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text: '[summary: A form of [1lz Bayes' rule] that uses relative [1rb odds].\n\nSuppose we're trying to solve a mysterious murder, and we [1rm start out] thinking the odds of Professor Plum vs. Miss Scarlet committing the murder are 1 : 2, that is, Scarlet is twice as likely as Plum to have committed the murder. We then observe that the victim was bludgeoned with a lead pipe. If we think that Plum, *if* he commits a murder, is around 60% likely to use a lead pipe, and that Scarlet, *if* she commits a murder, would be around 6% likely to us a lead pipe, this implies [1rq relative likelihoods] of 10 : 1 for Plum vs. Scarlet using the pipe.\n\nThe [1rp posterior] odds for Plum vs. Scarlet, after observing the victim to have been murdered by a pipe, are $(1 : 2) \\times (10 : 1) = (10 : 2) = (5 : 1)$. We now think Plum is around five times as likely as Scarlet to have committed the murder.]\n\nOne of the more convenient forms of [1lz Bayes' rule] uses [1rb relative odds]. Bayes' rule says that, when you observe a piece of evidence $e,$ your [1rp posterior] odds $\\mathbb O(\\boldsymbol H \\mid e)$ for your hypothesis [-vector] $\\boldsymbol H$ given $e$ is just your [1rm prior] odds $\\mathbb O(\\boldsymbol H)$ on $\\boldsymbol H$ times the [-56s] $\\mathcal L_e(\\boldsymbol H).$\n\nFor example, suppose we're trying to solve a mysterious murder, and we start out thinking the odds of Professor Plum vs. Miss Scarlet committing the murder are 1 : 2, that is, Scarlet is twice as likely as Plum to have committed the murder [1rm a priori]. We then observe that the victim was bludgeoned with a lead pipe. If we think that Plum, *if* he commits a murder, is around 60% likely to use a lead pipe, and that Scarlet, *if* she commits a murder, would be around 6% likely to us a lead pipe, this implies [1rq relative likelihoods] of 10 : 1 for Plum vs. Scarlet using the pipe. The [1rp posterior] odds for Plum vs. Scarlet, after observing the victim to have been murdered by a pipe, are $(1 : 2) \\times (10 : 1) = (10 : 2) = (5 : 1)$. We now think Plum is around five times as likely as Scarlet to have committed the murder.\n\n# Odds functions\n\nLet $\\boldsymbol H$ denote a [-vector] of hypotheses. An odds function $\\mathbb O$ is a function that maps $\\boldsymbol H$ to a set of [-1rb]. For example, if $\\boldsymbol H = (H_1, H_2, H_3),$ then $\\mathbb O(\\boldsymbol H)$ might be $(6 : 2 : 1),$ which says that $H_1$ is 3x as likely as $H_2$ and 6x as likely as $H_3.$ An odds function captures our *relative* probabilities between the hypotheses in $\\boldsymbol H;$ for example, (6 : 2 : 1) odds are the same as (18 : 6 : 3) odds. We don't need to know the absolute probabilities of the $H_i$ in order to know the relative odds. All we require is that the relative odds are proportional to the absolute probabilities:\n$$\\mathbb O(\\boldsymbol H) \\propto \\mathbb P(\\boldsymbol H).$$\n\nIn the example with the death of Mr. Boddy, suppose $H_1$ denotes the proposition "Reverend Green murdered Mr. Boddy", $H_2$ denotes "Mrs. White did it", and $H_3$ denotes "Colonel Mustard did it". Let $\\boldsymbol H$ be the vector $(H_1, H_2, H_3).$ If these propositions respectively have [1rm prior] probabilities of 80%, 8%, and 4% (the remaining 8% being reserved for other hypotheses), then $\\mathbb O(\\boldsymbol H) = (80 : 8 : 4) = (20 : 2 : 1)$ represents our *relative* credences about the murder suspects — that Reverend Green is 10 times as likely to be the murderer as Miss White, who is twice as likely to be the murderer as Colonel Mustard.\n\n# Likelihood functions\n\nSuppose we discover that the victim was murdered by wrench. Suppose we think that Reverend Green, Mrs. White, and Colonel Mustard, *if* they murdered someone, would respectively be 60%, 90%, and 30% likely to use a wrench. Letting $e_w$ denote the observation "The victim was murdered by wrench," we would have $\\mathbb P(e_w\\mid \\boldsymbol H) = (0.6, 0.9, 0.3).$ This gives us a [-56s] defined as $\\mathcal L_{e_w}(\\boldsymbol H) = P(e_w \\mid \\boldsymbol H).$\n\n# Bayes' rule, odds form\n\nLet $\\mathbb O(\\boldsymbol H\\mid e)$ denote the [1rp posterior] odds of the hypotheses $\\boldsymbol H$ after observing evidence $e.$ [1xr Bayes' rule] then states:\n\n$$\\mathbb O(\\boldsymbol H) \\times \\mathcal L_{e}(\\boldsymbol H) = \\mathbb O(\\boldsymbol H\\mid e)$$\n\nThis says that we can multiply the relative prior credence $\\mathbb O(\\boldsymbol H)$ by the likelihood $\\mathcal L_{e}(\\boldsymbol H)$ to arrive at the relative posterior credence $\\mathbb O(\\boldsymbol H\\mid e).$ Because odds are invariant under multiplication by a positive constant, it wouldn't make any difference if the _likelihood_ function was scaled up or down by a constant, because that would only have the effect of multiplying the final odds by a constant, which does not affect them. Thus, only the [-1rq relative likelihoods] are necessary to perform the calculation; the absolute likelihoods are unnecessary. Therefore, when performing the calculation, we can simplify $\\mathcal L_e(\\boldsymbol H) = (0.6, 0.9, 0.3)$ to the relative likelihoods $(2 : 3 : 1).$\n\nIn our example, this makes the calculation quite easy. The prior odds for Green vs White vs Mustard were $(20 : 2 : 1).$ The relative likelihoods were $(0.6 : 0.9 : 0.3)$ = $(2 : 3 : 1).$ Thus, the relative posterior odds after observing $e_w$ = Mr. Boddy was killed by wrench are $(20 : 2 : 1) \\times (2 : 3 : 1) = (40 : 6 : 1).$ Given the evidence, Reverend Green is 40 times as likely as Colonel Mustard to be the killer, and 20/3 times as likely as Mrs. White.\n\nBayes' rule states that this *relative* proportioning of odds among these three suspects will be correct, regardless of how our remaining 8% probability mass is assigned to all other suspects and possibilities, or indeed, how much probability mass we assigned to other suspects to begin with. For a proof, see [1xr].\n\n# Visualization\n\n[560 Frequency diagrams], [1wy waterfall diagrams], and [1zm spotlight diagrams] may be helpful for explaining or visualizing the odds form of Bayes' rule.',
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