Learning from limited signals

This post is largely a request for places to look for work in formal epistemology (broadly construed) that looks at what happens when the people with the evidence are constrained in how fine-grained a signal they can send to the people who need to make a decision.

The proximal inspiration for this question comes from The Ordinal Society and The Unaccountability Machine, both of which involve systems where there are limitations not exactly on who can talk to who, but on how much information can be passed along various channels. And I don’t have a good sense on how to best build that kind of limitation into the kind of models we normally use in formal/social epistemology.

Here’s an instance (in formal epistemology speak) of what I mean by a case where there is a limitation on the information that can be passed along.

There is an urn with some red balls and blue balls in it. Boss has to make a decision that turns on the proportion of balls that are red. Call this proportion r. For this case, make three simplifying assumptions:

1. Boss initially has a flat probability distribution over the possible values of r. So for any interval [x, y] in [0, 1], Boss’s credence that r is in [x, y] is y-x.

2. Boss is going to be asked a simple yes/no question of the form: Is r above or below some threshold?

3. Boss’s loss function is symmetric - all that we care about is maximising the probability that Boss says the true thing about whether r is above or below the threshold.

Boss can’t sample the urn. But Boss has 10 Underlings who can. Here are the constraints on the Underlings.

1. Each of them can sample the urn 10 times, with replacement.

2. None of them can communicate with any other Underling.

3. At the end, each of them can (separately) send Boss a 1-bit signal. That is, Boss can give them a yes/no question about their sample in advance, and at the end, each Underling will tell Boss what the answer is.

Here’s an example of that. Say we care about whether r is above or below 0.81. And we tell each Underling to answer the following question: Did you see at least 7 red balls in your sample of 10? And then Boss says that r is above 0.81 iff all 10 Underlings say yes.

That’s not a bad rule - it gets the right answer in 95% of cases. But we can do a bit better. Here are two other approaches we could use:

• Question to Underlings: Did you see at least 8 red balls?
  Decision Rule: Say r > 0.81 iff at least 8 Underlings say yes.

• Question to Underlings: Did you see at least 9 red balls?
  Decision Rule: Say r > 0.81 iff at least 5 Underlings say yes.

I’m not actually sure which of these rules is better. I ran 400,000 trials, and the second had 119 fewer errors. That’s not quite conclusive I think, but fairly good evidence that it’s the better approach.

What I am sure about is that one or other of these rules is the best rule satisfying two further constraints. First, each Underling is asked the same question. Second, that question is just about how many red balls they saw, and is insensitive to the order they saw them in. I have no idea what happens if you drop those constraints.

So that’s the kind of question I’m interested in: what is the optimal question to ask the Underlings? More generally, I’m interested in what patterns there might be in how the answer to this question varies depending on the initial setup, the credal distribution, or the loss function.

This is very abstract, and a long way from the real world cases I said were the inspiration for this. To make it somewhat more realistic, it would help to add in one or both of the following features:

• Different underlings get different kinds of evidence, or evidence from different parts of the subject matter. That would make the situation more like one where Boss is an actual boss of a complex organisation.

• The different questions that can get asked affect the rational credal distribution for r, e.g., if we’re looking for equilibrium solutions to a non-cooperative game where Adversary gets to set r after learning what questions get asked. That would make the situation more like one where how we manage complexity in the world, through things like rating systems, affects what the world ends up being like.

But the initial case is hard enough, and I’d be very interested in tips on where to look for more work on how to model these kinds of bandwidth constraints.