Edit: Looking back at this a few years later. It is pretty embarrassing, but I'm going to leave it up.
Why don't we start treating the log2 of the probability — conditional on every available piece of information — you assign to the great conjunction, as the best measure of your epistemic success? Let's call: log_2(P(the great conjunction|your available information)), your "Bayesian competence". It is a deductive fact that no other proper scoring rule could possibly give: Score(P(A|B)) + Score(P(B)) = Score(P(A&B)), and obviously, you should get the same score for assigning P(A|B) to A, after observing B, and assigning P(B) to B a priori, as you would get for assigning P(A&B) to A&B a priori. The great conjunction is the conjunction of all true statements expressible in your idiolect. Your available information may be treated as the ordered set of your retained stimulus.
It is standard LW doctrine that we should not name the highest value of rationality, and it is often defended quite brilliantly:
You may try to name the highest principle with names such as “the map that reflects the territory” or “experience of success and failure” or “Bayesian decision theory”. But perhaps you describe incorrectly the nameless virtue. How will you discover your mistake? Not by comparing your description to itself, but by comparing it to that which you did not name.
and of course also:
How can you improve your conception of rationality? Not by saying to yourself, “It is my duty to be rational.” By this you only enshrine your mistaken conception. Perhaps your conception of rationality is that it is rational to believe the words of the Great Teacher, and the Great Teacher says, “The sky is green,” and you look up at the sky and see blue. If you think: “It may look like the sky is blue, but rationality is to believe the words of the Great Teacher,” you lose a chance to discover your mistake.
These quotes are from the end of Twelve Virtues
Should we really be wondering if there's a virtue higher than bayesian competence? Is there really a probability worth worrying about that the description of bayesian competence above is misunderstood? Is the description not simple enough to be mathematical? What mistake might I discover in my understanding of bayesian competence by comparing it to that which I did not name, after I've already given a proof that bayesian competence is proper, and that the restrictions: score(P(B)*P(A|B)) = score(P(B)) + score(P(A|B)), and: must be a proper scoring rule, uniquely specify Logb?
I really want answers to these questions. I am still undecided about them; and change my mind about them far too often.
Of course, your bayesian competence is ridiculously difficult to compute. But I am not proposing the measure for practical reasons. I am proposing the measure to demonstrate that degree of rationality is an objective quantity that you could compute given the source code to the universe, even though there are likely no variables in the source that ever take on this value. This may be of little to no value to the most obsessively pragmatic practitioners of rationality. But it would be a very interesting result to philosophers of science and rationality.
Updated to better express view of author, and take feedback into account. Apologies to any commenter who's comment may have been nullified.
The comment below:
The general reason Eliezer advocates not naming the highest virtue (as I understand it) is that there may be some type of problem for which bayesian updating (and the scoring rule referred to) yields the wrong answer. This idea sounds rather improbable to me, but there is a non-negligible probability that bayes will yield a wrong answer on some question. Not naming the virtue is supposed to be a reminder that if bayes ever gives the wrong answer, we go with the right answer, not bayes.
has changed my mind about the openness of the questions I asked.