Simple Bayesian inference of the chance someone is infected with a disease.

bayes.hs

{--
 - These functions handle the common scenario presented in introductory classes on Bayesian Inference.
 -
 --}

type Probability = Double
type Prior       = Probability
type AccuracyPos = Probability
type AccuracyNeg = Probability
type BayesianInf = Probability


{-- pTruePositive gives the chance of an individual actually being infected given that they tested positive.
 -
 - Parameters:
 -
 - 1. The estimated frequency of infection within that individual's population
 - 2. The taken test's estimated accuracy for individuals that are infected
 - 3. The taken test's estimated accuracy for individuals that are not infected
 -
 --}


pTruePositive :: Prior -> AccuracyPos -> AccuracyNeg -> BayesianInf
pTruePositive prior p n = truePos / (truePos + falsePos)
                            where truePos  = prior       * p
                                  falsePos = (1 - prior) * (1 - n)



{-- pTruePositiveAfter gives the probability of an individual actually being infected given that x
 - tests have yielded positive results. Following the fundamental Bayesian model, the initial Prior
 - is iteratively updated with the information of a positive outcome, yielding the final result.
 -
 - The only difference here is that the first parameter is the number of positive results for this individual.
 -}
 
type Experiments = Integer

pTruePositiveAfter :: Experiments -> Prior -> AccuracyPos -> AccuracyNeg -> BayesianInf
pTruePositiveAfter x prior p n = case x of 1 -> pTruePositive prior p n
                                           _ -> pTruePositiveAfter (x - 1) (pTruePositive prior p n) p n