1
$\begingroup$

Bayes Theorem is derived using conditional probability:

$$P(A|B)\cdot P(B) = P(A) \cdot P(B|A)$$

Here terms are either probability or conditional probability. But later, in

$$P(A|B) = P(A)\cdot \frac{P(B|A)}{P(B)}$$

$P(B|A)$ is called likelihood.

Since likelihood and probability are not the same. Please help me understand: How did we end up with likelihood here and when do we use conditional probability and when do we use likelihood?

$\endgroup$

1 Answer 1

3
$\begingroup$

As you note, $P(B|A)$ plays two roles. The role depends on whether $A$ is fixed or $B$ is fixed.

When $A$ is fixed, $P(B|A)$ is the probability of $B$ given the value of $A$. Since probabilities add up to one, we have $\sum_B P(B|A) = 1$ for any value of $A$ for which the conditional probability is defined.

When $B$ is fixed, $P(B|A)$ is a function of $A$. This function is called the likelihood. Since likelihood is not probability, $\sum_A P(B|A)$ does not have to equal 1. It doesn't even have to be finite. Nevertheless, $\sum_A P(B|A)\,P(A) = P(B)$, which depends on $B$.

$B$ is fixed when Bayes theorem is used to compute the conditional probability $P(A|B)$: \begin{equation} P(A|B) = \frac{P(B|A)\,P(A)}{P(B)} . \end{equation} Therefore, $P(B|A)$ plays the likelihood role in this expression.

If instead we used Bayes theorem to compute the probability $P(B|A)$, the roles would be reversed and $P(A|B)$ would play the likelihood role: \begin{equation} P(B|A) = \frac{P(A|B)\,P(B)}{P(A)} . \end{equation}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.