Bayes’ Rule: False Positive Paradox

Bayesian statistics is a system for describing epistemological uncertainty using the mathematical language of probability. In the ’Bayesian paradigm,’ degrees of belief in states of nature are specified; these are non-negative, and the total belief in all states of nature is fixed to be one. Bayesian statistical methods start with existing ’prior’ beliefs, and update these using data to give ’posterior’ beliefs, which may be used as the basis for inferential decisions. Background In 1763, Thomas Bayes published a paper on the problem of induction, that is, arguing from the specific to the general. In modern language and notation, Bayes wanted to use Binomial data comprising r successes out of n attempts to learn about the underlying chance θ of each attempt succeeding. Bayes’ key contribution was to use a probability distribution to represent uncertainty about θ . This distribution represents ’epistemological’ uncertainty, due to lack of knowledge about the world, rather than ’aleatory’ probability arising from the essential unpredictability of future events, as may be familiar from games of chance. Modern ’Bayesian statistics’ is still based on formulating probability distributions to express uncertainty about unknown quantities. These can be underlying parameters of a system (induction) or future observations (prediction). Problem: A drug-screening test is used in a large population of people of whom 3% actually use drugs. Suppose that the false positive rate is 2% and the false negative rate is 4%. Thus a person who uses drugs tests positive for them 96% of the time and a person who does not use drugs tests negative 98% of the time. What is the probability that a randomly chosen person who tests positive for drugs actually uses drugs? #BayesRuleFalsePositiveParadox #BayesRule #FalsePositiveParadox #Probability #Statistics #Genetics #BayesTheorem #BayesLaw #NikolaysGeneticsLessons #Calculus #Education #math #Solution #Conditional #bayes #Theorem #BayesianInference #veritasium #BayesTheorem #rule #prior #conditionProbability #thomasBayes #bayesRule #bayesian #inference #mathematics #treeDiagrams #vennDiagrams #lawOfProbability