Uncertainty
STAT 240 - Fall 2025
Differing Perspectives
Differing Perspectives
All of the men in this room took a pregnancy test and all of the men tested as pregnant.
Frequentist uncertainty: “There’s something wrong with this pregnancy test.”
Bayesian uncertainty: “It’s a little ridiculous to say ALL of the men are pregnant.”
Challenge
A doctor is stuck in a room with \(3\) patients who have been exposed to a perfectly lethal (\(0\%\) survival) and perfectly transmissible (any exposure leads to inoculation) virus that only occurs in \(0.15\%\) of the population. The doctor tests all \(3\) patients for the pathogen and finds that one of them is positive while the other two are negative. If a patient is sick the test will always be positive but if the patient is healthy the test will be negative \(95\%\) of the time. The doctor has \(4\) doses of anti-viral available to them that will always cure the illness and prevent death, given that the pathogen is present. If the pathogen is not present the anti-viral will always kill the recipient. Who should the doctor administer the anti-viral to?
Frequentist
Methods using the perspective that the probability of an event is defined as the relative frequency of that event’s occurence across infinite trials.
Empirical probability
“Data driven”
- Data dependent
Central Limit Theorem
Simple in application
Bayesian
Methods using the perspective that the probability of an event is the result of comparing the likelihood and prior knowledge of the event with the observed evidence.
Prior knowledge
“Assumption driven”
- Open to assumptions
Horrendous in application
Bayes’ Rule
\[P(A|B)=\frac{P(B|A)P(A)}{P(B)}\]
\[\text{Posterior}\ = \ \frac{\text{Likelihood} \times \text{Prior}}{\text{Evidence}}\]
Bayes’ Rule
\[P(B)=P(B|A)P(A)+P(B|A^c)P(A^c)\]
\[P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^c)P(A^c)}\]
Bayes’ Rule
\[ \begin{aligned} \text{Let} \quad & D^+ \equiv \text{Disease Positive} \\ & D^- \equiv \text{Disease Negative} \\ & T^+ \equiv \text{Test Positive} \\ & T^- \equiv \text{Test Negative} \\ \end{aligned} \]
Bayes’ Rule
\[ \begin{aligned} & P(D^+) = 0.0015\\ & P(D^-) = 0.9985 \\ & P(T^+|D^+) = 1.00 \\ & P(T^-|D^-) = 0.95 \\ & P(T^+|D^-) = 0.05 \end{aligned} \]
Bayes’ Rule
\[ \begin{aligned} P(D^+|T^+)& =\frac{P(T^+|D^+)P(D^+)}{P(T^+|D^+)P(D^+)+P(T^+|D^-)P(D^-)} \\ \\ & = \frac{1 \times 0.0015}{1 \times 0.0015+0.05 \times 0.9985}\\ \\ & = 0.02917 \end{aligned} \]
Errors
“All models are wrong, but some are useful” - George Box
In Estimation
Are we sure we did this all right?
All of the patients have been exposed to the virus.
The virus is perfectly transmissible, any exposure leads to inoculation.
In Inference
Confidence Intervals
Frequentist
“the method (the test for the pathogen) is going to produce the correct result.”
Credible Intervals
Bayesian
“the probability that the estimate (the proposition that the patient is disease positive) is the truth.”
Making Assumptions
Nothing is free of assumptions. What are some of the assumptions behind our known methods?
Mean / Median / Mode?
Variance / Standard Deviation?
z-Scores?
Least squares?
Interval Estimation
What is the value of pi?
We could (hypothetically) express is as an interval.
\[ \begin{aligned} \text{Lower Bound} = & \pi / 2 = 3.1/2 = 1.55 \\ \\ \text{Point Estimate} = & \pi / 2 = 3.14/2 = 1.57 \\ \\ \text{Upper Bound} = & \pi / 2 = 3.141592653589793/2 = 1.570796 \\ \end{aligned} \]
