Relatedly, there's Conservation of Expected Evidence. A rational person can't seek to confirm their beliefs, only to test them. You should expect that, on average, a test will leave your beliefs unchanged. If not, you should update your beliefs now based on how you expect the test to turn out.
This appears to be wrong:
Shake a box containing a coin. What is your belief that the coin landed heads? 50% . Will your belief change if you open the box and look inside it? Sure it will.
http://en.wikipedia.org/wiki/Law_of_total_expectation
Expectation of your belief E(X) is not the same as your belief X.
Recently, I completed my first systematic read-through of the sequences. One of the biggest effects this had on me was considerably warming my attitude towards Bayesianism. Not long ago, if you'd asked me my opinion of Bayesianism, I'd probably have said something like, "Bayes' theorem is all well and good when you know what numbers to plug in, but all too often you don't."
Now I realize that that objection is based on a misunderstanding of Bayesianism, or at least Bayesianism-as-advocated-by-Eliezer-Yudkowsky. "When (Not) To Use Probabilities" is all about this issue, but a cleaner expression of Eliezer's true view may be this quote from "Beautiful Probability":
The practical upshot of seeing Bayesianism as an ideal to be approximated, I think, is this: you should avoid engaging in any reasoning that's demonstrably nonsensical in Bayesian terms. Furthermore, Bayesian reasoning can be fruitfully mined for heuristics that are useful in the real world. That's an idea that actually has real-world applications for human beings, hence the title of this post, "Bayesianism for Humans."
Here's my attempt to make an initial list of more directly applicable corollaries to Bayesianism. Many of these corollaries are non-obvious, yet eminently sensible once you think about them, which I think makes for a far better argument for Bayesianism than Dutch Book-type arguments with little real-world relevance. Most (but not all) of the links are to posts within the sequences, which hopefully will allow this post to double as a decent introductory guide to the parts of the sequences that explain Bayesianism.