Stoch is the category of Measurable spaces and stochastic maps. It is the Klesli category of the Giry monad. Deterministic theories form a subcategory of Stoch. Specifically, the objects are just objects in Stoch and the morphisms are all isomorphisms. Call this category Det-Stoch. Is this category a factorization of the Giry Monad? Does it restrict the Giry monad to additive Gaussian noise? The reason I ask this, is that the probability distributions we should expect to see when taking measurements of systems that obey deterministic theories should just be additive noise. This is in contrast to quantum theory, where the theory itself has a delicate interplay between the physics and information and randomness.
$\begingroup$
$\endgroup$
2
-
2$\begingroup$ What do you mean by "a factorization of the Giry monad"? Are you asking whether there is an adjunction $Pol^\to_\leftarrow Det-Stoch$ (where $Pol$ is the category of Polish spaces) whose induced monad on $Pol$ is the Giry monad? If that's what you mean, then the answer is obviously no: because $Det-Stoch$ has no noninvertible morphisms, any functor that factors through $Det-Stoch$ will send all morphisms to isomorphisms, and the Giry monad does not do this. $\endgroup$– Tim CampionCommented Mar 5, 2018 at 13:35
-
$\begingroup$ Hi Tim, Yes, I what I meant by a factorization of the Giry Monad. I guess you have answered my question. $\endgroup$– Ben SprottCommented Mar 5, 2018 at 17:17
Add a comment
|