In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$.

Will Sawin described the monad structure of this endofunctor in an answer, and in the comments thread Tom Leinster brought up the Giry monad, saying, "it's almost inconceivable that there could be two different reasonable monad structures on this endofunctor, so I bet it's the same."

Here's the citation to Giry's original paper:

Michèle Giry, A categorical approach to probability theory. In B. Banaschewski, editor, Categorical Aspects of Topology and Analysis, Springer LNM 915, 1982.

It is clear that monads are very useful in general, but I am not familiar enough with the concept to see how they are useful in this context. Fortunately, as of December 2012, there are 167 citations to Giry's paper on Google Scholar, so clearly many researchers have already recognized her work. There is also a discussion on the nLab page on probability theory.

For the benefit of future researchers, I've created this community wiki thread to aggregate possible applications of the Giry monad in probability and statistics. My hope is that this thread might be a place for the structuralist and probability communities to come together and learn from each other.

If you see any interesting applications of the Giry monad, please post them here.