Baesyan Inference

Conditional Probability

• Events are independent when their probability is unaffected by other events.
  • ex. when flipping a coin: you always have 50% of getting the tail.
• Events are dependent when their probability changes as conditions change.
• P(A|B) is called Conditional Probability and expresses the likelihood of an event occurring, assuming a different one has already happened.
• If A and B are independent: P(A|B) = P(A)P(B)/P(B) = P(A)
• If A and B are dependent: P(A|B) = P(A)∩P(B)/P(B)

Considering the formula favorable outcomes/sample space, for A|B to happen, B must happen (P(A)∩P(B)) and the sample space is all B since we know that B is given.

It is also important to observe that P(A|B) is generally different from P(B|A)

The Law of Total Probability

• If A = B1∪B2∪B3∪…∪Bn
• P(A) = B1P(A|B1) + B2P(A|B2) + … + Bn*P(A|Bn)
  • ex. the probability of being vegan is the probability of being a vegan man + the probability of being a vegan woman

Additive Law

• AUB = A+B - A∩B otherwise, we would count the intersection twice.
• ⇒ P(A∩B) = P(A) + P(B) - AUB

Multiplication Rule

• Since P(A|B) = P(A)∩P(B)/P(B)
• and considering the multiplication rule
• ⇒ P(A)∩P(B) = P(A|B)*P(B)
  • ex. if you know that when B happens P(B)=0.5, P(A|B) = 0.8,
  • then you can obtain P(A)∩P(B) = 0.5*0.8

Bayes's Law

• Take two events A and B
• Considering that P(A|B) = P(A)∩P(B)/P(B)
• Using the Multiplication rule we get that P(A|B) = P(B|A)*P(A)/P(B)

This rule can be used to find a relationship between different conditional probabilities of two events.