31:[] 2f:{"title":"Bayes' theorem for events","cells":[{"id":0,"value":"# Bayes' theorem for events","pinned":false,"hidden":false,"mode":"md"},{"id":165,"value":"### Bayes' theorem","pinned":false,"hidden":false,"mode":"md"},{"id":155,"value":"Bayes' theorem is a simple but fundamental result in probability. At it its core, it makes it possible to compute the conditional probability of event ${tex`A`} given event ${tex`B`}, ${tex`P(A\\vert B)`}, from the **reverse** conditional probability, ${tex`P(B\\vert A)`}.
\nBayes' theorem can therefore be said to *reverse a conditioning*.","pinned":false,"hidden":false,"mode":"md"},{"id":126,"value":">

**Bayes theorem**
\nThe probability of the event ${tex`A`} conditional on the occurrence of event ${tex`B`} is\n\n>

${tex`P(A \\vert B) = \\frac{P(B \\vert A)P(A)}{P(B)} = \\frac{P(B \\vert A)P(A)}{P(B \\vert A)P(A)+P(B \\vert A^c)P(A^c)}`}\n\n>

where the last equality follows from the *law of total probability*:\n \n>

${tex`P(A) = \\frac{P(B \\vert A)P(A)}{P(B \\vert A)P(A)+P(B \\vert A^c)P(A^c)}`}","pinned":false,"hidden":false,"mode":"md"},{"id":153,"value":"### Application - can a selftest detect Covid accurately?","pinned":false,"hidden":false,"mode":"md"},{"id":12,"value":"A person took a rapid self test for Covid-19. The test came out positive. What is the probability that the person actually has Covid? We are therefore interested in the conditional probability P(Covid | Positive test). We can solve this using Bayes' theorem by defining the events:\n- ${tex`A`} = {covid}\n- ${tex`B`} = {positive test}\n\nWe are interested in the conditional probability\n\n${tex`P(A|B)`} = P(covid | positive test)\n\nWe need to know\n- ${tex`P(B\\vert A)`} = P(positive test | covid)\n- ${tex`P(B\\vert A^c)`} = P(positive test | not covid) or P(negative test | not covid)\n- ${tex`P(A)`} = P(covid)\n\nsince ${tex`P(A^c) = 1 - P(A)`} we can easily compute ${tex`P(A^c)`} from ${tex`P(A)`}.","pinned":false,"hidden":false,"mode":"md"},{"id":21,"value":"The company that developed the rapid Covid test did a study with a number of randomly selected individuals, and summarizes the result from the study in the test instruction leaflet:\n- **Sensitivity** (also called **true positive rate**) is the probability of a positive test result, given that the individual truly has Covid. The company reports P(positive test | Covid) = ${tex`P(B | A) = 0.9677`} based on a rather limited study.\n- **Specificity** (also called **true negative rate**) is the probability of a negative test result, given that the individual does not have Covid. The company reports this as P(negative test | not Covid) = ${tex`P(B^c | A^c) = 0.992`}. \n- **Prevalence** is the proportion of persons with Covid in the population. Let us assume this is 5%. So ${tex`P(A )=0.05`}.","pinned":false,"hidden":false,"mode":"md"},{"id":140,"value":"The requested probability ${tex`P(A|B)`} = P(covid | positive test) is computed below. You can change the different probabilities by moving the sliders. You can also change the names of event A and B, so that you can use this widget for any other problem of interest.","pinned":false,"hidden":false,"mode":"md"},{"id":93,"value":"viewof Labels = Inputs.form([\n Inputs.text({label: \"Event A:\", placeholder: \"enter name of event A\", value: \"cov\"}),\n Inputs.text({label: \"Event B:\", placeholder: \"enter name of event B\", value: \"pos\"})\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":27,"value":"viewof probs = Inputs.form([\n Inputs.range([0, 1], {value: 0.9677, step: 0.0001, label: tex`P(\\text{${Labels[1]} | ${Labels[0]})}`}),\n Inputs.range([0, 1], {value: 0.9920, step: 0.0001, label: tex`P(\\text{not ${Labels[1]} | not ${Labels[0]})}`}),\n Inputs.range([0, 1], {value: 0.05, step: 0.0001, label: tex`P(\\text{${Labels[0]}})`})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":49,"value":"result_printout = md`

${tex`\\text{P(${Labels[0]} | ${Labels[1]} )}= ${prob_covid_pos_test.toPrecision(4)}`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":219,"value":"I think you will agree that this is surprisingly low. There are several reasons for this:\n1. the so called *base rate* P(covid) is very low so that the absolute majority of people tested are people who do not have Covid, and most persons with a positive test are therefore actually people without Covid. This [article](https://www.varsity.co.uk/science/21149) explain this well. Try changing P(covid) to higher numbers and see how P(covid | pos) quickly becomes larger.\n2. the calculation is using the prevalence of covid in the population as the base probability, P(covid). This would realistic if we took a random person from the population and tested her/him. But when you actually take the test you may already have some symptoms of Covid (fever, coughing etc). Conditional on these other symptoms, the base rate P(covid) is most likely much higher.","pinned":false,"hidden":false,"mode":"md"},{"id":80,"value":"
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\n\n---\n## 🙈 Appendix 🙈\nHere is just some code and data details that you probably don't care about.","pinned":false,"hidden":false,"mode":"md"},{"id":70,"value":"prob_covid_pos_test = probs[0]*probs[2]/(probs[0]*probs[2] + (1-probs[1])*(1-probs[2]))","pinned":false,"hidden":false,"mode":"ojs"},{"id":128,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":true,"hidden":false,"mode":"ojs"},{"id":146,"value":"viewof result_colour = Inputs.color({label: \"Highlight color cells\", value: \"#dae6f0\"})","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}