Lecture on Bayes' Theorem and Bayesian Inference

Introduction to Bayes' Theorem

• Bayes' Theorem is a significant concept in probability and statistics.
• It forms the foundation of Bayesian inference.
• The theorem reshaped the understanding of probabilities and statistics.

Conditional Probability Reminder

• Conditional probability: ( P(A|B) = \frac{P(A \cap B)}{P(B)} )
• Switching the condition: ( P(B|A) = \frac{P(B \cap A)}{P(A)} )
• Intersection is commutative: ( A \cap B = B \cap A )

Deriving Bayes' Theorem

• By substituting the conditional probabilities:
  • ( P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} )
• This formula helps interchange ( P(A|B) ) and ( P(B|A) ) using the ratio ( \frac{P(A)}{P(B)} ).

Application of Bayes' Theorem

• Easier computation: If either ( P(A|B) ) or ( P(B|A) ) is easy to compute, use Bayes' Theorem to find the other.

Example: Probability with Children

• Scenario: Given a couple with two children, at least one is a girl. Find the probability both are girls.
• Bayes' Theorem Application:
  • ( P(2G|1G) = \frac{P(1G|2G) \cdot P(2G)}{P(1G)} )
  • ( P(1G|2G) = 1 ) (100% certainty if there are two girls)
  • ( P(2G) = \frac{1}{4} ) (one scenario out of four possible child combinations)
  • ( P(1G) = \frac{3}{4} ) (three scenarios include at least one girl)
  • Result: ( \frac{1}{3} ), consistent with previous probability calculations.

Conclusion

• Bayes' Theorem is validated and useful for probability calculations.
• Demonstrates its application in real-world scenarios and more complex statistical problems.