Saul Youssef has a collection of links to papers on exotic variations to probability theory. These are forms of probability theory that share many of the usual axioms of probability theory but in which the probabilities themselves lie in a set other than the non-negative reals eg. the complex numbers, the quaternions, or even the p-adics. The primary motivation is that classical mechanics plus complex probabilities looks a lot like quantum mechanics, and so if you believe in complex probabilities you no longer have to worry about things like wavefunction collapse. Unfortunately it’s all a bit confusing if you’re a frequentist.

Such exotic variations of probability theory are not its only rivals — the Kolmogorov axioms for probability theory (KPT) (using real numbers) are not uncontested, although a person could study mathematical statistics for many years before learning this (as I know from personal experience). Most statisticians seem remarkably resistant to teaching alternatives to KPT.

Although the arguments with the standard approach to PT go back to its earliest days (the decade around 1665), and have been repeated in each century since, in modern times the main criticisms have come from researchers in Artificial Intelligence. KPT does not adequately or intuitively represent all forms of uncertainty, in particular, uncertainty about uncertainty. As a consequence, Possibility theory and Dempster-Shafer Theory have arisen as alternatives to KPT. Although many of these alternative approaches can be shown to be formally equivalent to second- or higher-order versions of KPT (eg, they involve random variables distributed according to functions with parameters which are themselves random variables), humans typically find them more intuitive than KPT.

A frequentist? Are there any left?

Is anyone in the world

notfirst taught probability theory from a frequentist perspective?Joke didn’t work, huh?

Maybe you should have gone for:

“Frequentists? How often do you see those?”