Complex (i.e., Imaginary) Probability

Question

I’ve been doing some numerical approximation of probability distributions. For continuous $\operatorname{PDF}$s (or $\operatorname{CDF}$s) greater smoothness can be exploited to achieve more efficient approximations (i.e., greater accuracy with fewer parameters).

In numerical analysis the extreme class of smoothness is often a set of functions that are analytic over some significant portion of the complex plane. For example, the Discrete Fourier Transform converges like $O(1/n)$ for functions with discontinuities on the real line, but if the function is analytic on a strip about the real line (with a bounded integral around the boundary of the domain) then the same Fourier Methods will converge like $O(e^{-\sqrt{n}})$. Similar exploitation of smoothness exists for orthogonal polynomials, Sinc-Methods, and some types of wavelet approximation.

I have examples where assuming that the $\operatorname{CDF}$ or $\operatorname{PDF}$ has an “analytic continuation” onto a region of the complex plane (i.e., strictly containing the original domain) yields significantly more efficient numerical approximations.

But this raises a philosophical question, to say that $\operatorname{PDF}(x)$ is analytic in some domain (of positive measure) implies that both $x$ and the probability density are complex valued.

For this to make sense do I need complex probabilities?

I think the answer is yes. I’d like the answer to be yes. For the functions describing a distribution to be analytic (i.e., over a non-trivial domain) the probabilities will have to take-on values over a non-trivial portion of the complex plane. That is, the image of an analytic function over a non-trivial domain must be a non-trivial region.

But at some level I don’t know what this would mean.

I see papers on real valued probabilities of complex signals or complex waveforms but am looking for the concept of complex probability.

Answer 1

Since the question did not end up being closed, I will give an answer extending the comments. You do not need complex probabilities. An analytic function might amount to probability on some of its domain, and be something else otherwise.

However, since we get analytic continuations, it is natural to ask whether the resulting values can reasonably be considered (in some sense) complex probabilities or are otherwise meaningful. Since analytic continuation is a natural concept, I suspect that in some cases the values are meaningful/useful, and (inspired by this question) I asked this separately, Interpretations of analytic continuations of CDFs to complex probabilities. However, there are good reasons for such meanings to be probability-distribution-specific rather than being a general "complex probability".

While CDFs are one-to-one for nonzero densities, their analytic continuations might not be, and even worse, they might be multivalued in that the value depends on the branch taken. Moreover, an $ε$-change to CDF (or just $ε$-change to PDF on a bounded set) can (through large values of higher derivatives) have an unbounded effect on the analytic continuation of CDF, which is contrary to how probability works.

That said it is difficult to rule out a natural concept for complex numbers that for real numbers in [0,1] amounts to probability. Quantum amplitudes are complex (though real quantum amplitudes are not probabilities). Extrapolations of probabilities beyond [0,1] can be viewed as (signed) measures. In nonstandard analysis, probabilities can be infinitesimal (though they are still ordered and additive). And there is even a noncommutative probability theory.