Gaussians definitely are not the only distribution. For example, the Poisson distribution is uniquely determined by its mean, so it also forms a monoid by the

Gaussians definitely are not the only distribution. For example, the Poisson distribution is uniquely determined by its mean, so it also forms a monoid by the same reasoning. The Categorical distribution is also implemented in the HLearn library as a monoid. The beta distribution is uniquely determined by two parameters called alpha and beta. Typically, these represent the sum of observations we've seen of some events. (On the same website is an example with heads and tails from coin flips.) The beta distribution forms a monoid simply by adding these parameters from each distribution. This is very similar to the Categorical distribution, and also generalizes to the Dirichlet distribution. Things get really interesting when you stop talking about distributions, and start using them in the context of actual machine learning algorithms. If you know how the Naive Bayes algorithm works, for example, it should be plausible that it forms a monoid in the same way that the Gaussian forms a monoid. The future posts will cover how to apply this to some of the "important" learning algorithms like this.