Algebraic probability

If you liked it then you prob’ly put a ring on it

June 15, 2017 — June 24, 2021


Commonly used algebraic structures over probability, as seen in, for example, Free probability.

1 Algebraic probability

Figure 1

In algebraic probability we do not take the Kolmogorov axioms as foundational, but do away with measure theory and event spaces, starting rather from RVs and expectations.

George Lowther introduces this and a connection to quantum probablility in characteristically plain-talk style 1, 2, which is one useful generalization. We can also get a handle on “non-commutative” probability this way, and are especially interested in free probability in that context. But my knowledge is exhausted now. If you wish to know more, here are some people who actually know stuff about

2 Group structures which arise in classic probability

Figure 2

There is obviously a lot going on . But I do not know it. See, however, John Baez’s category theory lists.

3 References

Aji, and McEliece. 2000. The Generalized Distributive Law.” IEEE Transactions on Information Theory.
Almost. 2011. “Semimartingales and Stochastic Integration.”
Applebaum. 2009. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 116.
Ressel. 1991. Semigroups in Probability Theory.” In Probability Measures on Groups X.
———. 2011. A Revision of Kimberling’s Results — With an Application to Max-Infinite Divisibility of Some Archimedean Copulas.” Statistics & Probability Letters.
Ruzsa, and Székely. 1988. Algebraic Probability Theory.
Speicher. 2019. Lecture Notes on ‘Free Probability Theory’.” arXiv:1908.08125 [Math].
Xia. 2019. A Simple Introduction to Free Probability Theory and Its Application to Random Matrices.” arXiv:1902.10763 [Math].