The n-Category Café

Now a morphism $\alpha$ between monads, $P$ and $Q$, gives rise to a functor, $F$, between Kleisli categories. This functor is the identity on objects, and if $f$ is in $Hom_{Kleisli P}(X, Y)$, so that it is a map between $X$ and $P(Y)$, then $F(f) = \alpha(Y) \cdot f$, which is between $X$ and $Q(Y)$, and so in $Hom_{Kleisli Q}(X, Y)$.

So we might want to look at morphisms between entries either of the right or left hand columns:

$\array{ \boldsymbol{Monad} & \boldsymbol {Kleisli category} \\ Identity & Set \\ +1 & Partial function \\ Powerset & Rel \\ Probability & Conditional distributions \\ R-module & Matrices over R}$

But where does the geometry, e.g., Fisher information metric, get in on the act? And how is it passed between different Kleisli categories? Well, there is a distance (or better, family of divergences) for unnormalised densities (eqn. (2), page 5 of this), which passes to the ordinary one for normalised densities on restriction. And there is a Fisher metric in quantum information geometry. It would be worth seeing how this relates to the probabilistic case.

Changing tack, to give an example where the geometry may do some work in Progic, imagine we have a large data base containing incidence of disease. We have someone who smokes over 40 a day, but who is also vegetarian. Now we don’t have figures for heavy smoking vegetarians, but we do have them for vegetarians and for heavy smokers individually. Now we learn the probabilities for each class of person of the four conditions {$\pm$ heart disease $\& \pm$ cancer}. So we have (estimates) for $Pr(\pm H \& \pm C | S)$ and $Pr(\pm H \& \pm C | V)$. The question is what should we say about $Pr(\pm H \& \pm C | S \& V)$?

The temptation is to draw a ‘straight line’ between the extreme points of the distributions for vegetarian and smoker. But what counts as straight? Here the idea of geodesics in the space of probability distributions appears. In these case, probability distributions over our two binary variables form a three dimensional surface, satisfying $\sum p(\pm H \& \pm C) = 1$. How would we join two points in it? There’s a good argument for taking logarithms of the coordinates, $$(log p(+H \& +C), log p(+H \& -C), log p(-H \& +C), log p(-H \& -C),$$ and looking for straight lines there.

Let me hint at one reason. If heart disease and lung cancer are independent conditional on being a vegetarian, and they are also independent conditional on being a heavy smoker, then you might think that intermediate distributions should continue possessing this independence property. But independence is expressed using a product, e.g $Pr(+H \& +C | V) = Pr(+H | V) \cdot Pr(+C | V)$, so we get

$$log Pr(+H \& +C | V) = log Pr(+H | V) + log Pr(+C | V), etc.$$

Drawing a straight line between two independent distributions represented by log coordinates, gives us a family of independent distributions.