From Bayesian Math to Communicating Risk
To evaluate a medical test, we start with its core properties. Click on each card to reveal the mathematical notation.
How common is the disease in the population? (The "prior probability").
If a person *has* the disease, how often is the test positive? (True Positive Rate).
If a person is healthy, how often is the test negative? (True Negative Rate).
These properties don't answer the patient's crucial question: "I tested positive. What is the chance I actually have the disease?" That requires calculating the predictive values.
Probability of disease, given a positive test.
Probability of no disease, given a negative test.
Now, let's explore how these values change in the Interactive Explorer.
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This interactive example walks through a classic scenario from Gerd Gigerenzer's book, "Calculated Risks," using natural frequencies to make the math intuitive.
The diagnostic test calculations you've been exploring are actually a specific application of Bayes' theorem—the fundamental framework for updating beliefs with evidence.
This appendix shows you how diagnostic testing fits into the broader landscape of Bayesian methods, helping you understand the deeper statistical principles at work.
Diagnostic testing is just a specific case of Bayes' theorem. Click each card to see the mathematical relationship:
The fundamental framework for updating beliefs with evidence.
What we want to know: probability of disease given a positive test.
The performance properties of the diagnostic test.
What we believe before seeing the test result.
Using multiple tests to refine diagnosis over time.
Comparing diagnostic test performance.
The key insight of Bayesian methods is prior updating: each piece of evidence transforms today's posterior probability into tomorrow's prior. This is exactly what happens when a patient gets multiple diagnostic tests.
Start with Prior: Initial prevalence (your belief before testing)
Observe Evidence: Test result (positive or negative)
Calculate Posterior: Updated probability using Bayes' theorem
Use as New Prior: For the next test or decision
The worked example in this demonstration comes from Gerd Gigerenzer's excellent book on communicating statistical risk:
Calculated Risks: How to Know When Numbers Deceive You
by Gerd Gigerenzer
Essential reading for understanding how to communicate Bayesian reasoning clearly and effectively.