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"viewCount": 116609, "text": "[summary: When an observation is more likely given one state of the world than another, we should increase our credence that we're in the world that was more likely to have produced that observation. \n\nSuppose that Professor Plum and Miss Scarlet are two suspects in a murder, and that we start out thinking that Professor Plum is twice as likely to have committed the murder as Miss Scarlet. We then discover that the victim was poisoned. We think that, on occasions where they do commit murders, Professor Plum is around one-fourth as likely to use poison as Miss Scarlet. Then after observing the victim was poisoned, we should think Professor Plum is around half as likely to have committed the murder as Miss Scarlet: $2 \\times \\dfrac{1}{4} = \\dfrac{1}{2}.$\n\nThe quantitative rule at work here, in its various forms, is known as Bayes' rule.]\n\n[summary(Technical): Bayes' rule (aka Bayes' theorem) is the quantitative [1bv law] governing how to [1ly revise probabilistic beliefs in response to observing new evidence]. Suppose we previously thought that the probability of $h_1$, denoted $\\mathbb {P}(h_1)$, was twice as great as $\\mathbb {P}(h_2)$. Now we see a new piece of evidence, $e_0$, such that the probability of our seeing $e_0$ if $h_1$ is true (denoted by $\\mathbb {P}(e_0\\mid h_1)$) is one-fourth as great as $\\mathbb {P}(e_0\\mid h_2)$ (the probability of seeing $e_0$ if $h_2$ is true). After observing $e_0$, we should think that $h_1$ is now half as likely as $h_2$:\n\n$$\\frac{\\mathbb {P}(h_1\\mid e_0)}{\\mathbb {P}(h_2\\mid e_0)} = \\frac{\\mathbb {P}(h_1)}{\\mathbb {P}(h_2)} \\cdot \\frac{\\mathbb {P}(e_0\\mid h_1)}{\\mathbb {P}(e_0\\mid h_2)}$$\n\n[1zj More generally], Bayes' rule states: $\\mathbb P(\\mathbf{H}\\mid e) \\propto \\operatorname{\\mathbb {P}}(e\\mid \\mathbf{H}) \\cdot \\operatorname{\\mathbb {P}}(\\mathbf{H}).$]\n\nBayes' rule (aka Bayes' theorem) is the quantitative law of [1bv probability theory] governing how to [1ly revise] probabilistic beliefs in response to observing new evidence.\n\nYou may want to start at the [1zq Guide] or the [693 Fast Intro].\n\n# The laws of reasoning\n\nImagine that, as part of a clinical study, you're being tested for a rare form of cancer, which affects 1 in 10,000 people. You have no reason to believe that you are more or less likely than average to have this form of cancer. You're administered a test which is 99% accurate, both in terms of [-specificity] and [-sensitivity]: It _correctly_ detects the cancer (in patients who have it) 99% of the time, and it _incorrectly_ detects cancer (in patients who don't have it) only 1% of the time. The test results come back positive. What's the chance that you have cancer?\n\nBayes' rule says that the answer is precisely a 1 in 102 chance, which is a probability a little below 1%. The remarkable thing about this is that there is only one answer: the odds of you having that type of cancer, given the above information, is _exactly_ 1 in 102; no more, no less.\n\n%comment: (999,900 * 0.99 + 100 * 0.99) / (100 * 0.99) = (10098 / 99) = 102. Please leave this comment here so the above paragraph is not edited to be wrong.%\n\nThis is one of the key insights of Bayes' rule: Given what you knew, and what you saw, the maximally accurate state of belief for you to be in is completely pinned down. While that belief state is quite difficult to find in practice, we know how to find it in principle. If you want your beliefs to become more accurate as you observe the world, Bayes' rule gives some hints about what you need to do.\n\n# Learn Bayes' rule\n\n- [1x5 __Bayes' rule: Odds form.__] Bayes' rule is simple, if you think in terms of relative odds.\n- [1zm __Bayes' rule: Proportional form.__] The fastest way to say something both convincing and true about belief-updating.\n- [1zh __Bayes' rule: Log-odds form.__] A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.\n- [554 __Bayes' rule: Probabilistic form.__] The original formulation of Bayes' rule.\n- [1zj __Bayes' rule: Functional form.__] Bayes' rule for continuous variables.\n- [1zg __Bayes' rule: Vector form.__] For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go.\n\n# Implications of Bayes' rule\n\n- [220 __A Bayesian view of scientific virtues.__] Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.\n- [update_by_inches __Update by inches.__] It's virtuous to change your mind in response to overwhelming evidence. It's even more virtuous to shift your beliefs a little bit at a time, in response to *all* evidence (no matter how small).\n- [1y6 __Belief revision as probability elimination.__] Update your beliefs by throwing away large chunks of probability mass.\n- [552 __Shift towards the hypothesis of least surprise.__] When you see new evidence, ask: which hypothesis is *least surprised?*\n- [21v __Extraordinary claims require extraordinary evidence.__] The people who adamantly claim they were abducted by aliens do provide *some* evidence for aliens. They just don't provide quantitatively *enough* evidence.\n- [ __Ideal reasoning via Bayes' rule.__] Bayes' rule is to reasoning as the [https://en.wikipedia.org/wiki/Carnot_cycle Carnot cycle] is to engines: Nobody can be a perfect Bayesian, but Bayesian reasoning is still the theoretical ideal.\n\n# Related content\n\n- [4vr __Subjective probability.__] Probability is in the mind, not the world. If you don't know whether a tossed coin came up heads or tails, that's a fact about you, not a fact about the coin.\n- [1bv __Probability theory.__] The quantification and study of objects that represent uncertainty about the world, and methods for making those representations more accurate.\n- [3qq __Information theory.__] The quantification and study of information, communication, and what it means for one object to tell us about another.\n\n# Other articles and introductions\n\n- [http://en.wikipedia.org/wiki/Bayes%27_rule Wikipedia]\n- [http://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayes-theorem/ Better Explained]", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 3, "maintainerCount": 3, "userSubscriberCount": 0, "lastVisit": "2016-02-27 20:27:39", "hasDraft": false, "votes": [], "voteSummary": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": true, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": false, "reason": "You don't have domain permission to delete this page" }, "comment": { "has": false, "reason": "You can't comment in this domain because you are not a member" }, "proposeComment": { "has": true, "reason": "" } }, "summaries": { "Summary": "When an observation is more likely given one state of the world than another, we should increase our credence that we're in the world that was more likely to have produced that observation. \n\nSuppose that Professor Plum and Miss Scarlet are two suspects in a murder, and that we start out thinking that Professor Plum is twice as likely to have committed the murder as Miss Scarlet. We then discover that the victim was poisoned. We think that, on occasions where they do commit murders, Professor Plum is around one-fourth as likely to use poison as Miss Scarlet. Then after observing the victim was poisoned, we should think Professor Plum is around half as likely to have committed the murder as Miss Scarlet: $2 \\times \\dfrac{1}{4} = \\dfrac{1}{2}.$\n\nThe quantitative rule at work here, in its various forms, is known as Bayes' rule.", "Technical": "Bayes' rule (aka Bayes' theorem) is the quantitative [1bv law] governing how to [1ly revise probabilistic beliefs in response to observing new evidence]. Suppose we previously thought that the probability of $h_1$, denoted $\\mathbb {P}(h_1)$, was twice as great as $\\mathbb {P}(h_2)$. Now we see a new piece of evidence, $e_0$, such that the probability of our seeing $e_0$ if $h_1$ is true (denoted by $\\mathbb {P}(e_0\\mid h_1)$) is one-fourth as great as $\\mathbb {P}(e_0\\mid h_2)$ (the probability of seeing $e_0$ if $h_2$ is true). After observing $e_0$, we should think that $h_1$ is now half as likely as $h_2$:\n\n$$\\frac{\\mathbb {P}(h_1\\mid e_0)}{\\mathbb {P}(h_2\\mid e_0)} = \\frac{\\mathbb {P}(h_1)}{\\mathbb {P}(h_2)} \\cdot \\frac{\\mathbb {P}(e_0\\mid h_1)}{\\mathbb {P}(e_0\\mid h_2)}$$\n\n[1zj More generally], Bayes' rule states: $\\mathbb P(\\mathbf{H}\\mid e) \\propto \\operatorname{\\mathbb {P}}(e\\mid \\mathbf{H}) \\cdot \\operatorname{\\mathbb {P}}(\\mathbf{H}).$" }, "creatorIds": [ "2", "1", "32", "6jm", "7th" ], "childIds": [ "1wt", "1wy", "1x5", "1xr", "1y6", "1y9", "1zg", "1zh", "1zj", "1zm", "1zq", "207", "552", "553", "554", "560", "5f3", "693" ], "parentIds": [ "1r8" ], "commentIds": [ "58r", "6jn" ], "questionIds": [], "tagIds": [ "4yd" ], "relatedIds": [], "markIds": [], "explanations": [ { "id": "6512", "parentId": "1lz", "childId": "1lz", "type": "subject", "creatorId": "2", "createdAt": "2016-10-04 06:10:09", "level": 2, "isStrong": true, "everPublished": true }, { "id": "2075", 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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." }, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [ "4yd" ], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "1x7": { "likeableId": "850", "likeableType": "page", "myLikeValue": 0, "likeCount": 1, "dislikeCount": 0, "likeScore": 1, "individualLikes": [], 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"", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": null, "summaries": { "Summary": "The odds form of Bayes' rule is a form of [1lz Bayes' rule] that uses [1rq relative likelihoods] and [-1rb]. It says that in general, the [1rm prior] odds times the [56t likelihood ratio] equals the [1rp posterior] odds.\n\n[22s Suppose] that the odds of a particular patient being sick are about $(1 : 4),$ i.e., they may be drawn from a population where there is 1 sick patient for every 4 healthy ones. Suppose we administer a test which comes back positive 90% of the time if the patient is _actually_ sick, but also comes back positive 30% of the time even when the patient is healthy; this means relative likelihoods of $(90 : 30) = (3 : 1)$: the test is 3x as likely to come back positive if the patient is sick. Bayes' rule then says that the odds of the patient being sick _after_ observing a positive test are $(1 : 4) \\times (3 : 1) = (3 : 4),$ for a final probability of about $3/(3+4) \\approx 43\\%.$" }, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [ "4yd" ], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "1x9": { "likeableId": "852", "likeableType": "page", "myLikeValue": 0, "likeCount": 2, "dislikeCount": 0, "likeScore": 2, "individualLikes": [], "pageId": "1x9", "edit": 17, "editSummary": "This page is redundant after refactoring the other page. 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"Summary": "To denote some of the quantities used in Bayes' rule, we'll need [1rj conditional probabilities]. The conditional probability $\\mathbb{P}(X\\mid Y)$ means \"The [1rf probability] of $X$ given $Y$.\" That is, $\\mathbb P(\\mathrm{left}\\mid \\mathrm{right})$ means \"The probability that $\\mathrm{left}$ is true, assuming that $\\mathrm{right}$ is true.\"" }, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [ "4yd" ], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "1yd": { "likeableId": "887", "likeableType": "page", "myLikeValue": 0, "likeCount": 1, "dislikeCount": 0, "likeScore": 1, "individualLikes": [ "3fd" ], "pageId": "1yd", "edit": 7, "editSummary": "Orphaning this page, because it's not sufficiently different (nor sufficiently easy to read) from the main lens. 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It should probably be deleted.]" }, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "1zg": { "likeableId": "921", "likeableType": "page", "myLikeValue": 0, "likeCount": 8, "dislikeCount": 0, "likeScore": 8, "individualLikes": [ "1ch", "1yq", "1zv", "2cl", "31k", "32", "5pp", "827" ], "pageId": "1zg", "edit": 29, "editSummary": "", "prevEdit": 28, "currentEdit": 29, "wasPublished": true, "type": "wiki", "title": "Bayes' rule: Vector form", "clickbait": "For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)", "textLength": 11401, "alias": "bayes_rule_multiple", "externalUrl": "", "sortChildrenBy": "likes", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "8d0", "editCreatedAt": "2017-05-23 00:31:28", "pageCreatorId": "2", "pageCreatedAt": "2016-02-13 19:54:09", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 5702, "text": "", "metaText": "", "isTextLoaded": false, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 2, "maintainerCount": 2, "userSubscriberCount": 0, "lastVisit": "2016-02-27 17:45:50", "hasDraft": false, "votes": [], "voteSummary": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": null, "summaries": { "Summary": "The [1x5 odds form of Bayes' rule] works for odds ratios between more than two hypotheses, and applying multiple pieces of evidence. Suppose there's a bathtub full of coins. 1/2 of the coins are fair and have a 50% probability of producing heads on each coinflip; 1/3 of the coins produce 25% heads; and 1/6 produce 75% heads. You pull out a coin at random, flip it 3 times, and get the result HTH. You may calculate:\n\n$$\\begin{array}{rll}\n(1/2 : 1/3 : 1/6) = & (3 : 2 : 1) & \\\\\n\\times & (2 : 1 : 3) & \\\\\n\\times & (2 : 3 : 1) & \\\\\n\\times & (2 : 1 : 3) & \\\\\n= & (24 : 6 : 9) & = (8 : 2 : 3)\n\\end{array}$$" }, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [ "4y7" ], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "1zh": { "likeableId": "922", "likeableType": "page", "myLikeValue": 0, "likeCount": 14, "dislikeCount": 0, "likeScore": 14, "individualLikes": [ "1yq", "2ln", "2t2", "32", "35m", "3f9", "5pp", "5pw", "5xm", "63t", "7g3", "7qk", "8l3", "942" ], "pageId": "1zh", "edit": 43, "editSummary": "", "prevEdit": 39, "currentEdit": 43, "wasPublished": true, "type": "wiki", "title": "Bayes' rule: Log-odds form", "clickbait": "A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.", "textLength": 14241, "alias": "bayes_log_odds", "externalUrl": "", "sortChildrenBy": "likes", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "414", "editCreatedAt": "2017-07-10 00:25:14", "pageCreatorId": "2", "pageCreatedAt": "2016-02-13 20:16:06", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 1, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 7475, "text": "", "metaText": "", "isTextLoaded": false, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 5, "maintainerCount": 2, "userSubscriberCount": 0, "lastVisit": "2016-02-27 20:35:09", "hasDraft": false, "votes": [], "voteSummary": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": null, "summaries": { "Summary": "[1x5 Bayes's Rule] states that the [1rm prior odds] times the [56t likelihood ratio] equals the [1rp posterior odds]. We can take the logarithms of both sides of the equation to get an equivalent rule which uses addition instead of multiplication. Letting $H_i$ and $H_j$ denote hypotheses and $e$ denote evidence:\n\n$$\n\\log \\left ( \\dfrac\n {\\mathbb P(H_i\\mid e)}\n {\\mathbb P(H_j\\mid e)}\n\\right )\n=\n\\log \\left ( \\dfrac\n {\\mathbb P(H_i)}\n {\\mathbb P(H_j)}\n\\right )\n+\n\\log \\left ( \\dfrac\n {\\mathbb P(e\\mid H_i)}\n {\\mathbb P(e\\mid H_j)}\n\\right ).\n$$\n\nTo use a real-life example, a study of Chinese blood donors found that roughly 1 in 100,000 of them had HIV (as determined by a very reliable gold-standard test). The non-gold-standard test used for initial screening had a sensitivity of 99.7% and a specificity of 99.8%, meaning that it was 500 times as likely to return positive for infected as non-infected patients. Then our prior belief is -5 orders of magnitude against HIV, and if we then observe a positive test result, this is evidence of strength +2.7 orders of magnitude for HIV. Our posterior belief is -2.3 orders of magnitude, or odds of less than 1 to a 100, against HIV." }, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [ "4yd", "5g2" ], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "1zj": { "likeableId": "923", "likeableType": "page", "myLikeValue": 0, "likeCount": 5, "dislikeCount": 0, "likeScore": 5, 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"viewCount": 1892, "text": "", "metaText": "", "isTextLoaded": false, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 3, "maintainerCount": 2, "userSubscriberCount": 0, "lastVisit": "2016-02-27 17:26:06", "hasDraft": false, "votes": [], "voteSummary": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": null, "summaries": { "Summary": "[1lz Bayes' rule] generalizes to continuous [3jy functions], and says, \"The [1rp posterior] probability [density](https://en.wikipedia.org/wiki/Probability_density_function) is *proportional* to the [1rq likelihood] function times the [1rm prior] probability [density](https://en.wikipedia.org/wiki/Probability_density_function).\"\n\n$$\\mathbb P(H_x\\mid e) \\propto \\mathcal L_e(H_x) \\cdot \\mathbb P(H_x)$$" }, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [ "4yd" ], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "1zm": { "likeableId": "926", "likeableType": "page", "myLikeValue": 0, "likeCount": 7, 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"anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 8795, "text": "", "metaText": "", "isTextLoaded": false, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 4, "maintainerCount": 2, "userSubscriberCount": 0, "lastVisit": "2016-02-27 16:05:45", "hasDraft": false, "votes": [], "voteSummary": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": null, "summaries": { "Summary": "Suppose that Professor Plum and Miss Scarlet are two suspects in a murder, and that we start out thinking that Professor Plum is twice as likely to have committed the murder as Miss Scarlet. We then discover that the victim was poisoned. We think that Professor Plum is around one-fourth as likely to use poison as Miss Scarlet. Then after observing the victim was poisoned, we should think Plum is around half as likely to have committed the murder as Scarlet: $2 \\times \\dfrac{1}{4} = \\dfrac{1}{2}.$ This reasoning is valid by [1lz Bayes' rule]." }, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [ "4y7" ], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "1zq": { "likeableId": "929", "likeableType": "page", "myLikeValue": 0, "likeCount": 45, "dislikeCount": 0, "likeScore": 45, "individualLikes": [ "1", "1yq", "2", "22m", "22q", "26f", "27f", "27l", "289", "299", "2fz", "2k3", "2ky", "2y2", "2yr", "2z7", "31t", "31x", "32", "3b9", "3bh", "3hm", "3xl", "47b", "5bm", "5dv", "5p9", "5pp", "63n", "65d", "71v", "74m", "76", "7nt", "7tk", "887", "8gd", "8yc", "8z2", "908", "91f", "93j", "950", "95t", "9h5" ], "pageId": "1zq", "edit": 119, "editSummary": "", "prevEdit": 118, "currentEdit": 119, "wasPublished": true, "type": "wiki", "title": "Bayes' rule: Guide", "clickbait": "The Arbital guide to Bayes' rule", "textLength": 3820, "alias": "bayes_rule_guide", "externalUrl": "", "sortChildrenBy": "likes", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "1", "editCreatedAt": "2016-10-25 23:43:14", "pageCreatorId": "2", "pageCreatedAt": "2016-02-14 00:00:33", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": true, "todoCount": 0, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 113057, "text": "Bayes' rule or Bayes' theorem is the law of probability governing *the strength of evidence* - the rule saying *how much* to revise our probabilities (change our minds) when we learn a new fact or observe new evidence.\n\nYou may want to learn about Bayes' rule if you are:\n\n- A professional who uses statistics, such as a scientist or doctor;\n- A computer programmer working in machine learning;\n- A human being.\n\nAs Philip Tetlock found when studying \"superforecasters\", people who were especially good at predicting future events:\n\n> The superforecasters are a numerate bunch: many know about Bayes' theorem and could deploy it if they felt it was worth the trouble. But they rarely crunch the numbers so explicitly. What matters far more to the superforecasters than Bayes' theorem is Bayes' core insight of gradually getting closer to the truth by constantly updating in proportion to the weight of the evidence.

— Philip Tetlock and Dan Gardner, [https://en.wikipedia.org/wiki/Superforecasting _Superforecasting_]\n\n## Learning Bayes' rule\n\nThis guide to Bayes' rule uses Arbital's technology to allow for multiple flavors of introduction. They vary by technical level, speed, and topics covered. After you pick your path, remember that you can still switch between pages, in particular by using the \"Say what?\" and \"Go faster\" buttons.\n\n[multiple-choice(q_wants_angles): Which case fits you best?\na: I want to have a basic theoretical and practical understanding of the Bayes' rule.\n-wants: [62d],[62f]\nb: I can easily read algebra and don't mind the explanation moving at a fast pace. Just give me the basics, quick!\nwants: [62d]\n-wants: [62f]\nc: I want the basics, but I'm also interested in reading more about the theoretical implications and the reasons why Bayes' rule is considered so important.\nwants: [62f]\n-wants: [62d]\nd: I'd like to read everything! I want to have a *deep* theoretical and practical understanding of the Bayes' rule.\nwants: [62d],[62f]\n]\n\n%%%wants-requisite([62d]):\n%%wants-requisite([62f]):\n%box:\n*Your path will go over all forms of Baye's Rule, along with developing deep appreciation for its scientific usefulness. Your path will contain 12 pages:*\n\n * *Frequency diagrams: A first look at Bayes*\n * *Waterfall diagrams and relative odds*\n * *Introduction to Bayes' rule: Odds form*\n * *Bayes' rule: Proportional form*\n * *Extraordinary claims require extraordinary evidence*\n * *Ordinary claims require ordinary evidence*\n * *Bayes' rule: Log-odds form*\n * *Shift towards the hypothesis of least surprise*\n * *Bayes' rule: Vector form*\n * *Belief revision as probability elimination*\n * *Bayes' rule: Probability form*\n * *Bayesian view of scientific virtues*\n%\n\n%start-path([61b])%\n%%\n%%!wants-requisite([62f]):\n%box:\nNo time to waste! 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What would you say is your general math level?\n\n\nI'm bad at math\\.\n\nI'm not bad at math, but I'm not especially good at it either\\.\n\nI can have fun with math, and I won't be anxious if I see a word problem I don't already know how to solve\\.\n\nI am a programmer, physical engineer, or some other profession that calls on me to sometimes think in algebra\\.\n\nI do research\\-level math\\. Hit me with the LaTeX formulas\\.\n\n\n\n\n", "anchorText": "I'm bad at math\\.\n\nI'm not bad at math, but I'm not especially good at it either\\.", "anchorOffset": 155, "mergedInto": "", "isDeleted": false, "viewCount": 89707, "text": "Joe made a good point about the way this is phrased not sorting people quite right:\n\njoe [11:50 AM] \n“bad at math” = out of Arbital’s range\n\neric_bruylant [11:50 AM] \ncurrently, yes\nthe bad at math we're talking about is significantly a psychological aversion, not lack of background\n\njoe [11:51 AM] \nI’d say one of the things you might want to do\n\nis to … oh\n\neric_bruylant [11:51 AM] \nand we can't do therapy yet\n\njoe [11:51 AM] \nin that case, I think it’s somewhat poorly worded\n\nbecause some people who are not psychologically averse might still consider themselves “bad at math”\n\njust because they never really put any effort into it\n\nlike, they can’t multiply two-digit numbers, but they’d whip out a calculator if they had to\n\nanyway: I’d say one of the things you might want to do is to have a list of problems that those people should be able to understand the full meaning of, although not necessarily solve\n\neric_bruylant [11:52 AM] \nhm, yea. 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"discussionSubscriberCount": 1, "maintainerCount": 1, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": null, "summaries": { "Summary": "Odds express relative belief: we write \"the odds for X versus Y are $17 : 2$\" when we think that proposition X is 17/2 = 8.5 times as likely as proposition Y.%%note: The colon denotes that we are forming a set of odds. It does not denote division, as it might in French or German.%%" }, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [ "4y7" ], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "6jn": { "likeableId": "0", "likeableType": "page", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "pageId": "6jn", "edit": 1, "editSummary": "", "prevEdit": 0, "currentEdit": 1, "wasPublished": true, "type": "comment", "title": "\"I'm confused, and surely wrong, about the cance...\"", "clickbait": "", "textLength": 1039, "alias": "6jn", "externalUrl": "", "sortChildrenBy": "recentFirst", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "6jm", "editCreatedAt": "2016-11-16 05:14:21", "pageCreatorId": "6jm", "pageCreatedAt": "2016-11-16 05:14:21", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": true, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 78281, "text": "I'm confused, and surely wrong, about the cancer example.\n\n1 in 10000 people are sick. 1 sick person : 9999 well persons\nmultiply by 100: 100 sick people : 999900 well persons\n99% of the sick people have positive tests: (0.99 * 100 = ) 99 Positive tests \n 1% of the well people have false positive tests: (0.01 * 999900 = 9999)\n\nUsing the odds view: number of sick persons with positive tests / total number of persons with positive tests: (99 / (99 + 9999) = 99 / 10098. Multiply top and bottom by (1/99) => (99/99) / (10098/99) = 1 / 102. The text says the answer is 1 / 101.010101... which is 99/10000.\n\nSo, try the waterfall method.\n\nprior odds of being sick: 1 in 10000.\nBeing sick: 1\nBeing well: 9999\n\nchance of having positive test while sick: 99\nchance of having positive test while well: 1\n\nodds of being sick given positive test: (1 / 9999) * (99 / 1) = 99 / 9999 = 0.00990099\nprobability of being sick given positive test: 99 / (99+9999) = 1 / 102 from above.\n\nWhere did I go wrong? Thanks in advance for any time you have!", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 1, "maintainerCount": 1, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": false, "reason": "You don't have domain permission to delete this page" }, "comment": { "has": false, "reason": "You can't comment in this domain because you are not a member" }, "proposeComment": { "has": true, "reason": "" } }, "summaries": {}, "creatorIds": [], "childIds": [], "parentIds": [ "1lz" ], "commentIds": [], "questionIds": [], "tagIds": [], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": true, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "91t": { "likeableId": "0", "likeableType": "page", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "pageId": "91t", "edit": 3, "editSummary": "", "prevEdit": 2, "currentEdit": 3, "wasPublished": true, "type": "group", "title": "Jordan Bennett", "clickbait": "Automatically generated page for Jordan Bennett", "textLength": 273, "alias": "JordanBennett", "externalUrl": "", "sortChildrenBy": "likes", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "91t", "editCreatedAt": "2018-03-30 04:35:26", "pageCreatorId": "91t", "pageCreatedAt": "2018-03-30 04:30:38", "seeDomainId": "0", "editDomainId": "3062", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": false, "isApprovedComment": false, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 1, "text": "", "metaText": "", "isTextLoaded": false, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 0, "maintainerCount": 0, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": [], "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": 0, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": null, "summaries": {}, "creatorIds": [], "childIds": [], "parentIds": [], "commentIds": [], "questionIds": [], "tagIds": [], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": false, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "91v": { "likeableId": "0", "likeableType": "page", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "pageId": "91v", "edit": 1, "editSummary": "", "prevEdit": 0, "currentEdit": 1, "wasPublished": true, "type": "comment", "title": "\"a.) As Neil Tyson Degrasse expresses, science i...\"", "clickbait": "", "textLength": 658, "alias": "91v", "externalUrl": "", "sortChildrenBy": "recentFirst", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "91t", "editCreatedAt": "2018-03-30 04:42:26", "pageCreatorId": "91t", "pageCreatedAt": "2018-03-30 04:42:26", "seeDomainId": "0", "editDomainId": "3062", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": false, "isApprovedComment": false, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 30149, "text": "a.) As Neil Tyson Degrasse expresses, science is true regardless of belief:\n\nb.) Source: https://www.youtube.com/watch?v=WtBnm0X50VQ\n\n1.) I no longer subscribe to the concept of belief.\n\n2.) 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