Probability Paradigms Dec 23, 2024

The Scenario

Imagine we’re dealing with a new diagnostic test for a rare genetic condition that affects 1 in 1000 people. The test has the following characteristics:

• If someone has the condition, the test will be positive 99% of the time (sensitivity)
• If someone doesn’t have the condition, the test will be negative 95% of the time (specificity)

Now, let’s say Sarah gets tested and receives a positive result. What’s the probability that she actually has the condition? Each probability paradigm approaches this question differently.

The Frequentist Approach

The frequentist views probability as the long-run frequency of events in repeated trials. They might say:

“If we were to run this test on a very large number of people, and look only at the positive test results, what proportion of those people would actually have the condition?”

The frequentist would focus on the sampling distribution and confidence intervals. They might resist making a direct probability statement about Sarah’s individual case, arguing that she either has the condition or doesn’t – there’s no probability about it. Instead, they would frame their analysis in terms of error rates and long-term frequencies.

In this case, they would calculate the false positive and true positive rates across a hypothetical population:

• In 1000 people, 1 person would have the condition
• That 1 person would test positive 99% of the time = 0.99 true positives
• The 999 people without the condition would test positive 5% of the time = 49.95 false positives
• Total positive tests = 50.94
• Proportion of positive tests that are true positives ≈ 1.94%

The Bayesian Approach

The Bayesian sees probability as a degree of belief that gets updated with new evidence. They would start with a prior probability (the base rate of 1/1000) and update it with the new evidence (the positive test result) using Bayes’ Theorem.

The Bayesian would say: “Before the test, we believed Sarah had a 0.1% chance of having the condition. The positive test result gives us new information to update this belief.”

Using Bayes’ Theorem: P(Condition|Positive) = P(Positive|Condition) × P(Condition) / P(Positive) = 0.99 × 0.001 / (0.99 × 0.001 + 0.05 × 0.999) ≈ 0.0194 or about 1.94%

The Bayesian is comfortable making a direct probability statement about Sarah ‘s specific case, saying there’s a 1.94% chance she has the condition.

The Propensity Approach

The propensity theorist views probability as an objective physical property – a tendency or disposition of a system to produce certain outcomes. They might say:

“The test result reflects the physical properties of Sarah’s genes and the testing mechanism. The probability emerges from the actual physical setup of the situation.”

They would focus on the causal mechanisms involved: the biological factors that cause the condition, the chemical reactions in the test, and how these physical properties interact to produce results. They might analyze the molecular basis of the test’s accuracy and how it interacts with the genetic markers it’s designed to detect.

These different approaches can lead to different practical recommendations:

• A frequentist might emphasize the need for additional testing, pointing out that in a large population, most positive results are false positives.

• A Bayesian might focus on gathering more prior information about Sarah (family history, symptoms, etc.) to refine the probability estimate.

• A propensity theorist might advocate for understanding the biological mechanism better, perhaps recommending a different type of test that targets the condition’s physical manifestations more directly.