What does Bayes' Theorem say in plain English?

Question

What does Bayes’ Theorem mean? What do the different parts of the equation say in plain language?

Answer

Let’s first recall that Bayes’ theorem is stated as follows

P(A|B) = P(B|A) * P(A) / P(B)

In plain language, Bayes’ theorem is saying

If we know that B is true and would like to find when A is also true, we should

  • focus our attention on the outcomes where B is true
  • and then once focused there, look at all the outcomes in this area where A is true.

The first point first. Since all probabilities are between 0 and 1, if we consider the number 1 to mean all possible outcomes, then P(B) means all outcomes where B is true. So this is where the denominator of P(B) in the equation comes from.

For the second point, we’re asking here about the intersection of A and B which we can write as P(B|A) * P(A), explaining the numerator of the equation.

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Suppose you know the probability of A (and consequently of not A), and for each of these outcomes you know the probability of B/not B.

You want actually to reverse the prediction and figure out the proba of A if B subsequently happens. This corresponds to a fraction of the ensemble where B happens, because this ensemble also includes the event that A doesn’t happen and B does.

Consequently, the proba of this event you look for is equal to the probability of the event that A happens and then B divided by the overall probability of B happening.

This can be depicted by the scheme below


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