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Stream: theory: applied category theory

Topic: Risk aversion & convex algebras

Jules Hedges (Apr 07 2020 at 10:06):

In economics you often need to "rate" a risky outcome, modelled as a probability distribution on $\mathbb R$. The most obvious way is to take the expectation, which is an algebra $\mathbb E : \mathcal D \mathbb R \to \mathbb R$ of a probability monad $\mathcal D$ (there's multiple options, standard choices are the finite distribution monad on $\mathbf{Set}$ and the Giry monad on $\mathbf{Meas}$). In economics this is called risk neutral. But in most situations people don't actually take the expectation, most commonly humans assign greater weight to potential losses than to potential gains, which is called risk aversion. I've implicitly assumed for years that this amounts to taking a different $\mathcal D$-algebra structure on $\mathbb R$. But I never actually checked whether the algebra axioms would hold. Does anyone have any quick insight that would save me some work?

Jules Hedges (Apr 07 2020 at 10:07):

For example, I wonder whether expectation is the only possible $\mathcal D$-algebra structure (aka convex structure) on $\mathbb R$?

Soichiro Fujii (Apr 07 2020 at 12:54):

Jules Hedges said:

For example, I wonder whether expectation is the only possible $\mathcal D$-algebra structure (aka convex structure) on $\mathbb R$?

Assuming that $\mathcal{D}$ is the finite distribution monad on $\mathbf{Set}$, here is a rather trivial counterexample.

Any pointed set $(X,x)$ gives rise to a $\mathcal{D}$-algebra, with the structure map $\mathcal{D}X\longrightarrow X$ mapping every trivial convex combination $1\cdot x’$ to $x’$ and every other convex combination to the base point $x$. (This construction extends to a fully faithful functor from the category of pointed sets to the category of convex algebras, and is induced by the suitable monad morphism from $\mathcal{D}$ to the lift monad.)

To preclude such trivial counterexamples, one could ask the following: let $w\colon \mathbb{R}\longrightarrow \mathbb{R}_{>0}$ be a positive-real-valued “weight function”, and consider the function $\mathbb{E}_{w}\colon \mathcal{D}\mathbb{R}\longrightarrow \mathbb{R}$ defined by $\mathbb{E}_w(\oplus (p_i\cdot x_i))=\frac{\sum w(x_i)p_ix_i}{\sum w(x_i)p_i}$.

Q. Suppose $\mathbb{E}_w$ defines a $\mathcal{D}$-algebra. Is $w$ necessarily a constant function (so that $\mathbb{E}_w=\mathbb{E}$)?

(I don’t know the answer. :slight_smile: )

Ben MacAdam (Apr 07 2020 at 16:17):

Jules Hedges said:

In economics you often need to "rate" a risky outcome, modelled as a probability distribution on $\mathbb R$. The most obvious way is to take the expectation, which is an algebra $\mathbb E : \mathcal D \mathbb R \to \mathbb R$ of a probability monad $\mathcal D$ (there's multiple options, standard choices are the finite distribution monad on $\mathbf{Set}$ and the Giry monad on $\mathbf{Meas}$). In economics this is called risk neutral. But in most situations people don't actually take the expectation, most commonly humans assign greater weight to potential losses than to potential gains, which is called risk aversion. I've implicitly assumed for years that this amounts to taking a different $\mathcal D$-algebra structure on $\mathbb R$. But I never actually checked whether the algebra axioms would hold. Does anyone have any quick insight that would save me some work?

I hope I'm not derailing anything here, but is there a formal definition of a probability monad? I've seen constructions that yield measures and distributions (e.g. functional analytic contexts) but I haven't seen anything characterizing what a probability monad is.

Jules Hedges (Apr 07 2020 at 16:19):

I don't believe there's a general definition of "probability monad", no. You know it when you see it. Maybe that's an interesting question...

Jules Hedges (Apr 07 2020 at 16:20):

I can think of 2 properties that every probability monad has: they're all "affine monads" - they take the terminal object to itself, because there's one distribution on a point; and their strength is always commutative, which means their kleisli category admits a monoidal structure

Jules Hedges (Apr 07 2020 at 16:22):

Maybe it's possible to take the definition of a "Markov category" that Tobias Fritz worked out, and reverse engineer conditions on a monad so that its kleisli category is Markov

Jules Hedges (Apr 07 2020 at 16:23):

(That's in https://arxiv.org/abs/1908.07021)

Ben MacAdam (Apr 07 2020 at 16:25):

Oh okay - that seems quite reasonable. That does line up with the sort of intuition you were describing, and the Kleisli category of an affine monad will always give rise to a Markov category.

Ben MacAdam (Apr 07 2020 at 16:27):

Thanks! I will spend some time digging into that paper.

Paolo Perrone (Apr 08 2020 at 16:05):

Here's one thing that could be useful: lax morphisms of algebras. On categories of ordered spaces (metric, topological...) probability monads are actually 2-monads, and concave and convex functions to R (with its usual order) are exactly lax and oplax morphisms.
(https://ncatlab.org/nlab/show/Radon+monad#lax_morphisms_are_concave_maps)

Arthur Parzygnat (Apr 26 2020 at 21:48):

@Jules Hedges , @Ben MacAdam Just to clarify, Fritz' definition of a Markov category was also discovered by Cho and Jacobs, who called them CD (copy-discard) categories (https://arxiv.org/abs/1709.00322). Fritz mentions this and also work of Golubtsov (though I am unfamiliar with this latter work). Also, @Ben MacAdam, although only an opinion, whatever definition one obtains for a "probability monad," I suspect it should also include A. Westerbaan's monad in the non-commutative setting (https://arxiv.org/abs/1501.01020). Ordinary Markov categories cannot describe this structure (due to the no-cloning theorem), but I think positive subcategories may have a chance! (Note: positive categories were introduced by Fritz in the paper already mentioned.) Indeed, positive subcategories that contain the "copy" and "discard" maps (and a few tensor products) are automatically commutative (Theorem 4.20 in https://arxiv.org/pdf/2001.08375.pdf). Even if you don't care about the quantum side of the story, it might be possible that with less structure, you obtain more insight into the essence of what a probability monad is (at least, that's one of the reasons I'm exploring it). Anyway, if you happen to come across a candidate definition, please let me know. This has been on the back of my mind for quite a while now...

Arthur Parzygnat (Apr 26 2020 at 21:58):

@Paolo Perrone , Wow, I had no idea you and Fritz talked about concave and convex morphisms using lax and oplax morphisms. I did the same thing in my thesis and have been revisiting these ideas lately! I'll have to check out that paper! I do have a question though before I read it in case your response comes faster: what guarantees a fixed "directionality" for a morphism of the form $f(\lambda x+(1-\lambda)y)\rightarrow\lambda f(x)+(1-\lambda)f(y)$ for all x,y and $\lambda$? For example, in BR (reals viewed as one-object category with + as composition), you might have the arrow representing a positive number or a negative number, but that's not adequate. You just want one of them. Do you discuss something like this in your work? (oh, is that what the "ordered" part of your title is about!?)

Paolo Perrone (Apr 26 2020 at 22:11):

Yep, precisely, it's the order of R, or of any other space of values you may want :)

Arthur Parzygnat (Apr 26 2020 at 22:26):

Paolo Perrone said:

Yep, precisely, it's the order of R, or of any other space of values you may want :)

Ah, so it's a space equipped with an order, which is then viewed as a poset category?

Joe Moeller (Apr 27 2020 at 17:18):

Is the idea for Markov/copy-delete that you have basically everything for cartesian monoidal except for naturality of duplication?

Tobias Fritz (Apr 27 2020 at 17:49):

Arthur Parzygnat said:

Paolo Perrone , Wow, I had no idea you and Fritz talked about concave and convex morphisms using lax and oplax morphisms. I did the same thing in my thesis and have been revisiting these ideas lately! I'll have to check out that paper! I do have a question though before I read it in case your response comes faster: what guarantees a fixed "directionality" for a morphism of the form $f(\lambda x+(1-\lambda)y)\rightarrow\lambda f(x)+(1-\lambda)f(y)$ for all x,y and $\lambda$? For example, in BR (reals viewed as one-object category with + as composition), you might have the arrow representing a positive number or a negative number, but that's not adequate. You just want one of them. Do you discuss something like this in your work? (oh, is that what the "ordered" part of your title is about!?)

Interesting! I see that this is Example 4.4.16 of your thesis. So you actually did that before us, and in some sense in greater generality. You've considered convex functors between categories on which the operad of convex combinations acts and worked out all the relevant coherences. We've done the lower categorical thing of only considering ordered spaces, but we've instead also taken some analytic (or rather metric) on our spaces into account.

Evan Patterson (Apr 27 2020 at 17:51):

Joe Moeller said:

Is the idea for Markov/copy-delete that you have basically everything for cartesian monoidal except for naturality of duplication?

Right, that's all there is to it. So in various guises the notion goes way back. What's impressive about the work by Cho & Jacobs, Fritz, and others is that they show you can do a surprising amount of probabilistic reasoning, involving disintegration, independence, and conditional independence, under these very minimal axioms.

Tobias Fritz (Apr 27 2020 at 18:06):

Jules Hedges said:

For example, I wonder whether expectation is the only possible $\mathcal D$-algebra structure (aka convex structure) on $\mathbb R$?

Well, an obvious but uninteresting way to get other convex structures on $\mathbb{R}$ is to take any bijection $\phi : \mathbb{R} \to \mathbb{R}$ and conjugate the usual convex structure by $\phi$. But there's a more interesting sense in which the answer to your question is "no": namely for many well-behaved probability monads $P$, every $P$-algebra can be realized as a convex subset of a vector space. For example this happens for the Radon monad.

For the finite distribution monad $\mathcal{D}$, it's well-known that there also are other algebras having a flavour like the example given by @Soichiro Fujii. However, every cancellative $\mathcal{D}$-algebra, meaning one in which binary convex combinations satisfy the cancellation law, again embeds into a vector space. This is a classic theorem of Stone.

So the answer is no, $\mathcal{D}$-algebras cannot be used to model risk aversion. But perhaps (op)lax $\mathcal{D}$-algebras in the 2-category of posets can; probably @Arthur Parzygnat and @Paolo Perrone are in a better position to comment on this.

Tobias Fritz (Apr 27 2020 at 18:13):

Joe Moeller said:

Is the idea for Markov/copy-delete that you have basically everything for cartesian monoidal except for naturality of duplication?

Yes, that's exactly right! And while the aspect of "except for naturality of duplication" may sound strange and, ahem, unnatural from the categorical point of view, this unnaturality is really what probability theory is all about. A probability distribution on a product space $X \times Y$ is a morphism $I \to X \otimes Y$ in the corresponding Markov category. Composing with the deletion maps means that we can marginalize it to $I \to X$ and $I \to Y$. Now if the duplication was natural, then the monoidal structure would be cartesian, and thus the probability distribution would be uniquely determined by its marginals. But all the nontrivial statements of probability theory are concerned with how things are correlated, and the phenomenon of correlation means exactly that a distribution is not determined by its marginals.

Arthur Parzygnat (Apr 27 2020 at 18:40):

Tobias Fritz said:

Arthur Parzygnat said:

Paolo Perrone , Wow, I had no idea you and Fritz talked about concave and convex morphisms using lax and oplax morphisms. I did the same thing in my thesis and have been revisiting these ideas lately! I'll have to check out that paper! I do have a question though before I read it in case your response comes faster: what guarantees a fixed "directionality" for a morphism of the form $f(\lambda x+(1-\lambda)y)\rightarrow\lambda f(x)+(1-\lambda)f(y)$ for all x,y and $\lambda$? For example, in BR (reals viewed as one-object category with + as composition), you might have the arrow representing a positive number or a negative number, but that's not adequate. You just want one of them. Do you discuss something like this in your work? (oh, is that what the "ordered" part of your title is about!?)

Interesting! I see that this is Example 4.4.16 of your thesis. So you actually did that before us, and in some sense in greater generality. You've considered convex functors between categories on which the operad of convex combinations acts and worked out all the relevant coherences. We've done the lower categorical thing of only considering ordered spaces, but we've instead also taken some analytic (or rather metric) on our spaces into account.

I actually spent a part of today reading Perrone's thesis, and the approaches do seem a bit different and focus on different aspects. It's definitely a cool perspective and I'm really enjoying reading and learning from it. I suspect it will answer some questions I've had in the back of my mind. :slight_smile:
I never published that chapter in my thesis because that work is still in progress (so take what's in that chapter with a grain of salt). But I'm getting back into it these days, so I hope to polish that chapter and hopefully learn about quantum entropies.

Joe Moeller (Apr 27 2020 at 18:51):

If getting rid of naturality is a good idea for duplication, why isn't it for deletion?

Joe Moeller (Apr 27 2020 at 19:00):

I'm wondering if it makes sense to think about the "cartesian center" of a copy-delete monoidal category.

Joe Moeller (Apr 27 2020 at 19:00):

Meaning the maximal subcategory on which the duplication is natural.

Tobias Fritz (Apr 27 2020 at 19:06):

Joe Moeller said:

If getting rid of naturality is a good idea for duplication, why isn't it for deletion?

That's a great question, and I'm not sure if the matter is settled yet; for example, Arthur's paper does indeed get rid of naturality of deletion as well. I like keeping it around, because it's the categorical counterpart of the normalization of probability, and assuming normalization has arguably (at least) not impeded the development of probability theory. It's also a very convenient property to have in string diagram computations, as it lets you truncate all those parts of a diagram that do not have outgoing wires.

Tobias Fritz (Apr 27 2020 at 19:10):

Joe Moeller said:

I'm wondering if it makes sense to think about the "cartesian center" of a copy-delete monoidal category.

Yes, it does! Those are exactly the "deterministic morphisms", i.e. those which do not involve any randomness. They make up a cartesian monoidal subcategory which plays a central role in the theory. See Definition 10.1 in my paper for some discussion and references to earlier work on this. An interesting aspect is that the defining equation for a morphism $f$ to be deterministic says exactly that $f$ must be independent of itself. That generalizes a famous method for proving that an event has probability $0$ or $1$: you prove that it's independent of itself.

John Baez (Apr 27 2020 at 19:11):

For someone reason I'd never heard anyone say "independent of itself" before. Very nice.

Evan Patterson (Apr 27 2020 at 19:52):

Joe Moeller said:

If getting rid of naturality is a good idea for duplication, why isn't it for deletion?

I think it depends on the context and what phenomena you want to allow or disallow. An old spinoff of bicategories of relations keeps naturality for duplication, but not deletion, leading to a "bicategory of partial maps": http://www.numdam.org/article/CTGDC_1987__28_2_111_0.pdf

Joe Moeller (Apr 27 2020 at 23:26):

Great, so there's like a little lattice of increasing naturality between cartesian monoidal and (every object is cocomm comonoid) monoidal. I guess a natural question is what are group objects like in the motivating examples of these categories?