The prominent Bayesian statistician Donald Rubin of Harvard has served as a consultant for tobacco companies facing lawsuits for damages from smoking. It reminds me of the theory of evolution, another idea that seems tautologically simple or dauntingly deep, depending on how you view it, and that has inspired abundant nonsense as well as profound insights. Or, to put it another way, all of a statistical model needs to be understood and evaluated.
I object to the attitude that the data model is assumed correct while the prior distribution is suspect. Are Brains Bayesian? The views expressed are those of the author s and are not necessarily those of Scientific American. For many years, he wrote the immensely popular blog Cross Check for Scientific American. Follow John Horgan on Twitter. Already a subscriber? Sign in. Thanks for reading Scientific American. Create your free account or Sign in to continue. See Subscription Options. Go Paperless with Digital.
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Bayes' theorem, named after 18th-century British mathematician Thomas Bayes, is a mathematical formula for determining conditional probability. Conditional probability is the likelihood of an outcome occurring, based on a previous outcome occurring.
Bayes' theorem provides a way to revise existing predictions or theories update probabilities given new or additional evidence. In finance, Bayes' theorem can be used to rate the risk of lending money to potential borrowers. Bayes' theorem is also called Bayes' Rule or Bayes' Law and is the foundation of the field of Bayesian statistics. Applications of the theorem are widespread and not limited to the financial realm.
As an example, Bayes' theorem can be used to determine the accuracy of medical test results by taking into consideration how likely any given person is to have a disease and the general accuracy of the test. Bayes' theorem relies on incorporating prior probability distributions in order to generate posterior probabilities. Prior probability, in Bayesian statistical inference, is the probability of an event before new data is collected.
This is the best rational assessment of the probability of an outcome based on the current knowledge before an experiment is performed. Posterior probability is the revised probability of an event occurring after taking into consideration new information. Posterior probability is calculated by updating the prior probability by using Bayes' theorem.
In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred. This is a conditional probability. It is the probability of the hypothesis being true, if the evidence is present. Think of the prior or "previous" probability as your belief in the hypothesis before seeing the new evidence. If you had a strong belief in the hypothesis already, the prior probability will be large.
The prior is multiplied by a fraction. Think of this as the "strength" of the evidence. The posterior probability is greater when the top part numerator is big, and the bottom part denominator is small. The numerator is the likelihood. This is another conditional probability. It is the probability of the evidence being present, given the hypothesis is true.
Remember, the "probability of the evidence being present given the hypothesis is true" is not the same as the "probability of the hypothesis being true given the evidence is present". Now look at the denominator. This is the marginal probability of the evidence. Receive full access to our market insights, commentary, newsletters, breaking news alerts, and more. I agree to TheMaven's Terms and Policy. What Is Bayes Theorem? TheStreet Recommends. By Scott Rutt. By TheStreet Staff. By Vidhi Choudhary.
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