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I understand that Bayesians interpret probability as belief, which can be updated with evidence. We happen to compute the posterior with Bayes theorem.

What is the difference between Bayes Theorem and Bayesian Statistics? Are they just named after the same guy?

Can a frequentist approach incorporate Bayes theorem (not for updating the probability)?

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I can't find the quote but I read somewhere that: using Bayes Theorem doesn't make you a Bayesian, using Bayes Theorem for everything does.

Bayes Theorem is used by frequentists all the time. See the examples at the Bayes Theorem Wikipedia page. Scroll down to the Interpretation section and you'll notice that there is a Bayesian Interpretation and a Frequentist Interpretation section.

Contrast this with Bayesian Inference.

So yes, a frequentist can use Bayes Theorem. Bayesian inference views probabilities and uncertainty differently than frequentist inference, and Bayes Theorem is sort of the universal engine used in that inference.

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