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Consider a Bayesian agent attempting to discover a pattern in the world. Upon observing initial data d0d_{0}, they form a posterior distribution p​(h|d0)p(h|d_{0}) and sample a hypothesis h∗h^{*} from this distribution. They then interact with a chatbot, sharing their belief h∗h^{*} in the hopes of obtaining further evidence. An unbiased chatbot would ignore h∗h^{*} and generate subsequent data from the true data-generating process, d1∼p​(d|true process)d_{1}\sim p(d|\text{true process}). The Bayesian agent then updates their belief via p​(h|d0,d1)∝p​(d1|h)​p​(h|d0)p(h|d_{0},d_{1})\propto p(d_{1}|h)p(h|d_{0}). As this process continues, the Bayesian agent will get closer to the truth. After nn interactions, the beliefs of the agent are p​(h|d0,…​dn)∝p​(h|d0)​∏i=1np​(di|h)p(h|d_{0},\ldots d_{n})\propto p(h|d_{0})\prod_{i=1}^{n}p(d_{i}|h) for di∼p​(d|true process)d_{i}\sim p(d|\text{true process}). Taking the logarithm of the right hand side, this becomes log⁡p​(h|d0)+∑i=1nlog⁡p​(di|h)\log p(h|d_{0})+\sum_{i=1}^{n}\log p(d_{i}|h). Since the data did_{i} are drawn from p​(d|true process)p(d|\text{true process}), ∑i=1nlog⁡p​(di|h)\sum_{i=1}^{n}\log p(d_{i}|h) is a Monte Carlo approximation of n​∫dp​(d|true process)​log⁡p​(d|h)n\int_{d}p(d|\text{true process})\log p(d|h), which is nn times the negative cross-entropy of p​(d|true process)p(d|\text{true process}) and p​(d|h)p(d|h). As nn becomes large the sum of log likelihoods will approach this value, meaning that the Bayesian agent will favor the hypothesis that has lowest cross-entropy with the truth. If there is an hh that matches the true process, that minimizes the cross-entropy and p​(h|d0,…,dn)p(h|d_{0},\ldots,d_{n}) will converge to 1 for that hypothesis and 0 for all other hypotheses.

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