{
  localUrl: '../page/bayes_rule_probability_proof.html',
  arbitalUrl: 'https://arbital.com/p/bayes_rule_probability_proof',
  rawJsonUrl: '../raw/56j.json',
  likeableId: '2990',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '2',
  dislikeCount: '0',
  likeScore: '2',
  individualLikes: [
    'EricBruylant',
    'NateSoares'
  ],
  pageId: 'bayes_rule_probability_proof',
  edit: '3',
  editSummary: '',
  prevEdit: '2',
  currentEdit: '3',
  wasPublished: 'true',
  type: 'wiki',
  title: 'Proof of Bayes' rule: Probability form',
  clickbait: '',
  textLength: '1901',
  alias: 'bayes_rule_probability_proof',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'EliezerYudkowsky',
  editCreatedAt: '2016-10-08 20:26:54',
  pageCreatorId: 'NateSoares',
  pageCreatedAt: '2016-07-07 01:41:50',
  seeDomainId: '0',
  editDomainId: 'AlexeiAndreev',
  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: '353',
  text: 'Let $\\mathbf H$ be a [random_variable variable] in $\\mathbb P$ for the true hypothesis, and let $H_k$ be the possible values of $\\mathbf H,$ such that $H_k$ is [-1rd]. Then, Bayes' theorem states:\n\n$$\\mathbb P(H_i\\mid e) = \\dfrac{\\mathbb P(e\\mid H_i) \\cdot \\mathbb P(H_i)}{\\sum_k \\mathbb P(e\\mid H_k) \\cdot \\mathbb P(H_k)},$$\n\nwith a proof that runs as follows. By the definition of [-1rj],\n\n$$\\mathbb P(H_i\\mid e) = \\dfrac{\\mathbb P(e \\wedge H_i)}{\\mathbb P(e)} = \\dfrac{\\mathbb P(e \\mid  H_i) \\cdot \\mathbb P(H_i)}{\\mathbb P(e)}$$\n\nBy the law of [law_of_marginal_probability marginal probability]:\n\n$$\\mathbb P(e) = \\sum_{k} \\mathbb P(e \\wedge H_k)$$\n\nBy the definition of conditional probability again:\n\n$$\\mathbb P(e \\wedge H_k) = \\mathbb P(e\\mid H_k) \\cdot \\mathbb P(H_k)$$\n\nDone.\n\nNote that this proof of Bayes' rule is less general than the [1xr proof] of the [1x5 odds form of Bayes' rule].\n\n## Example\n\nUsing the [22s Diseasitis] example problem, this proof runs as follows:\n\n$$\\begin{array}{c}\n\\mathbb P({sick}\\mid {positive}) = \\dfrac{\\mathbb P({positive} \\wedge {sick})}{\\mathbb P({positive})} \\\\[0.3em]\n= \\dfrac{\\mathbb P({positive} \\wedge {sick})}{\\mathbb P({positive} \\wedge {sick}) + \\mathbb P({positive} \\wedge \\neg {sick})} \\\\[0.3em]\n= \\dfrac{\\mathbb P({positive}\\mid {sick}) \\cdot \\mathbb P({sick})}{(\\mathbb P({positive}\\mid {sick}) \\cdot \\mathbb P({sick})) + (\\mathbb P({positive}\\mid \\neg {sick}) \\cdot \\mathbb P(\\neg {sick}))}\n\\end{array}\n$$\n\nNumerically:\n\n$$3/7 = \\dfrac{0.18}{0.42} = \\dfrac{0.18}{0.18 + 0.24} = \\dfrac{90\\% * 20\\%}{(90\\% * 20\\%) + (30\\% * 80\\%)}$$\n\nUsing red for sick, blue for healthy, and + signs for positive test results, the proof above can be visually depicted as follows:\n\n![bayes theorem probability](https://i.imgur.com/H9im04o.png?0)\n\n%todo: if we replace the other Venn diagram for the proof of Bayes' rule, we should probably update this one too.%',
  metaText: '',
  isTextLoaded: 'true',
  isSubscribedToDiscussion: 'false',
  isSubscribedToUser: 'false',
  isSubscribedAsMaintainer: 'false',
  discussionSubscriberCount: '2',
  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: {
    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: 'Let $\\mathbf H$ be a [random_variable variable] in $\\mathbb P$ for the true hypothesis, and let $H_k$ be the possible values of $\\mathbf H,$ such that $H_k$ is [-1rd]. Then, Bayes' theorem states:'
  },
  creatorIds: [
    'NateSoares',
    'EricBruylant',
    'EliezerYudkowsky'
  ],
  childIds: [],
  parentIds: [
    'bayes_rule_proof',
    'bayes_rule_probability'
  ],
  commentIds: [],
  questionIds: [],
  tagIds: [
    'needs_clickbait_meta_tag',
    'b_class_meta_tag'
  ],
  relatedIds: [],
  markIds: [],
  explanations: [
    {
      id: '5769',
      parentId: 'bayes_rule_probability_proof',
      childId: 'bayes_rule_probability_proof',
      type: 'subject',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-08-01 23:45:42',
      level: '2',
      isStrong: 'true',
      everPublished: 'true'
    }
  ],
  learnMore: [],
  requirements: [
    {
      id: '5124',
      parentId: 'bayes_rule',
      childId: 'bayes_rule_probability_proof',
      type: 'requirement',
      creatorId: 'NateSoares',
      createdAt: '2016-07-10 21:27:49',
      level: '2',
      isStrong: 'true',
      everPublished: 'true'
    },
    {
      id: '5125',
      parentId: 'conditional_probability',
      childId: 'bayes_rule_probability_proof',
      type: 'requirement',
      creatorId: 'NateSoares',
      createdAt: '2016-07-10 21:27:56',
      level: '2',
      isStrong: 'true',
      everPublished: 'true'
    },
    {
      id: '5126',
      parentId: 'probability',
      childId: 'bayes_rule_probability_proof',
      type: 'requirement',
      creatorId: 'NateSoares',
      createdAt: '2016-07-10 21:28:18',
      level: '2',
      isStrong: 'true',
      everPublished: 'true'
    },
    {
      id: '5636',
      parentId: 'math2',
      childId: 'bayes_rule_probability_proof',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-07-26 16:56:09',
      level: '2',
      isStrong: 'true',
      everPublished: 'true'
    },
    {
      id: '5768',
      parentId: 'bayes_rule_probability',
      childId: 'bayes_rule_probability_proof',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-08-01 23:45:00',
      level: '2',
      isStrong: 'true',
      everPublished: 'true'
    }
  ],
  subjects: [
    {
      id: '5637',
      parentId: 'bayes_rule_probability',
      childId: 'bayes_rule_probability_proof',
      type: 'subject',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-07-26 16:57:32',
      level: '2',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '5638',
      parentId: 'bayes_rule',
      childId: 'bayes_rule_probability_proof',
      type: 'subject',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-07-26 16:58:20',
      level: '2',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '5641',
      parentId: 'bayes_rule_proof',
      childId: 'bayes_rule_probability_proof',
      type: 'subject',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-07-26 17:07:57',
      level: '2',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '5769',
      parentId: 'bayes_rule_probability_proof',
      childId: 'bayes_rule_probability_proof',
      type: 'subject',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-08-01 23:45:42',
      level: '2',
      isStrong: 'true',
      everPublished: 'true'
    }
  ],
  lenses: [],
  lensParentId: '',
  pathPages: [],
  learnMoreTaughtMap: {},
  learnMoreCoveredMap: {
    '554': [
      '555'
    ],
    '1lz': [
      '1xr',
      '1yc',
      '1zh',
      '1zm',
      '220',
      '552',
      '6cj'
    ]
  },
  learnMoreRequiredMap: {},
  editHistory: {},
  domainSubmissions: {},
  answers: [],
  answerCount: '0',
  commentCount: '0',
  newCommentCount: '0',
  linkedMarkCount: '0',
  changeLogs: [
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '19959',
      pageId: 'bayes_rule_probability_proof',
      userId: 'EliezerYudkowsky',
      edit: '3',
      type: 'newEdit',
      createdAt: '2016-10-08 20:26:55',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '18228',
      pageId: 'bayes_rule_probability_proof',
      userId: 'EricBruylant',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-08-03 16:34:51',
      auxPageId: 'needs_clickbait_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '18191',
      pageId: 'bayes_rule_probability_proof',
      userId: 'EricBruylant',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-08-02 18:56:08',
      auxPageId: 'b_class_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '18190',
      pageId: 'bayes_rule_probability_proof',
      userId: 'EricBruylant',
      edit: '2',
      type: 'newEdit',
      createdAt: '2016-08-02 18:54:43',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: 'added a link'
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17961',
      pageId: 'bayes_rule_probability_proof',
      userId: 'AlexeiAndreev',
      edit: '0',
      type: 'newTeacher',
      createdAt: '2016-08-01 23:45:42',
      auxPageId: 'bayes_rule_probability_proof',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17962',
      pageId: 'bayes_rule_probability_proof',
      userId: 'AlexeiAndreev',
      edit: '0',
      type: 'newSubject',
      createdAt: '2016-08-01 23:45:42',
      auxPageId: 'bayes_rule_probability_proof',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17960',
      pageId: 'bayes_rule_probability_proof',
      userId: 'AlexeiAndreev',
      edit: '0',
      type: 'newRequirement',
      createdAt: '2016-08-01 23:45:00',
      auxPageId: 'bayes_rule_probability',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17543',
      pageId: 'bayes_rule_probability_proof',
      userId: 'AlexeiAndreev',
      edit: '0',
      type: 'newSubject',
      createdAt: '2016-07-26 17:07:58',
      auxPageId: 'bayes_rule_proof',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17538',
      pageId: 'bayes_rule_probability_proof',
      userId: 'AlexeiAndreev',
      edit: '0',
      type: 'newSubject',
      createdAt: '2016-07-26 16:58:21',
      auxPageId: 'bayes_rule',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17536',
      pageId: 'bayes_rule_probability_proof',
      userId: 'AlexeiAndreev',
      edit: '0',
      type: 'newSubject',
      createdAt: '2016-07-26 16:57:32',
      auxPageId: 'bayes_rule_probability',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17534',
      pageId: 'bayes_rule_probability_proof',
      userId: 'AlexeiAndreev',
      edit: '0',
      type: 'newRequirement',
      createdAt: '2016-07-26 16:56:09',
      auxPageId: 'math2',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '16408',
      pageId: 'bayes_rule_probability_proof',
      userId: 'NateSoares',
      edit: '0',
      type: 'newRequirement',
      createdAt: '2016-07-10 21:28:18',
      auxPageId: 'probability',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '16407',
      pageId: 'bayes_rule_probability_proof',
      userId: 'NateSoares',
      edit: '0',
      type: 'newRequirement',
      createdAt: '2016-07-10 21:27:57',
      auxPageId: 'conditional_probability',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '16406',
      pageId: 'bayes_rule_probability_proof',
      userId: 'NateSoares',
      edit: '0',
      type: 'newRequirement',
      createdAt: '2016-07-10 21:27:49',
      auxPageId: 'bayes_rule',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15905',
      pageId: 'bayes_rule_probability_proof',
      userId: 'NateSoares',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-07-07 03:20:42',
      auxPageId: 'bayes_rule_probability',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15889',
      pageId: 'bayes_rule_probability_proof',
      userId: 'NateSoares',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-07-07 01:41:52',
      auxPageId: 'bayes_rule_proof',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15887',
      pageId: 'bayes_rule_probability_proof',
      userId: 'NateSoares',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-07-07 01:41:50',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}