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Benito comments on Looking for proof of conditional probability - Less Wrong
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I'm sorry, the comments on this post all seem to miss the point. Bayes' Theorem can be proven from basic logic, look at places like the Khan Academy, or Lukeprog's Intuitive Explanation of Yudkowsky's Intuitive Explanation. Once you understand that, the Kolmogorov axioms will be obvious. It's not assumed,