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Nathaniel Virgo (Apr 03 2021 at 07:55):A very useful property for a Markov category to have is that is has conditionals. Here's the definition, from Fritz' Markov categories paper:
A lot of Markov categories arise as the Kleisli categories of monads. However, not all the Markov categories that arise that way have conditionals, and in particular, the Kleisli category of the Giry monad doesn't.
Because of this, I'm wondering whether there are any 'nice' properties that a monad can have that guarantee that its Kleisli category is a Markov category with conditionals. Is anything like that known?
Of course, we can just write the definition directly in terms of the base category, and say that the Kleisli category of has conditionals if for every morphism there exists a morphism such that
but this seems weirdly complicated, and I'm wondering if there's anything simpler that implies it. (The notation is adapted from Fritz and Perrone's bimonoidal structures paper.)
Tobias Fritz (Apr 03 2021 at 09:18):That's a great question! As far as I know it's completely open, although @Paolo Perrone has recently found a nice characterization for when a Kleisli category is a positive Markov category (where "positive" refers to the positivity axiom from my paper). This is related insofar as the existence of conditionals implies positivity. So perhaps Paolo will already know more.
Concerning the Giry monad, it's true that the Kleisli cat of the Giry monad on all measurable spaces doesn't have conditionals. However, for the Giry monad on standard Borel spaces this does hold. and this is enough for most practical purposes. This is one of the reasons for why one often restricts to standard Borel spaces (or equivalently Polish spaces), and for why conditionals can still be assumed to exist in categorical probability.
In fact, that the existence of conditionals seems to be a powerful condition that gets us pretty close to measure-theoretic probability on standard Borel spaces. For example, the only non-cartesian Kleisli category with conditionals that I'm aware of which also has countably infinite tensor products (in the form of Kolmogorov products) is the Kleisli category of the Giry monad on standard Borel spaces. (In terms of the monad, the existence of countable Kolmogorov products probably amounts to the preservation of countable filtered limits, but I haven't checked the details.)
A stronger piece of evidence for my hypothesis that the existence of conditionals gets us close to probability is that we've just proven the de Finetti theorem from it in purely categorical terms, together with two more generic assumptions (namely countable Kolmogorov products and a.s.-compatible representability). See p.23-26 of these slides for the categorical statement.
By the way, I'm curious about what categorical probability does for you. Is it an independent interest of yours, or is it already relevant for your work on the origin of life or thermodynamics?
Nathaniel Virgo (Apr 03 2021 at 11:03):Thank you for those slides and the link to the Kolmogorov product paper. I've been puzzling over sample distributions from repeated experiments recently, so it's likely to be relevant for me!
I have one question about this slide:
Is that referring to only having countable Kolmogorov products in , or does one of the other things fail as well?
Tobias Fritz said:
By the way, I'm curious about what categorical probability does for you. Is it an independent interest of yours, or is it already relevant for your work on the origin of life or thermodynamics?
It's a bit of both. I'm also interested in information theory and control theory (specifically planning tasks) - I started reading about category-theoretic probability a bit more than a year ago with the intention of applying it to those things, but I've also started to get into it for its own sake, because it's given me a new perspective on loads of other things as well. I'm hoping to also to use it for some stuff on stochastic processes and statistical mechanics. (Apropos: do you know anything that gets anywhere close to doing large deviations theory within a category-theoretic framework?)
Tobias Fritz (Apr 03 2021 at 12:12):Nathaniel Virgo said:
Thank you for those slides and the link to the Kolmogorov product paper. I've been puzzling over sample distributions from repeated experiments recently, so it's likely to be relevant for me!
Great! That sounds a lot like a de-Finetti-type situation, right?
Is that referring to only having countable Kolmogorov products in , or does one of the other things fail as well?
Yep, that's one thing that fails, and which seems strange from the categorical perspective. Because why would countable cardinality be special? The other things that fail is that doesn't have supports, and its deterministic subcategory isn't cartesian closed either. So pretty much everything fails! Given all of this, looks like a pretty bad choice of a category to work with, and one can dream about trying to find more well-behaved ones, but it seems to be a very tricky problem.
As for supports, the Lebesgues measure on still has full measure on any set with finite complement, but the intersection of all of these sets of full measure is empty. So there isn't a well-behaved notion of support. There are other candidate categories for measure-theoretic probability that do have supports, based on locales or commutative W*-algebras, but if I recally correctly cartesian closure still fails and I'm not sure about conditionals either.
It's a bit of both. I'm also interested in information theory and control theory (specifically planning tasks) - I started reading about category-theoretic probability a bit more than a year ago with the intention of applying it to those things, but I've also started to get into it for its own sake, because it's given me a new perspective on loads of other things as well. I'm hoping to also to use it for some stuff on stochastic processes and statistical mechanics. (Apropos: do you know anything that gets anywhere close to doing large deviations theory within a category-theoretic framework?)
That's nice to hear, and it would be wonderful to see applications worked out (and more theory as well too of course)!
I don't know anything going in the direction of large deviations yet, but I can well imagine that this can be done. For example, there must be a Markov category for "rate kernels", defined in terms of probabilities taking values in the tropical reals rather than the ordinary reals, corresponding to the idea that two exponentials can be added by just keeping the dominating one ignoring the other one, and two exponentials can be multiplied by adding the exponents. Then perhaps there's a functor from or even to that category. But I'm just dreaming here...