nostalgebraist - bayes: a kinda-sorta masterpost

At a very rough guess, I think Bayesian thinking is helpful in 50-80% of nontrivial epistemic problems, more than p-adic analysis.

How might the law-type properties be indirectly relevant? Here are some cases:

• In game theory it's pretty common to assume that the players are Bayesian about certain properties of the environment (see Bayesian game). Some generality is lost by doing so (after all, reasoning about non-Bayesian players might be useful), but, due to the complete class theorems, less generality is lost than one might think, since (with some caveats) all policies that are not strictly dominated are Bayesian policies with respect to some prior.

• Sometimes likelihood ratios for different theories with respect to some test can be computed or approximated, e.g. in physics. Bayes' rule yields a relationship between the prior and posterior probability. Even in the absence of a way to determine what the right prior for the different theories is, if we can form a set of "plausible" priors (e.g. based on parsimony of the different theories and existing evidence), then Bayes' rule then yields a set of "plausible" posteriors, which can be narrow even if the set of plausible priors was broad.

• Bayes' rule implies properties about belief updates such as conservation of expected evidence [LW · GW]. If I expect my beliefs about some proposition to update in a particular direction in expectation, then I am expecting myself to violate Bayes' rule, which implies (by CCT) that, if the set of decision problems I might face is sufficiently rich, I expect my beliefs to yield some strictly dominated decision rule. It is not clear what to do in this state of knowledge, but the fact that my decision rule is currently strictly dominated does imply that I am somewhat likely to make better decisions if I think about the structure of my beliefs, and where the inconsistency is coming from. (In effect, noticing violations of Bayes' rule is a diagnostic tool similar to noticing violations of logical consistency)

I do think that some advocacy of Bayesianism has been overly ambitious, for the reasons stated in your post as well as those in this post [LW · GW]. I think Jaynes in particular is overly ambitious in applications of Bayesianism, such as in recommending maximum-entropy models as an epistemological principle rather than as a useful tool. And I think this post by Eliezer [LW · GW] (which you discussed) overreaches in a few ways. I still think that "Strong Bayesianism" as you defined it is a strawman, though there is some cluster in thoughtspace that could be called "Strong Bayesianism" that both of us would have disagreements with.

(as an aside, as far as I can tell, the entire $A_p$ section of Jaynes's Probability Theory: The Logic of Science is logically inconsistent)

You assume a creature that can't see all logical consequences of hypotheses [...] Then you make it realize new facts about logical consequences of hypotheses

This is not quite what is going on in section 7b. The agent isn't learning any new logical information. For instance, in jadagul's "US in 2100" example, all of the logical facts involved are things the agent already knows. " 'California is a US state in 2100' implies 'The US exists in 2100' " is not a new fact, it's something we already knew before running through the exercise.

My argument in 7b is not really about updating -- it's about whether probabilities can adequately capture the agent's knowledge, even at a single time.

This is in a context (typical of real decisions) where:

• the agent knows a huge number of logical facts, because it can correctly interpret hypotheses written in a logically transparent way, like "A and B," and because it knows lots of things about subsets in the world (like US / California)
• but, the agent doesn't have the time/memory to write down a "map" of every hypothesis connected by these facts (like a sigma-algebra). For example, you can read an arbitrary string of hypotheses "A and B and C and ..." and know that this implies "A", "A and C", etc., but you don't have in your mind a giant table containing every such construction.

So the agent can't assign credences/probabilities simultaneously to every hypothesis on that map. Instead, they have some sort of "credence generator" that can take in a hypothesis and output how plausible it seems, using heuristics. In their raw form, these outputs may not be real numbers (they will have an order, but may not have e.g. a metric).

If we want to use Bayes here, we need to turn these raw credences into probabilities. But remember, the agent knows a lot of logical facts, and via the probability axioms, these all translate to facts relating probabilities to one another. There may not be any mapping from raw credence-generator-output to probabilities that preserves all of these facts, and so the agent's probabilities will not be consistent.

To be more concrete about the "credence generator": I find that when I am asked to produce subjective probabilities, I am translating them from internal representations like

• Event A feels "very likely"
• Event B, which is not logically entailed by A or vice versa, feels "pretty likely"
• Event (A and B) feels "pretty likely"

If we demand that these map one-to-one to probabilities in any natural way, this is inconsistent. But I don't think it's inconsistent in itself; it just reflects that my heuristics have limited resolution. There isn't a conjunction fallacy here because I'm not treating these representations as probabilities -- but if I decide to do so, then I will have a conjunction fallacy! If I notice this happening, I can "plug the leak" by changing the probabilities, but I will expect to keep seeing new leaks, since I know so many logical facts, and thus there are so many consequences of the probability axioms that can fail to hold. And because I expect this to happen going forward, I am skeptical now that my reported probabilities reflect my actual beliefs -- not even approximately, since I expect to keep deriving very wrong things like an event being impossible instead of likely.

None of this is meant disapprove of using probability estimates to, say, make more grounded estimates of cost/benefit in real-world decisions. I do find that useful, but I think it is useful for a non-Bayesian reason: even if you don't demand a universal mapping from raw credences, you can get a lot of value out of saying things like "this decision isn't worth it unless you think P(A) > 97%", and then doing a one-time mapping of that back onto a raw credence, and this has a lot of pragmatic value even if you know the mappings will break down if you push them too hard.