Here's my current best guess at how Infra-Bayes works:
- We want to get worst-case guarantees for an agent using a Bayesian-like framework.
- So, let our agent be a Bayesian which models the environment as containing an adversary which chooses worst-case values for any of the things over which we want worst-case guarantees.
- That's just a standard two-player zero-sum game between the agent and the adversary, so we can import all the nice intuitive stuff from game theory.
- ... but instead of that, we're going to express everything in the unnecessarily-abstract language of measure theory and convex sets, and rederive a bunch of game theory without mentioning that that's what we're doing.
This bounty is for someone to write an intuitively-accessible infrabayes explainer in game theoretic language, and explain how the game-theoretic concepts relate to the concepts in existing presentations of infra-bayes. In short: provide a translation.
Here's a sample of the sort of thing I have in mind:
Conceptually, an infrabayesian agent is just an ordinary Bayesian game-theoretic agent, which models itself/its environment as a standard two-player zero-sum game.
In the existing presentations of infra-bayes, the two-player game is only given implicitly. The agent's strategy solves the problem:
In game-theoretic terms, the "max" represents the agent's decision, while the "min" represents the adversary's.
Much of the mathematical tractability stems from the fact that is a convex set of environments (i.e. functions from policy to probability distributions). In game-theoretic terms, the adversary's choice of strategy determines which "environment" the agent faces, and the adversary can choose from any option in . Convexity of follows from the adversary's ability to use mixed strategies: because the adversary can take a randomized mix of any two strategies available to it, the adversary can make the agent face any convex combination of (policy -> distribution) functions in . Thus, is closed under convex combinations; it's a convex set.
I'd like a writeup along roughly these conceptual lines which covers as much as possible of the major high-level definitions and results in infra-bayes to date. On the other hand, I give approximately-zero shits about all the measure theory; just state the relevant high-level results in game-theoretic language, say what they mean intuitively, maybe mention whether there's some pre-existing standard game-theory theorem which can do the job or whether the infra-bayes version of the theorem is in fact the first proof of the game-theoretic equivalent, and move on.
Alternatively, insofar as core parts of infrabayes differ from a two-player zero-sum game, or the general path I'm pointing to doesn't work, an explanation of how they differ and what the consequences are could also qualify for prize money.
Most of the headache in administering this sort of bounty is the risk that some well-intended person will write something which is not at all what I want, expecting to get paid, and then I will either have to explain how/why it's not what I want (which takes a lot of work), or I have to just accept it. To mitigate that failure mode, I'll run this as a contest: to submit, write up your explanation as a lesswrong post, then send me a message on lesswrong to make sure I'm aware of it. Deadline is end of April. I will distribute money among submissions based on my own highly-subjective judgement. If people write stuff up early, I might leave feedback on their posts, but no promises.
I will count the "sample" above as a submission in its own right - i.e. I will imagine that three-paragraph blurb were instead a three-paragraph post in its own right, and someone submitted it. That will provide a baseline for prizes to be paid out at all: if no submission adds value not already included in the three-paragraph blurb, then the three-paragraph blurb gets the prize money, i.e. I don't pay anyone.
Note that the $500 prize is probably not enough to fully pay for the amount of effort which I expect will be involved in doing this well. Others are welcome to add to the prize pool; please leave a comment if you'd like to do so.
Adding $200 to the pool. Also, I endorse the existence of more bounties/contests like this.
I'll add $500 to the pot.
adding $200 to the pool.
I wonder what other math distillations are worth funding.
Hmm, update: I've heard from someone who looked closely at infrabayes that it might not be that insightful anyway? I do not retract this bounty, but I am now expecting to be disappointed and find that infra-bayes basically simply boils down to game theory in general, rather than identifying new general shapes in the implications.
Adding $80 to the pool.
Edited from $50 to $80 after realizing market price I’m usually willing to pay for excellent math distillations.
Aside: Vanessa mentioned in person at one point that the game-theoretic perspective on infra-bayes indeed basically works, and she has a result somewhere about the equivalence. So that might prove useful, if you're looking to claim this prize.
Are there any specific results of infra-Bayesiansm that you would like to see derived using the worst case game-theoretical approach?