Jaynes gives an example of an experiment in which a psychic predicts cards, getting n out of m correct. In classical hypothesis testing H0 would be "the psychic got lucky" and H1 is "the psychic has mystic powers". Jaynes first attempts to use Bayesian reasoning to work backward to determine your prior belief in psychics. That is, how much evidence would it take to convince you, compare that to the raw likelihood and what you have is your prior beliefs quantified.
He then points out that he personally would never believe the subject was psychic, this is because there are not just H0 and H1, but H2 "the psychic is deceiving the experimenters", H3 "the experimenters are making an error" , H4 "the experiments are fudging the results", H5 ... Each of these has their own prior.
If your prior for believing in psychics is low enough and your prior for "the experimenters are fraud" is high enough, the more extreme the evidence the more you will be convinced that the experimenters are disreputable con artists, and subsequently the less you will believe the subject is psychic.
This is actually Jaynes' solution to a huge problem with Bayesian reasoning as a form of human reasoning: "If more data should override a bad prior, then why in the 'age of information' does nobody argee on anything!" This example shows, according to Jaynes, that while we can certainly have irrational priors we can still explain human reason in Bayesian terms and still get a situation where two people faced with plentiful information will arrive at contradictory conclusions.