The statement, “You should expect that, on average, a test will leave your beliefs unchanged,” means that you cannot expect an unbiased test to change you beliefs in a particular direction, as is clear from the original post.
Actually you can: Shake a box with a coin you know to be biased. Before you look into the box, your belief for heads is, say, 80%. You expect that is more likely that, when you open the box, your belief will change to 100% heads rather than 0%.
I don’t think there is an useful way to patch the statement without making explicit reference to the technical definition of Bayesian belief.
Actually you can:
Shake a box with a coin you know to be biased. Before you look into the box, your belief for heads is, say, 80%. You expect that is more likely that, when you open the box, your belief will change to 100% heads rather than 0%.
I don’t think there is an useful way to patch the statement without making explicit reference to the technical definition of Bayesian belief.