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text: 'Bayes' rule (in the [1x5 odds form]) says that, for every pair of hypotheses $H_i$ and $H_j$ and piece of evidence $e,$\n\n$$\\dfrac{\\mathbb P(H_i)}{\\mathbb P(H_j)} \\times \\dfrac{\\mathbb P(e \\mid H_i)}{\\mathbb P(e \\mid H_j)} = \\dfrac{\\mathbb P(H_i \\mid e)}{\\mathbb P(H_j \\mid e)}.$$\n\nBy the definition of [1rj conditional probability], $\\mathbb P(e \\land H)$ $=$ $\\mathbb P(H) \\cdot \\mathbb P(e \\mid H),$ so\n\n$$ \\dfrac{\\mathbb P(H_i)}{\\mathbb P(H_j)} \\times \\dfrac{\\mathbb P(e\\mid H_i)}{\\mathbb P(e\\mid H_j)} = \\dfrac{\\mathbb P(e \\wedge H_i)}{\\mathbb P(e \\wedge H_j)} $$\n\nDividing both the numerator and the denominator by $\\mathbb P(e),$ we have\n\n$$ \\dfrac{\\mathbb P(e \\wedge H_i)}{\\mathbb P(e \\wedge H_j)} = \\dfrac{\\mathbb P(e \\wedge H_i) / \\mathbb P(e)}{\\mathbb P(e \\wedge H_j) / \\mathbb P(e)} $$\n\nInvoking the definition of conditional probability again,\n\n$$ \\dfrac{\\mathbb P(e \\wedge H_i) / \\mathbb P(e)}{\\mathbb P(e \\wedge H_j) / \\mathbb P(e)} = \\dfrac{\\mathbb P(H_i\\mid e)}{\\mathbb P(H_j\\mid e)}.$$\n\nDone.\n\n---\n\nOf note is the equality\n\n$$\\frac{\\mathbb P(H_i\\mid e)}{\\mathbb P(H_j\\mid e)} = \\frac{\\mathbb P(H_i \\land e)}{\\mathbb P(H_j \\land e)},$$\n\nwhich says that the posterior odds (on the left) for $H_i$ (vs $H_j$) given evidence $e$ is exactly equal to the prior odds of $H_i$ (vs $H_j$) in the parts of $\\mathbb P$ where $e$ was already true. $\\mathbb P(x \\land e)$ is the amount of probability mass that $\\mathbb P$ allocated to worlds where both $x$ and $e$ are true, and the above equation says that after observing $e,$ your belief in $H_i$ relative to $H_j$ should be equal to $H_i$'s odds relative to $H_j$ _in those worlds._ In other words, Bayes' rule can be interpreted as saying: "Once you've seen $e$, simply throw away all probability mass except the mass on worlds where $e$ was true, and then continue reasoning according to the remaining probability mass." See also [1y6].\n\n## Illustration (using the Diseasitis example)\n\nSpecializing to the [22s Diseasitis] problem, using red for sick, blue for healthy, and + signs for positive test results, the proof above can be visually depicted as follows:\n\n\n\nThis visualization can be read as saying: The ratio of the initial sick population (red) to the initial healthy population (blue), times the ratio of positive results (+) in the sick population to positive results in the blue population, equals the ratio of the positive-and-red population to positive-and-blue population. Thus we can divide both into the proportion of the whole population which got positive results (grey and +), yielding the posterior odds of sick (red) vs healthy (blue) among only those with positive results.\n\n\nThe corresponding numbers are:\n\n$$\\dfrac{20\\%}{80\\%} \\times \\dfrac{90\\%}{30\\%} = \\dfrac{18\\%}{24\\%} = \\dfrac{0.18 / 0.42}{0.24 / 0.42} = \\dfrac{3}{4}$$\n\nfor a final probability $\\mathbb P(sick)$ of $\\frac{3}{7} \\approx 43\\%.$\n\n## Generality\n\nThe [1x5 odds] and [1zm proportional] forms of Bayes' rule talk about the *relative* probability of two hypotheses $H_i$ and $H_j.$ In the particular example of Diseasitis it happens that [1rd every patient is either sick or not-sick], so that we can [1rk normalize] the final odds 3 : 4 to probabilities of $\\frac{3}{7} : \\frac{4}{7}.$ However, the proof above shows that even if we were talking about two different possible diseases and their total prevalances did not sum to 1, the equation above would still hold between the *relative* prior odds for $\\frac{\\mathbb P(H_i)}{\\mathbb P(H_j)}$ and the *relative* posterior odds for $\\frac{\\mathbb P(H_i\\mid e)}{\\mathbb P(H_j\\mid e)}.$\n\nThe above proof can be specialized to the probabilistic case; see [56j].\n\n',
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