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In this paper, we provide a rational analysis of the effects of sycophantic AI, considering how a Bayesian agent would respond to confirmatory evidence. Our analysis shows that such an agent will not get any closer to the truth, but will increase in their certainty about an incorrect hypothesis. We test this model in an online experiment where users are made to interact with an AI agent as they complete a rule discovery task. Our results show that the default interactions of a popular chatbot resemble the effects of providing people with confirmatory evidence, increasing confidence but bringing them no closer to the truth. These results provide a theoretical and empirical demonstration of how conversations with generative AI chatbots can facilitate delusion-like epistemic states, producing beliefs markedly divergent from reality.