Bayes’ theorem

Dmitry Khlestkin


Bayes’ theorem was named after the Reverend Thomas Bayes (1701–61). His ideas were first published in 1763. Bayes’ works were significantly edited by Richard Price before being read at the Royal Society. The French mathematician and physicist Pierre-Simon Laplace developed the idea and published the theorem in its modern edition in 1812.

This is one of the fundamental theorems of elementary probability theory.

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Using Bayesian formula, one can fairly accurately recalculate the probability of an event, taking into account both the pre-existing data and data from the new observations. Bayes’ formula is based on the definition of a conditional probability – the probability of a particular event, provided that another event already occurred.

Thus, after the event occurred it is possible to calculate the probability that it was caused by a specific cause. Bayes’ theorem shows the relationship between the probabilitiy of the event A and the probability of the event B, the conditional probability of an event A with the existing B and vice versa. When calculating the Bayesian probability, you should identify the variables yourself. The practical application of Bayes’ theorem requires a large amount of calculations, so Bayesian method became actively used only after the revolution in computer and network technology.

An example of calculating the probability of rain

So, suppose you estimate that there is a 30 percent chance to rain tomorrow.

And you know that on an average day there is a 50 percent chance of clouds in the sky.

You also know that the likelihood of clouds is 100 percent given that rain is 100 percent (there is no rain without clouds).

You have the following information:

  • P(A)= Probability of rain = 30 percent
  • P(B)= Probability of clouds = 50 percent
  • P(B|A)= Probability of clouds given rain = 100 percent

You wake up in the morning and find there are clouds in the sky. You should now perform a Bayesian update on the probability of it raining, which is calculated as follows

So, as we recall, P(A|B)=P(A)*P(B|A)/P(B)= chance of rain * chance of clouds given rain/chance of clouds=30 %*100%/50%=60%

So, the probability of raining is 60 percent.

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