The document discusses Bayes' rule using a case study of a man named Adam who tested positive for COVID-19. It details the probabilities of false positives and negatives, and calculates the likelihood that Adam actually has COVID given the test result, which is found to be only 0.89%. This illustrates the importance of considering prior probabilities, also referred to as the base rate fallacy.

1. Behavioural Decision Making
23710
Instructor:
Dr Elif Incekara Hafalir

2. Bayes rule example
• Consider Adam from Australia: He tested positive for Covid. Assume
that the reliability of the test 90%, meaning that
• 10% of the time it gives false positive and 10% of the time it gives false
negative)
• Assume that the rate of Covid is 0.1% in Australia. What is the
likelihood that Adam actually has Covid?
• Let’s solve this using the Bayes rule formula

3. Bayes rule and Bayesian Updating
• Given that a person has covid, the probability to test positive is 90% (correct positive)
• P(T+|D+) = .90
• Given that a person does not have covid, the probability to test positive is 10% (false positive)
• P(T+|D-) = .10
• 0.1% of the population in Australia has Covid (The Bayes rate, aka prior probability)
• P(D+) = .001
• 99.9% of the population does not have Covid (I derive this number from the previous bullet point) by using the complement rule
• P(D-) = .999
• If someone (e.g. Adam in our example) tests positive, what is the probability that this person actually has Covid?
• P(D+|T+)?
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)

4. Bayes rule and Bayesian Updating
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)
Without even testing anyone, we know that the 0.1%
of the population has Covid
This is our prior.

5. Bayes rule and Bayesian Updating
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)
Without even testing anyone, we know that the .1%
of the population has Covid
This is our prior.
BASE RATE
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?

6. Bayes rule and Bayesian Updating
We know that the person tested positive. This is new information.
We need to update our prior after this new evidence.
Given that the person tested positive, what is the probability that person actually has Covid.
This is our posterior .
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)
Without even testing anyone, we know that the .1%
of the population has Covid
This is our prior.
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?

7. Bayes rule and Bayesian Updating
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)
There are two ways for a person to test positive:
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?

8. Bayes rule and Bayesian Updating
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)
There are two ways for a person to test positive:
The person has Covid and tests positive
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?

9. Bayes rule and Bayesian Updating
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)
There are two ways for a person to test positive:
The person has Covid and tests positive The person does NOT have Covid, but tests positive
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?

10. Bayes rule and Bayesian Updating
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)
Note that you want to find the
probability that the person has Covid
given that they are tested positive
The person has the disease and tests positive The person does NOT have the disease, but test positive
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?

11. Bayes rule and Bayesian Updating
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?

12. Bayes rule and Bayesian Updating
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)

13. Bayes rule and Bayesian Updating
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)

14. Bayes rule and Bayesian Updating
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?
𝑃 𝐷 + 𝑇+) =
𝑃 𝑇 + 𝐷+) 𝑃(𝐷+)
𝑃 𝑇 + 𝐷+) 𝑃 𝐷 + + 𝑃 𝑇 + 𝐷−) 𝑃(𝐷−)
1 − 𝑃(𝐷+)

15. Bayes rule and Bayesian Updating
𝑃 𝐷 + 𝑇+) =
.90 ∗ .001
.90 ∗ .001 + .10 ∗ .999
= .0089 = .89%
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?

16. Bayes rule and Bayesian Updating
Despite the fact that the test gives 90% true positive,
The posterior probability that the person has Covid given
that they are tested positive is only .89%
This finding may surprise you
if you do not pay enough
attention to
prior probability or Base rate.
BAYE RATE FALLACY
• P(T+|D+) = .90
• P(T+|D-) = .10
• P(D+) = .001
• P(D+|T+)?
𝑃 𝐷 + 𝑇+) =
.90 ∗ .001
.90 ∗ .001 + .10 ∗ .999
= .0089 = .89%