A transformer trained on sequences whose source parameters are drawn afresh each time can learn only what the sequences share: the source family and its prior. The Bayesian predictor that knows both, which we call informed Bayes, is the unique minimizer of the training loss and is computable from the family alone; we use it as a quantitative, falsifiable reference. Where earlier work compared transformers with Bayes on sources whose contexts can be learned independently, we build Markov sources with up to 64 symbols in which observations in one context inform prediction in another: product sources with known or hidden encodings, transition rows sharing an unknown base law and depth, and a mixture of three families. Our main result is that informed Bayes predicts, from the family and prior alone, how a trained transformer behaves in four tests that go beyond the learning curve: the loss that remains under a source outside the training family (predicted 2.237 bits, measured 2.309), the doubling of the extra loss when two independent components change rather than one (0.345 predicted, 0.350 measured), the benefit of other contexts when a token appears for the first time, and the depth at which copying an earlier continuation starts to pay. On the learning curves themselves, small transformers match informed Bayes when the shared structure is visible in the tokens, and hidden structure needs more layers and training, with gaps below 0.01 bits per token for the largest models. We explain why the imitation is possible and where it must fail: informed Bayes decomposes into sufficient statistics that attention selects and accumulates and an estimator, fixed by the prior, that the feed-forward network learns; training pays the complexity of informed Bayes under the architecture's prior, and finite-memory universal prediction bounds how closely a bounded network can reach it.

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