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I have been studying Artificial Intelligence and I have noticed that the Bayes' rule allows us to infer the posterior probability if a variable. But, my question is, what does the word, or phrase, 'posterior' mean in this context with regard to the Bayes' rule?

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The word posterior stems from the latin word posterior (base form posterus), which means later, after, behind.

In the context of probability and Bayes' theorem it denotes a probability, often written as a symbol $P(A|B)$, where $A$ and $B$ are random variables. This symbol $P(A|B)$ means the probability of $A$ after you have observed $B$. Hence the naming posterior probability, or short posterior.

In this context the probability of observing a certain value of the random variable $B$ is denoted by $P(B)$ and is called prior probability, or just prior for short sometimes. The word prior also stems from a Latin word, pri, which means before. The name hints that you will gather observations of $B$ before you calculate $P(A|B)$.

In this way you use prior information to improve your model of $A$, expressed by the posterior probability $P(A|B)$. By this your probability of $A$ is based on the observed outcome of $B$. Like it has been said by John Maynard Keynes "When the facts change, I change my mind. What do you do, sir?".

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