the axioms of probability theory says that all probabilities must lie in the range zero to one. However, we could imagine relaxing this rule...​

Negative probability is a logical contradiction unless probability is redefined, in which case I'm not sure why you insist on still calling it probability.

 

[Dmitry67]

You realize the link you posted contradicts what you just said, right?

We start with tweaking probability theory a bit. One of the axioms of probability theory says that all probabilities must lie in the range zero to one. However, we could imagine relaxing this rule...​

Negative probability is a logical contradiction unless probability is redefined, in which case I'm not sure why you insist on still calling it probability.

We are talking about PHYSICS not MATHEMATICS here
Nature does not care about what "one of the axioms says"
If square fi is negative sometimes, then it is bad for the probability theory or for the Born rule, or for the both, but not for the Nature.

 

[Tac-Tics]

You realize the link you posted contradicts what you just said, right?

It does not contradict what I said, because what I said was that there is no standard nomenclature when talking about the subject. Both Dirac and Feynman had no problem calling them probabilities. As long as we're all on the same page, what does it matter?

The restriction that the probability of an event lie between zero and one need not be part of the definition. Working in a more general theory of probability (one which allows for negative probabilities), we simply label sample spaces where this property holds as "classical" and all the regular probability theory applies to it. On the other hand, non-classical sample spaces obey slightly weaker rules, which allow for more interesting behavior.

 

[kote]

We are talking about PHYSICS not MATHEMATICS here
Nature does not care about what "one of the axioms says"
If square fi is negative sometimes, then it is bad for the probability theory or for the Born rule, or for the both, but not for the Nature.

Math, by definition, cannot be wrong. Physics is the mathematical modeling of nature. The only question is which formulas best represent nature. You can't question valid formulas themselves, only their applicability. Physics won't and can't ever tell us anything about math.

But I'll let you get back to the original discussion.

 

[Dmitry67]

kote, math is a tricky thing. Let's take set theory for example.

At first, you have ZF axiomatic vs more general sets and proper classes axiom set.
Then you can accept of deny AC.
Then you can accept or deny GCH (Generalized Continium Hypotesis)
There are other statetements you can accept or deny
So you end with dozens of different and mutually inconsistent set theories
What is a right one? :)

Second, you are still ignoring my argument. Using your approach you can prove any bs in physics. So, r is vector in euclidean space. Euclidean space has curvature=0 by definition. Hence, space in our Universe is flat! Do you see the place where you make a trick?

 

[Cthugha]

The negative probability result is a reductio ad absurdum proof of this. Assume a definite position and momentum (garbage in) and you get negative probabilities (garbage out). It's the same as when I used to screw up on my math homework and end with the equation 1=-1. I didn't disprove logic, I just screwed up in one of my assumptions.

Sure, the funny thing is that when you integrate over small areas in this phase space (as small as the HUP allows) you always get the correct results - sometimes because of the small areas of negative probability you have integrated over. I was pretty stunned, when I first realized that. However, all I wanted to say is that "strange" kinds of probability distributions can give strange probability densities.

 

[Fra]

I just noticed this thread linked from the other one, and I would add my support to the original question which is justified.

The problem is not wether probability can be negative or not in the context of axiomatic mathematics - that objection is a bit simple minded I think, the problem is wether the application of standard probability theory and statistics to physics is correct or wether our understanding is still weak.

I think there is a lot to wish here, which I have advocated in a few of my posts where I refer to a reconstruction of probability theory rather as a reconstruction of the measure theory whose purpose is what we think probability theory as we know it fulfills.

Deep questions here are the physical basis of statistics. It has become common practice to ponder totally ambigous things like statistical ensembles of universes etc that apparently have no clear physical basis. Mathematically well defined perhaps, but apparently without clear physical motivation and often hardly scientific.

While I see it differently than Dmitry, and I wouldn't phrase it as negative probability, I think the spirit of intent to search for a better abstraction of physics is good.

Actually the simple proability theory, with probability spaces and normalised distributions does not always have a physical correspondance. There are more than one example of that. So if you still force this framwork ontop of physics, weird stuff can happen I agree. The question is I think to understand what his means, and if standard probability theory apparently fails (which I think it does) then what do we replace it with?

/Fredrik

 

[kote]

Actually the simple proability theory, with probability spaces and normalised distributions does not always have a physical correspondance. There are more than one example of that. So if you still force this framwork ontop of physics, weird stuff can happen I agree. The question is I think to understand what his means, and if standard probability theory apparently fails (which I think it does) then what do we replace it with?

/Fredrik

I can agree with this, but I think we have to be careful when replacing something like probability. There are conceptual rules to how we interpret reality in addition to the perhaps arbitrary way nature acts. We can't, for example, ever have 5i apples. Imaginary numbers don't have direct physical significance. I view probability the same way. An event can have no chance of happening, it can definitely be happening, or its chance might possibly be somewhere between these two options.

We can't replace probability. We may be able to come up with some other math to model QM, but it won't refer to the odds of an event happening if it isn't compatible with the probability we have now.

 

[Fra]

I can agree with this, but I think we have to be careful when replacing something like probability. There are conceptual rules to how we interpret reality in addition to the perhaps arbitrary way nature acts. We can't, for example, ever have 5i apples. Imaginary numbers don't have direct physical significance. I view probability the same way. An event can have no chance of happening, it can definitely be happening, or its chance might possibly be somewhere between these two options.

We can't replace probability. We may be able to come up with some other math to model QM, but it won't refer to the odds of an event happening if it isn't compatible with the probability we have now.

This is complex indeed, and it would take a longer response to elaborate, and I noticed this tread is old already, but it's exacly the physical basis of what and odds is, that is the question.

- For example, is the concept of odds (as described probabilistically) objective?
- What does the process where by odds are inferred or calculated look like? Is this process objective?
- What is the difference the conpcet of odds make anyway? Surely we are not talking about forming odds based on history just to write books about the frequency of things in the past, it's all about the future. So the odds make a difference to our actions, and intrinsic probabilities makes a different to physical actions. So the plain view as an odds as a simple relative frequency is not quite satisfactory.

All these questions really suggest that the odds of something happening, is not so clear after all. It is reasonably clear for repetitive experiments, but not all things are suited for this abstraction - take cosmological models. The entire notion of repeating evolving universes may be a neat mathematical abstraction, but it's physical motivation is doutful I think.

Of course, standard probability will remain. It's not that we are redefining probability, in that sense I agree with you. But that's more about naming. I think the question is more like what is the physical theory of computing odds in a rational manner, and how are rational actions formed from these computations - as we already know, plain probability doesn't do the job, this is why we have quantum mechanics. Noone really so far has IMO produced a satisfactory explanation/understanding of this, the born rule and all that.

But IMO that is just one example. The issues raised by Smoling when talking about evolving law, is another point in this discussion. He never ever mentions negative probabilities of course, but the problems are I think related and worth of discussion.

/Fredrik

 

[kote]

Of course, standard probability will remain. It's not that we are redefining probability, in that sense I agree with you. But that's more about naming. I think the question is more like what is the physical theory of computing odds in a rational manner, and how are rational actions formed from these computations - as we already know, plain probability doesn't do the job, this is why we have quantum mechanics. Noone really so far has IMO produced a satisfactory explanation/understanding of this, the born rule and all that.

Plain probability does the job quite well ! Locality and such are part of classical mechanics, and that has definitely failed. QM uses plain probability. The fact that any probability is required in QM is the problem.

Probability can't ever explain anything physical. Using probability is either an admission that our theory is incomplete or that nature is inherently random and there's nothing there to explain.

- For example, is the concept of odds (as described probabilistically) objective?

Yes. Probability is axiomatic and analytic, making it objective.

- What does the process where by odds are inferred or calculated look like? Is this process objective?

No, it's not objective! The process is called science and it is empirical, subjective, and inductive. This is, of course, when you are referring to the probability of real events. I'll add that the pure calculation part is objective but the inference part is not.

- What is the difference the conpcet of odds make anyway? Surely we are not talking about forming odds based on history just to write books about the frequency of things in the past, it's all about the future. So the odds make a difference to our actions, and intrinsic probabilities makes a different to physical actions. So the plain view as an odds as a simple relative frequency is not quite satisfactory.

The problem of induction in all of science, not just probabilistic models, is that we are always looking at the frequency of historical events. We assume that future events will behave with the same probabilities (whether the historical probability is 1 or some other number). Agreed, it doesn't seem satisfactory.

 

[nitin.jain]

Sure, the funny thing is that when you integrate over small areas in this phase space (as small as the HUP allows) you always get the correct results - sometimes because of the small areas of negative probability you have integrated over. I was pretty stunned, when I first realized that. However, all I wanted to say is that "strange" kinds of probability distributions can give strange probability densities.

Is that not the very reason they (for example, the Wigner function) are 'quasi-probability' distributions? As in, the value of W(Xa,Pa), with (Xa,Pa) being some point in the phase space may or may not make sense... however, any physical region that Nature (in other words, HUP) allows you to consider, shall yield a classical probability?

 

[Demystifier]

I wonder why negative probabilities should be avoided.

Positive probabilities can be measured, as relative frequencies. How would you measure a negative probability?

 

[Dmitry67]

Can events be more then 100% correlated? You know the answer: EPR.

So, if something have probability of 150% then when you do trials, you get 100%. You won't suspect anything wrong until you try to decrease the probability. You will find that still you get 100% dispite all odds, odds which must decrease the probability -garanteed!

 

[Fra]

Positive probabilities can be measured, as relative frequencies. How would you measure a negative probability?

In the direct sense one can't, but one way is if you see probability not as a mathematical definition, but in context as a measure of degree of belief or as a way to rate evidence.

Then, during an inference process, evidence is COMBINED, and then questions arises howto combine two pieces of evidence, sometimes an outcome is the result of inference, rather than "direct counting of first line evidence".

In this sense, one is easily lead to generalisations of such a measure of evidence, and the rules of howto merge different evidence, during constraints of limited resources, where storing complete time histories just doesn't work.

In this sense, things happens where the properties of this "measure of evidence" can not have the properties that the standard probability have.

I think this is what Dmitry means, and the question makes sense to me if interpreted in this sense (in context; as as tool of inference) rather just as in a pure mathematical context, where the question easily seems silly.

/Fredrik