140. Bayes' Rule
140.1. Beyes’ Rule
Start with the multiplication rule in both directions:
Set them equal:
Solve for :
So Bayes’ Rule is the multiplication rule rearranged.
Start: Definition of conditional probability
- Numerator: (Multiplication Rule)
Using the multiplication rule:
This expresses the joint probability as prior likelihood
So the numerator comes directly from multiplying:
- prior
- likelihood
- Denominator: (Total Probability Theorem)
If form a partition, then the total probability theorem gives:
This decomposes the probability of the “evidence” into all possible scenarios
- Put numerator & denominator together
Substitute both results into the definition:
Interpretation
- Numerator: how strongly scenario explains the evidence (likelihood prior)
- Denominator: total probability of seeing the evidence at all (sum of all scenario contributions)
So Bayes’ Rule is literally:
Example
Identifying Source of a Defective Part
A company sources a critical component from three suppliers:
-
Supplier A_1: 50% of parts
-
Supplier A_2: 30% of parts
-
Supplier A_3: 20% of parts
Each supplier has a different defect rate:
Here:
-
: part comes from supplier i
-
: part is defective
1. Total Probability Theorem: What is the overall defect rate?
We compute
Plug in numbers:
Overall defect rate = 2.4%
2. Multiplication Rule: Joint probability from each supplier
- Supplier 1:
- Supplier 2:
- Supplier 3:
This term appears in the numerator of Bayes’ Rule.
3. Bayes’ Rule: If a part is defective, what is the probability it came from supplier 3?
Substitute:
About a 41.7% chance the defective part came from supplier 3.
Even though supplier 3 supplies only 20% of parts, it becomes the most likely source of a defect because its defect rate is highest.