The Vocabulary of Bayes

Larry Jones Larry Jones

Larry Jones

发布日期: 2025年1月18日
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Like many ideas, Bayes' Theorem introduces a new vocabulary and some adapted vocabulary. The new vocabulary consists of three terms

  • Prior probability - The probability that a hypothesis is true before we've seen any data.
  • Likelihood - The degree to which a hypothesis is true given that some data is observed.
  • Posterior Probability - The probability that a hypothesis is true after we've seen the data.

These definitions introduce our adapted vocabulary

  • Data - What we actually observe.
  • Hypothesis - Something which may or may not be true.
  • Model - A mechanism that we posit to generate the data we see.

Bayes' theorem can now be expressed as:

The conditional probability that the hypothesis given that the data is true equals the product of the probability of the hypothesis independent of the data and the likelihood of the data being true given that the hypothesis is true all divided by the probability of the data under any hypotheses.

Seems simple, right? It is. The difficulty is, of course, is assigning the (appropriate) numbers to the prior probability, the likelihood, and the evidence (the denominator in Bayes' theorem). In some situations, we can assign specific numbers or distributions to these values and perform the calculation "by hand." In most real world situations, we cannot calculate these values but must use a computer to help us.

But we'll discuss that problem in future.

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