AHL 4.13—Bayes theorem

Bayes' Theorem

Bayes' theorem, named after the 18th-century British mathematician Thomas Bayes, is a fundamental concept in probability theory and statistics. It provides a way to update the probability of an event based on new evidence or information.

Basic Formulation

The basic form of Bayes' theorem is:

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Where:

• $P(A|B)$ is the posterior probability of A given B
• $P(B|A)$ is the likelihood of B given A
• $P(A)$ is the prior probability of A
• $P(B)$ is the marginal probability of B

Note

This formula allows us to calculate the probability of an event A occurring, given that we know event B has occurred, by using our prior knowledge of the probabilities of A and B, and the probability of B occurring given A.

Application to Multiple Events

In the context of the IB Math AA HL curriculum, students are expected to apply Bayes' theorem to a maximum of three events. This extension of the basic formula involves considering multiple conditional probabilities.

For three events A, B, and C, Bayes' theorem can be expressed as:

$$P(A|B,C) = \frac{P(B,C|A) \cdot P(A)}{P(B,C)}$$

Where $P(A|B,C)$ is the probability of A given both B and C have occurred.

Example

Suppose we have a medical test for a rare disease. Let:

• A: Patient has the disease

• B: Test result is positive

Given:

• P(A) = 0.01 (1% of the population has the disease)

• P(B|A) = 0.95 (95% chance of a positive test if you have the disease)

• P(B|not A) = 0.10 (10% chance of a false positive)

We can calculate P(A|B), the probability of having the disease given a positive test:

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B|A) \cdot P(A) + P(B|not A) \cdot P(not A)}$$

$$= \frac{0.95 \cdot 0.01}{0.95 \cdot 0.01 + 0.10 \cdot 0.99} \approx 0.0876$$

So, even with a positive test, there's only about an 8.76% chance of actually having the disease.

Connection to Independent Events

Bayes' theorem has a special relationship with independent events. Two events A and B are considered independent if the occurrence of one does not affect the probability of the other.

For independent events:

• $P(A|B) = P(A)$
• $P(B|A) = P(B)$

In this case, Bayes' theorem simplifies to:

$$P(A|B) = \frac{P(B) \cdot P(A)}{P(B)} = P(A)$$

This reinforces the definition of independence, as the conditional probability is equal to the prior probability.

Common Mistake

Students often confuse independence with mutual exclusivity. Independent events can occur together, while mutually exclusive events cannot.

Medical Risk Assessment

One of the most practical applications of Bayes' theorem is in medical risk assessment. It allows healthcare professionals to update the probability of a patient having a certain condition based on test results or symptoms.

Example

Consider a screening test for a genetic disorder:

• A: Patient has the disorder

• B: Test is positive

• C: Patient has a family history of the disorder

Given:

• P(A) = 0.001 (0.1% prevalence in general population)

• P(B|A) = 0.99 (99% sensitivity)

• P(B|not A) = 0.02 (2% false positive rate)

• P(C|A) = 0.30 (30% of those with the disorder have a family history)

• P(C) = 0.05 (5% of the population has a family history)

We can use Bayes' theorem to calculate P(A|B,C), the probability of having the disorder given a positive test and family history:

$$P(A|B,C) = \frac{P(B,C|A) \cdot P(A)}{P(B,C)}$$

$$= \frac{P(B|A) \cdot P(C|A) \cdot P(A)}{P(B,C)}$$

(Assuming B and C are conditionally independent given A)

$$\approx 0.0294$$

This means there's about a 2.94% chance of having the disorder given these conditions.

Tip

When solving complex Bayes' theorem problems, it often helps to draw a probability tree or use a contingency table to organize the given information.

Philosophical Implications

The application of Bayes' theorem raises interesting questions about the nature of knowledge and its applicability across different domains. In the context of Theory of Knowledge (TOK), students might consider:

1. How does the applicability of probabilistic knowledge vary across different areas of knowledge (e.g., natural sciences vs. human sciences)?
2. What are the implications of measuring the value of knowledge solely in terms of its applicability?
3. How does Bayesian reasoning influence our understanding of scientific method and the process of updating beliefs based on new evidence?

Note

These philosophical considerations highlight the broader implications of Bayes' theorem beyond its mathematical formulation, encouraging students to think critically about the nature of knowledge and probability.

Conclusion

Bayes' theorem is a powerful tool for updating probabilities based on new information. Its applications range from medical diagnostics to machine learning algorithms. Understanding how to apply this theorem to multiple events and recognizing its connection to independent events is crucial for IB Math AA HL students. Moreover, considering its philosophical implications can lead to deeper insights into the nature of knowledge and probability across various disciplines.