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where the transition from the second to the third line reflects the fact that the sum over h∗h^{*} is the same as the sum over hh in the denominator of Bayes’ rule, so the two terms cancel. As a consequence, after entering their hypothesis h∗h^{*} and receiving data d1d_{1}, the probability that an agent selects a particular hypothesis is exactly the same as before they interacted with the chatbot. By induction, the same result holds across subsequent interactions – the hypothesis an agent enters will be drawn from p(h|d0)p(h|d_{0}) and the process will repeat.
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Now consider the consequences of a sycophantic AI that generates responses by sampling examples consistent with the user’s hypothesis: d1∼p(d|h∗)d_{1}\sim p(d|h^{*}) rather than from the true data-generating process, d1∼p(d|true process)d_{1}\sim p(d|\text{true process}). The user, unaware of this bias, treats d1d_{1} as independent evidence and performs a standard Bayesian update, p(h|d1,d0)∝p(d1|h)p(h|d0)p(h|d_{1},d_{0})\propto p(d_{1}|h)p(h|d_{0}). But this update is circular. Because d1d_{1} was sampled conditional on hh, the user is updating their belief in hh based on data that was generated assuming hh was true. To see this, we can ask what the posterior distribution would be after this additional observation, averaging over the selected hypothesis h∗h^{*} and the particular piece of data generated from p(d1|h∗)p(d_{1}|h^{*}). We have