Bayesian probability views likelihoods as probabilities, with subjective and objective schools. It emphasizes Bayes’ theorem and prior probability. False positives can result when the background incidence of a quality being tested is low.
Bayesian probability is an approach to statistics and inference that views likelihoods as probabilities rather than frequencies. There are two primary schools of Bayesian probability, the subjectivist school and the objectivist school, which consider subjective and objective probabilities, respectively. The subjective school views Bayesian probability as subjective states of belief, while the objectivist school, founded by Edwin Thompson Jaynes and Sir Harold Jeffreys, views Bayesian probabilities as objectively justified and in fact the only logically consistent form of inference. In the Objectivist school, Bayesian probability is seen as an extension of Aristotelian logic.
Today’s excitement with Bayesian methods began around 1950, when people started looking for independence from the more narrow Frequentist system, which sees probabilities as frequencies, say, a “1 in 10 chance.” Bayesian statisticians, on the other hand, consider likelihoods as probabilities, such as a “10% probability.” Bayesians emphasize the importance of Bayes’ theorem, a formal theorem that demonstrates a strict probabilistic relationship between the conditional and marginal probabilities of two random events. Bayes’ theorem places great emphasis on the prior probability of a given event: for example, when assessing the probability that a patient has cancer based on a positive test result, be sure to take into account the underlying probability that a random person has cancer at all.
Students of Bayesian probability have published thousands of papers revealing the further, and sometimes non-intuitive, consequences of Bayes’ theorem and related theorems. For example, consider a company is testing their employees for opium use and the test is 99% sensitive and 99% specific, meaning it correctly identifies a drug user 99% of the time and a non-user 99% of the time. If the background probability of a given addict using opium is only 0.5%, plugging the numbers into Bayes’ theorem shows that a positive test on a given addict gives only a 33% chance that he or she is an addict. When the background incidence of the quality being tested is very low, numerous false positives can result, even when the sensitivity and specificity of the test are high. In the medical world, lazy doctors’ interpretations of probability routinely cause healthy patients a great deal of distress, when they test positive for dangerous diseases but are unaware of the margin of error.
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