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text: '[summary: [1lz Bayes' rule] relates [1rm prior] belief and the [1rq likelihood] of evidence to [1rp posterior] belief.\n\nThese quantities are often denoted using [1rj conditional probabilities]:\n\n- [1rm Prior] belief in hypothesis: $\\mathbb P(H).$\n- [1rq Likelihood] of evidence, conditional on hypothesis: $\\mathbb P(e \\mid H).$\n- [1rp Posterior] belief: $\\mathbb P(H \\mid e).$\n]\n\n[1lz Bayes' rule] relates [1rm prior] belief and the [1rq likelihood] of evidence to [1rp posterior] belief.\n\nThese quantities are often written using [1rj conditional probabilities]:\n\n- [1rm Prior] belief in the hypothesis: $\\mathbb P(H).$\n- [1rq Likelihood] of evidence, conditional on the hypothesis: $\\mathbb P(e \\mid H).$\n- [1rp Posterior] belief in hypothesis, after seeing evidence: $\\mathbb P(H \\mid e).$\n\nFor example, Bayes' rule in the [1x5 odds form] describes the relative belief in a hypothesis $H_1$ vs an alternative $H_2,$ given a piece of evidence $e,$ as follows:\n\n$$\\dfrac{\\mathbb P(H_1)}{\\mathbb P(H_2)} \\times \\dfrac{\\mathbb P(e \\mid H_1)}{\\mathbb P(e \\mid H_2)} = \\dfrac{\\mathbb P(H_1\\mid e)}{\\mathbb P(H_2\\mid e)}.$$',
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