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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 view probabilities as subjective and objective 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 the fact the only form of inference that is logically consistent. In the objectivist school, Bayesian probability is viewed as an extension of Aristotelian logic.
The present-day enthusiasm with Bayesian methods began around 1950 when people began to seek independence from the narrower frequentist system, which sees likelihoods as frequencies, say, a "1 in 10 chance." Bayesian statisticians instead view likelihoods as probabilities, say, a "10% probability." Bayesians emphasize the importance of Bayes' theorem, a formal theorem that proves a rigid probabilistic relationship between the conditional and marginal probabilities of two random events. Bayes' theorem puts great emphasis on the prior probability of a given event -- for instance, in evaluating the probability that one patient has cancer based on a positive test result, one must be sure to take into account the background probability that any random person has cancer at all.
Students of Bayesian probability have published thousands of papers unraveling the further, and sometimes unintuitive consequences of Bayes' theorem and related theorems. For instance, consider a company is testing its 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 any given employee engaging in opium use is only 0.5%, plugging the numbers into Bayes' theorem shows that a positive test on any given employee only gives a probability that they are a drug user of 33%. When the background incidence of the quality being tested for is very low, numerous false positives may result, even when sensitivity and specificity of the test is high. In the medical world, lazy interpretations of probability by doctors routinely cause healthy patients a high degree of distress, when they test positive for dangerous diseases but are not aware of the margin of error.