# Can one use complex numbers in probability?

I have never thought about using complex numbers in probability. I am examining Bayes Theorem, and attempting to relate it to projective geometry and this question came to mind. I am not talking about probability amplitude or any application to quantuum theory. I am asking about pure math.

Could someone give a simple example where the answer to a question is eg 2+i?

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In my opinion: It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. – Git Gud May 9 '13 at 22:20
The characteristic function $E(e^{it X})$ is useful. – André Nicolas May 9 '13 at 22:33
i may be asking a very stupid question… i tend to do this… so a simple pointer in the right direction would help... – happyseaurchin May 9 '13 at 22:46
Easy $1+i+1=2+i$ – TylerHG Jun 1 '14 at 2:29
This is not a stupid question (actually no question is stupid) since it can lead to either of 2 outcomes: 1. a correct answer and thus illumination or, 2. a breakthrough. And in this case there is such a formulation (which can be a breakthrough), see my answer – Nikos M. Jun 1 '14 at 2:38

It's not a stupid question, but it's not the sort of question that has an answer.

The big exciting thing that's going on with complex numbers and probability is Schramm Lowener Evolution.

There was a slightly obscure idea in complex analysis called the Lowener differential equation. Essentially given any function on the real numbers I can associate a path in the complex plane that never crosses itself. It was clever, but never really took off as an idea.

Then a guy called Oded Schramm came along and solved the Lowener equation for a Brownian motion (A very random function) instead of a deterministic function.

The solutions of these are random paths in the complex plane that never cross themselves. It turns out they are also conformally invariant. (If I apply a differentiable complex function to the solution of the Lowener equation the probability distribution of the solution doesn't change.)

This area has seen a massive explosion of research in the last ten years. And there's an attempt to use these ideas to try and unite Quantum mechanics with gravity.

That's the only thing I can think of where there's a real union of complex analysis and probability theory. It's definitely not something I'm an expert on, and it's a seriously hard subject. (Two fields medals in the last ten years.) But it's probably worth your time having a look at what's going on in this area.

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In the approach to uncertainty in which expectation is primitive (see, e.g., Bruno de Finetti, Frank Lad, Peter Whittle), a probability is defined as the expectation of the indicator function of a proposition. Random quantities and expectations can certainty be complex-valued, but probabilities are always real-valued because the indicator function only takes the real values 0 and 1.

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Yes, quantum mechanics is an application of complex probability theory. If complex probabilities did not hold up in "pure math" they would continue to fall apart in QM and produce nonsensical results.

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Quantum probability is a bit more than adding complex numbers to probability theory. In classical probability theory if two events $A$ and $B$ are independent then $\mathbb P(A\cap B) = \mathbb P(A)\times\mathbb P(B)$. In quantum probability multiplication is not commutitive. So $\mathbb P(A)\times\mathbb P(B)\neq\mathbb P(B)\times\mathbb P(A)$ so the probability of $A$ and $B$ depends on the order in which you multiply (or observe) them. It is true that complex numbers play a crucial role in QM. But I think it's misleading to say this is an application of complex probability theory. – Tim May 9 '13 at 22:57
Ummmm...that's kind of off the mark. The Copenhagen interpretation of QM (to which Einstein famously did not subscribe) states that the squared magnitude of the normalized wave function is a probability. – Ron Gordon May 9 '13 at 22:57
No sane physicist would ever talk about complex probabilities. The term used is probability amplitude to reflect the fact that the magnitude is all that's relevant to compute an actual probability ( – Sharkos May 10 '13 at 0:40
Perhaps it is indeed misleading to call this an application of complex probability theory since many calculations in QM are more interested maintaing the phase portion of a formula than it's normalized physical component. An example of this is indeed (as Sharkos points out) probability amplitude. – Phy51x May 10 '13 at 5:15
@Sharkos Regarding sane physicists, one famous physicist once observed (regarding the first years of QM), that this is just madness. And another replied, but it has a method. Now no sane physicist would claim that QM is madness. But this same sane physicist (esp. if a famous one) might still object to some other formulation which may be years ahead (all is in the method) – Nikos M. Jun 1 '14 at 2:57

One can indeed do that, for example have (exotic) probabilities which can be negative, or even greater than one etc..

For a simple introduction check this link

As a bonus, even quantum mechanics can be formulated (at least to some extend) as complex probabilities

Check also this post (which covers a different aspect), but which can be a possible application

UPDATE:

adding a comment i made here, since it answers criticism on complex probability, but also puts the discussion into the correct framework.

The very starting words on Kolmogorov's landmark work on axiomatization of probability theory states (not exact quote): "A probability is a number assigned to an event in an event space..". Now use some other number (and adjust the arithmetic accordingly) and you have a probability theory (maybe exotic but not less real)

In fact something very similar to exotic probabilities already exists in conventional (lets say) probability theory. It is called random variables. It seems people who downvoted this answer do not have a grasp of the extend of the random variables concept. What are random variables? Random Variables are ways to assign arbitrary numbers, arithmetic and measure to probabilistic events. What is exotic probability? (e.g complex probability), exactly the same. As some say same difference.

One can ask here, so if random variables already cover the functionality of exotic probability and is already a part of traditional probability theory, why should we be concerned with exotic places? Nice question, it is not necesary, in the same sense it is not necesary to use complex numbers instead just use 2 real numbers with specific rules of combination. Or use an array of numbers with special rules of combination instead of matrices, and so on and so forth. One can indeed do that, they are isomoprhic representations. It just happens some are natural representations and provide intuition, flexibility (and potentialy greater understanding) while others are not.

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