You should expect that, on average, a test will leave your beliefs unchanged.
That happens to be not true. A test which ouputs useful information WILL change your beliefs. Especially given point 2, one can say “Any informative test will always change your beliefs”.
What’s tricky here is expectation. You expect your beliefs to change but you don’t know in which direction. So your expectation is for zero change even though you know that you’ll get some non-zero change.
This looks paradoxical, but is the entirely standard way in which statistics (in particular random variables) operate. Consider a toss of a fair coin. The expectation is half heads half tails which is guaranteed not to happen. You know you’ll get either heads or tail but not which one of those two. The expectation will not match the outcome—all it can do is be equidistant (appropriately weighted) from all possible outcomes.
I think that problem is in the sentence
That happens to be not true. A test which ouputs useful information WILL change your beliefs. Especially given point 2, one can say “Any informative test will always change your beliefs”.
What’s tricky here is expectation. You expect your beliefs to change but you don’t know in which direction. So your expectation is for zero change even though you know that you’ll get some non-zero change.
This looks paradoxical, but is the entirely standard way in which statistics (in particular random variables) operate. Consider a toss of a fair coin. The expectation is half heads half tails which is guaranteed not to happen. You know you’ll get either heads or tail but not which one of those two. The expectation will not match the outcome—all it can do is be equidistant (appropriately weighted) from all possible outcomes.