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Hopefully my naivety doesn't shine through in this question: Are random forests Bayesian?

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Not really.

Random Forests are closer to bootstrapping (there is a bootstrapping piece in the random forest fit).

Bootstrapping can be seen as a Bayes analysis with a very specific prior and only looking at summaries of the posterior, but using a specific prior (that matches frequentist output) and not looking at the whole posterior does not really follow the philosophy of Bayesian analysis.

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