2a:null 2b:["$","title",null,{"children":"Bayes' Theorem | Chris Wallace | Observable"}] 30:T4b2,md`There's a classic example from medical diagnosis that addresses the question > Given that a patient tests positive for a condition, what is the probability that they actually have the condition? If we can assume that patients contract the disease independently and every patient is equally likely to contract the disease, we can use Bayes' theorem to answer this question by mapping: - P(_A_) to the probability of a patient having the condition, - P(_B_) to the probability of a patient testing positive, and - P(_B_|_A_) to the probability of a patient testing positive _given_ that they have the condition. The P(_A_|_B_) is the answer to our question of interest: the probability that a patient actually has a disease when they test positive for it. Use the sliders below to set the probabilities and view the outcome of a simulated outbreak in a population of 400 dots (patients). Notice that if the probability of disease is very low, then even with a very accurate test, many people will test positive who do not have the disease. _(Note: these don't currently work in Chrome if you have [strict site isolation](https://www.chromium.org/Home/chromium-security/site-isolation) enabled)._`31:[] 2e:{"id":"6c6a754ab8ccf5b4","slug":"bayes-theorem","title":"Bayes' Theorem","thumbnail":"d6f866f13d8d9c5bc77e31968097ecee5e0a1eb8fb27e3a970e1373cd481a590","access_level":"public","create_time":"$D2018-02-02T11:06:45.706Z","update_time":"$D2021-07-16T10:58:29.261Z","view_time":null,"first_public_time":"$D2018-02-02T13:55:06.288Z","fork":false,"stars":7,"starred":false,"owner":{"type":"team","id":"0ac9abcf38e64f7b","login":"cjwallace","name":"Chris Wallace","avatar_url":"https://avatars-next.observableusercontent.com/ddff0ab0c8f4670e38fe8871e77257cde4558310a5516931b79aa56e27b59005","tier":"starter_2024"},"version":110,"publish_version":110,"role":null,"authored":false,"authors":[{"type":"individual","id":"b6d36fa8c41d8f92","login":"cjwallace","name":"Chris Wallace","avatar_url":"https://avatars-next.observableusercontent.com/ddff0ab0c8f4670e38fe8871e77257cde4558310a5516931b79aa56e27b59005","tier":null}],"members":[{"type":"individual","id":"b6d36fa8c41d8f92","login":"cjwallace","name":"Chris Wallace","avatar_url":"https://avatars-next.observableusercontent.com/ddff0ab0c8f4670e38fe8871e77257cde4558310a5516931b79aa56e27b59005","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":8,"collection":{"id":"a71d4c214fd19731","slug":"ml","title":"ml","owner":{"id":"0ac9abcf38e64f7b","login":"cjwallace","name":"Chris Wallace"}},"type":"notebook","body":{"title":"Bayes' Theorem","cells":[{"id":0,"value":"md`# Bayes' Theorem\n### Why you probably don't have a rare disease`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":11,"value":"md`Recall that for two events _A_ and _B_, Bayes' Theorem may be written,`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":9,"value":"tex`P(A|B) = P(B|A)P(A) / P(B)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":13,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":33,"value":"viewof prob_infected = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":35,"value":"md`The probability of a person being infected is ${prob_infected}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":38,"value":"viewof accuracy_of_test = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":40,"value":"md`The accuracy of the test is ${Number(100*accuracy_of_test).toFixed(0)}%. _(Thought experiment: why is this capped at 50%?)_`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":43,"value":"chart = {\n const viz = d3.select(DOM.svg(600, 180));\n viz.append('g')\n\t .selectAll('circle')\n .data(circles)\n .enter()\n .append('circle')\n \t.attr('cx', d => grid(d.x))\n \t.attr('cy', d => grid(d.y))\n \t.attr('r' , '4')\n \t.attr('fill', d => d.colour)\n \n return viz.node()\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":65,"value":"md` are people who don't have the disease and tested negative - the general populace.\n are people who don't have the disease and tested positive - \"false alarms\".\n are people who have the disease and tested positive - correct diagnoses.\n are people who have the disease and tested negative - missed diagnoses.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":53,"value":"md`_Below here is the stuff that makes this notebook tick. The built in reactive model is really cool. It means one can define a data generating process separately from the visualisation process, and still have the viz update automatically: there's no update() function in the chart above._`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":45,"value":"d3 = require('d3')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":47,"value":"grid = d3.scaleLinear()\n .domain([0, 40])\n .range([20, 580])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":49,"value":"circles = {\n var arr = []\n for (let i = 1; i <= 40; ++i) {\n for (let j = 1; j <= 10; ++j) {\n\t var col = '#eaeded' // light grey\n \n let infected = (Math.random() <= prob_infected) ? true : false\n \n if (infected) {\n\t let test_positive = Math.random() <= accuracy_of_test\n\t col = test_positive ?\n \t '#f5b7b1' : // light red\n \t '#a93226' // dark red\n } else {\n let test_positive = Math.random() > accuracy_of_test\n\t col = test_positive ?\n \t\t'#f7dc6f' : // yellow\n \t\t\tcol\t // still light grey\n }\n \n \t arr.push({x: i,\n y: j,\n colour: col\n })\n \t}\n }\n return arr\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 2f:"$2e:body"