# Why is Bayes' theorem important

## Bayes' theorem

1.* term*: Theorem from the calculus of probabilities for calculating conditional probabilities. Denote w (θ_{1}) and w (θ_{2}) the unconditional probabilities for the occurrence of the stochastically dependent events θ_{1} and θ_{2}, and further denotes w (θ_{2}│θ_{1}) the (conditional) probability for the event θ_{2} on condition that θ_{1} has occurred, the inverse conditional probability w (θ_{1}│θ_{2}) according to Bayes' theorem:

2.* application*: The Bayes' theorem is especially used in information processing (see also information value): Denotes θ_{2} a certain event (e.g. future bankruptcy of a borrower) and θ_{1} information about the event (e.g. credit check by the lender), then the adjustment of the probability judgment about θ follows_{2} (in the example: adjustment of the probability judgment about the bankruptcy of the borrower) rationally according to Bayes' theorem. The a priori judgment w (θ_{2}) into the a posteriori judgment w (θ_{1}) converted according to the formula given.

3.* Calculation example*: A lender estimates the likelihood that a borrower will become insolvent (event θ_{2}), to 5 percent. He also assumes that the credit check routinely carried out in-house provides a correct assessment with an 80 percent probability. The test turns out negative (event θ_{1}). The probability for this negative test result is w (θ_{1}) = 0.8 x 0.05 + 0.2 x 0.95 = 0.23. It is made up of two individual probabilities: the probability of a correct insolvency forecast (0.8 · 0.05) plus the probability of an incorrect insolvency forecast (0.2 · 0.95). According to Bayes' theorem, the lender adjusts its probability judgment as follows:

The probability of insolvency increases from an initial 5 percent (a priori judgment) to 17.4 percent (a posteriori judgment).

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