Maybe I should write an article about negative probabilities - or will I end up like Hippasus 

go for it, but it will be one of the many articles..
shudda, cudda, wudda

tbh I find AI agents more exciting.

Yeah, but all the articles I saw dance around the concept of Wigner function, hidden markov or other obscure to noncognoscenti abstract stuff. And in those particular cases only obscure it more by trying to force the abstract into the concrete.

Maybe the problem isn't that the ML practitioners ignore mathematics, but that mathematics ignores the practitioners: https://www.argmin.net/p/holding-out-for-an-explanation

Unfortunately, most post-Cold War mathematicians are not practical in my experience (e.g. number theorists, algebraic topology, groups, string theory ugh) and don't like/can't program Fortran. Fair enough. The chair of a maths group will take on a AI project if it fits with the team's weltanschauungen.

RKHS looks promising but how many in AI understand this article?

I’m just a finance guy, and can’t speak for the AI crowd, but that 4-pager (with the reference to EH Moore oddly suppressed) seems like very straightforward if slightly tedious functional analysis.

On a mostly unrelated note, I’ve had at least a couple of people working for me who took negative probabilities very seriously in practical credit derivatives applications. Although I can’t say I could wrap my head around it by myself.

The standard lore was that negative probabilities were sign of arbitrage lurking in the model, ie the prices you got weren't consistent.

That sounds realistic. Only callling them NP is a bit reckless IMHO.

Maybe the problem isn't that the ML practitioners ignore mathematics, but that mathematics ignores the practitioners: https://www.argmin.net/p/holding-out-for-an-explanation

Unfortunately, most post-Cold War mathematicians are not practical in my experience (e.g. number theorists, algebraic topology, groups, string theory ugh) and don't like/can't program Fortran. Fair enough. The chair of a maths group will take on a AI project if it fits with the team's weltanschauungen.

RKHS looks promising but how many in AI understand this article?

I’m just a finance guy, and can’t speak for the AI crowd, but that 4-pager (with the reference to EH Moore oddly suppressed) seems like very straightforward if slightly tedious functional analysis.

On a mostly unrelated note, I’ve had at least a couple of people working for me who took negative probabilities very seriously in practical credit derivatives applications. Although I can’t say I could wrap my head around it by myself.

Pity they didn’t take the so seriously in 2008!
I guess there are many reasons “negative probabilities” can pop up in such models because of incorrect assumptions or miscalibration. But then those are just error signals, correctable or not. No direct real-world interpretation. I agree with Cuchulainn.

No, this was meant to be much deeper than the usual conflation of state prices and “risk neutral” probabilities in internally inconsistent models. A related expositionary piece was published as a technical article in (drumroll) Wilmott Magazine by Gábor J. Székely. Something about flipping half coins. As I said, I couldn’t totally wrap my head around it.

Professor Székely was just a peripheral consultant. Deeper thinkers with far more academic and commercial (?) heft were at the heart of the project.

Oh, and wearing my (occasional) editor’s hat, I wouldn’t have accepted that article either.

No, this was meant to be much deeper than the usual conflation of state prices and “risk neutral” probabilities in internally inconsistent models. A related expositionary piece was published as a technical article in (drumroll) Wilmott Magazine by Gábor J. Székely. Something about flipping half coins. As I said, I couldn’t totally wrap my head around it.

https://www2.cs.uh.edu/~ml_kdd/EMOA/Ref ... /DE101.pdf

Mumbo Jumbo

Confusion is in the eye of the beholder. And, as indicated earlier, I don’t particularly disagree. But, I was not the smartest guy in the room when these things were discussed.

That looks interesting, half coins? It looks a little like imaginary numbers are making their introduction to probability

Not to mention the P&L

Dutch pension fund ABP sold Meta and Alphabet as well as Tesla

https://www.dutchnews.nl/2025/02/dutch- ... -as-tesla/

The paper bases its claims on an analysis of ABP’s investment strategy. ABP is the biggest pension fund in Europe and one of the biggest in the world with an investment portfolio totalling some €500 billion.

No, this was meant to be much deeper than the usual conflation of state prices and “risk neutral” probabilities in internally inconsistent models. A related expositionary piece was published as a technical article in (drumroll) Wilmott Magazine by Gábor J. Székely. Something about flipping half coins. As I said, I couldn’t totally wrap my head around it.

Oh, I remember someone else showed me that paper not long ago on the occasion of a similar discussion! Now it’s the second time I’ve read it, which is usually when I start to grasp things (-: It’s obscured by the rigorous mathematical theory of the half-coin, but I think the intuition is exactly what I had in mind.

An even simpler example that the paper gives could be a regular bank loan. Say Bob has a 70% probability to repay (because for institutions, there are no rock bottom numbers to the life - everything is probability derived from statistics based on data!). Then the default probability is 30%. But if Bob has collateral, it provides an extra guarantee of say 40%. And if he insures the loan, that adds another 10%. There can also be events that worsen his situation - eg if the insurer refuses to pay, let’s say that adds 5% risk.

So now, if we calculate the final default probability:
30 - 40 - 10 + 5 = -15%

:-O What happened to the missing 15%?

For a bank, risk isn’t a single absolute number - it’s an aggregation of multiple contributing factors. The bank doesn’t just look at a simple repay vs default calculation; it considers all risk adjustments, including insurance, guarantees, collateral, and external events.

This negative probability means that Bob’s loan is over-secured. the risk-mitigating factors exceed the expected default risk. In practical terms, this can mean:
1 the bank might offer Bob a lower interest rate, since his loan is more secure than usual.
2 excess security might be offloaded to other institutions, such as insurers or investors.
3 if Bob defaults, the bank might actually make money from collateral liquidation or insurance payouts - tempting!

This connects to the half-coin model from the paper. Just like a half-coin has negative probabilities but balances out when paired with another half-coin, the bank’s risk assessment isn’t just about default - it includes all mitigating factors that shift risk to other entities, making the total behave predictably.

So rather than being an error, the negative probability just represents hidden financial mechanisms at play. This risk has already been accounted for/“absorbed” elsewhere.

Banks don’t see risk as a single fixed number, but rather as a weighted balance of different components. Natural, intuitive and convenient. Is that what how your folks understand it?

Ps, I have it a bit more complicated at work and use what, upon literature research, turned out to be Choquet frequencies