1.3.1 Bayes' Theorem

Question 1

What is Bayes’ Theorem used for?

Answer 1

To calculate $P(A_i \mid B)$ when you know $P(B \mid A_i)$, the prior $P(A_i)$, and want the updated belief about $A_i$ after observing $B$.

Question 2

What’s the formula for Bayes’ Theorem (finite partitions)?

Answer 2

P(A _i|B)= P(B | A_i) * P(A_i) all over the sum from j=1 to n of P(B|A_j) * P(A_j)

Question 3

What are the conditions under which you should use Bayes’ Theorem?

Answer 3

1. P(B | A_i) is known.
2. P(B) is not.
3. P(A_i | B) is the goal.

Question 4

What theorem is Bayes’ Theorem derived from?

Answer 4

The Law of Total Probability and the definition of conditional probability.

Question 5

What’s the first step in applying Bayes’ Theorem?

Answer 5

Define events clearly (e.g. partitions $A_i$, and the observed event $B$).

Question 6

Why is the numerator in Bayes’ Theorem always part of the denominator?

Answer 6

Because the denominator is a sum over all possible ways $B$ can happen, and the numerator represents just one specific way.

Question 7

What does the denominator in Bayes’ Theorem represent?

Answer 7

The total probability of observing event $B$, regardless of which $A_i$ occurred.

Question 8

Why is Bayes’ Theorem important in probability?

Answer 8

It lets us flip conditionality: find $P(A \mid B)$ when we only know $P(B \mid A)$.

Question 9

What intuition underlies Bayes’ Theorem?

Answer 9

Update your beliefs about the world after getting new info. Prior $\rightarrow$ Posterior.

Question 10

What does the denominator in Bayes’ Theorem normalize?

Answer 10

It ensures that the probabilities across all $A_i$ sum to 1 — that you have a valid distribution.