Day 11: Bayes' Theorem: The Mathematics of Updating Beliefs

A Deep Dive into Probability and Belief Updates

What is Bayes' Theorem?

Basic Definition

Bayes' Theorem calculates the probability of an event based on prior knowledge of conditions related to the event. It's expressed as:

P(A|B) = [P(B|A) × P(A)] / P(B)

Where:

P(A|B) = Posterior probability
P(B|A) = Likelihood
P(A) = Prior probability
P(B) = Total probability

Movie Theater Example:

Component	Value	Meaning
P(Late)	30%	Bookings in last 2 hours
P(Full\|Late)	90%	Full shows when booked late
P(Full)	40%	Overall full shows

Practical Applications

Medical Diagnosis Example

Component	Description
Prior	Disease prevalence in population
Likelihood	Test accuracy
Posterior	Actual probability after test

Spam Detection Example

Component	Description
Prior	General spam frequency
Likelihood	Word patterns in spam
Posterior	Email spam probability

Key Points to Remember

Base Rate Matters: Always start with prior probabilities
Evidence Updates Beliefs: New information refines probability
Context is Critical: Same evidence can mean different things
Systematic Approach: Break complex problems into steps

Conclusion

Bayes' Theorem isn't just a formula - it's a powerful framework for updating beliefs with new evidence. Whether in medicine, technology, or daily decisions, it helps us think more clearly about probability and uncertainty.

Remember: Probability isn't about certainty; it's about being systematically less wrong over time.