What is the difference between prior, likelihood, and posterior probabilities in Bayesian Theorem?

In the Bayesian Theorem, the terms prior, likelihood, and posterior probabilities represent different stages of the belief update process. Let's break down each term:

• Prior Probability (P(H)): This is your initial belief or degree of certainty about a hypothesis (H) before considering any new evidence. It's called "prior" because it represents your belief before the new information comes in. This can be based on previous knowledge, experience, or even intuition. In the Bayesian formula, P(H) is the initial probability of the hypothesis.

• Likelihood (P(E|H)): This measures how likely you are to observe the evidence (E) given that the hypothesis (H) is true. It's a way of saying, "If the hypothesis is correct, how likely is it that we would see this evidence?" In the formula, P(E|H) is the probability of the evidence given the hypothesis.

• Posterior Probability (P(H|E)): This is your updated belief about the hypothesis (H) after considering the new evidence (E). It's called "posterior" because it represents your belief after the new information has been taken into account. The Bayesian theorem provides a way to calculate this updated belief using the prior and likelihood. In the formula, P(H|E) is the probability of the hypothesis given the evidence.

In essence, the Bayesian theorem helps you update your prior belief (P(H)) into a posterior belief (P(H|E)) by incorporating the likelihood of the evidence (P(E|H)). This process is often visualized as a cycle, where your beliefs are continually updated as new evidence comes in.