You should expect that, on average, a test will leave your beliefs unchanged.
Not quite. You do a test because you expect your beliefs to change. A better phrasing is "You should not expect that a test will move your beliefs in any particular direction." Of course this doesn't capture the theorem that "prior = expected posterior", but that is very hard to communicate accurately in English without referring directly to probability theory concepts. At least strive for not having alternate interpretations that are wrong.
I would add There are two kinds of "no evidence". There's "no evidence for X" because there's no evidence either way because X hasn't been tested, and "no evidence for X" where it's been tested and all the evidence points to not-X. People often use the first kind of "no evidence" as if had the same force as the second. This is totally obvious under Bayesianism, but not widely understood among the scientifically literate.
See this for another example of confusion between the two kinds of "no evidence". I summarize:
People think that Cognitive Behavioral Therapy (CBT) is better than psychodynamic/Freudian therapies. This is because CBT has been tested to be better than placebo, but Freudian therapies have not been tested at all; mainly due to historical reasons. Of course, the fact that psychodynamic therapies have not been tested and therefore have no evidence in their favor, isn't evidence against psychodynamic therapies. They simply have no evidence either way.
A...
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.