Solved – Bayes’ Theorem Intuition

bayesianintuitionlikelihood

I've been trying to develop an intuition based understanding of Bayes' theorem in terms of the prior, posterior, likelihood and marginal probability. For that I use the following equation:
$$P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$
where $A$ represents a hypothesis or belief and $B$ represents data or evidence.
I've understood the concept of the posterior – it's a unifying entity that combines the prior belief and the likelihood of an event. What I don't understand is what does the likelihood signify? And why is the marginal probability in the denominator?
After reviewing a couple of resources I came across this quote:

The likelihood is the weight of event $B$ given by the occurrence of $A$ … $P(B|A)$ is the posterior probability of event $B$ , given that event $A$ has occurred.

The above 2 statements seem identical to me, just written in different ways. Can anyone please explain the difference between the two?

Best Answer

Although there are four components listed in Bayes' law, I prefer to think in terms of three conceptual components:
$$ \underbrace{P(B|A)}_2 = \underbrace{\frac{P(A|B)}{P(A)}}_3 \underbrace{P(B)}_1 $$

  1. The prior is what you believed about $B$ before having encountered a new and relevant piece of information (i.e., $A$).
  2. The posterior is what you believe (or ought to, if you are rational) about $B$ after having encountered a new and relevant piece of information.
  3. The quotient of the likelihood divided by the marginal probability of the new piece of information indexes the informativeness of the new information for your beliefs about $B$.
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