Bayesian Risk Assessment
Log in to starLast updated 2mo ago
Bayesian analysis updates a prior probability (based on pedigree or population data) with new conditional information (test results, family history, unaffected children) to calculate a posterior probability. It is one of the most useful quantitative tools in personalized risk assessment.
| Hypothesis 1 (e.g., Carrier) | Hypothesis 2 (e.g., Non-carrier) | |
|---|---|---|
| Prior probability | P(H1) | P(H2) |
| Conditional probability | P(evidence | H1) | P(evidence | H2) |
| Joint probability | Prior x Conditional | Prior x Conditional |
| Posterior probability | Joint1 / (Joint1 + Joint2) | Joint2 / (Joint1 + Joint2) |
Example 1: Carrier risk updated by negative test
Problem: A woman has a 1/2 prior probability of being a carrier for CF. She undergoes carrier testing that detects 90% of pathogenic variants in her population. The test is negative. What is her updated carrier risk?
Solution:
| Carrier | Non-carrier | |
|---|---|---|
| Prior | 1/2 | 1/2 |
| Conditional (negative test) | 0.10 (10% of variants missed) | 1.00 |
| Joint | 1/2 x 0.10 = 0.05 | 1/2 x 1.00 = 0.50 |
| Posterior | 0.05 / 0.55 = 1/11 | 0.50 / 0.55 = 10/11 |
- Updated carrier risk: 1/11 (approximately 9%)
- The residual risk is 1/11, not zero, because the test does not detect all variants
Example 2: Combining family history with test results
Problem: A woman's brother has DMD. Her prior carrier risk is 1/2. She has 2 unaffected sons and a normal CK level (about two-thirds of carriers have elevated CK, so ~1/3 of carriers have a normal CK). What is her updated carrier risk?
Solution:
| Carrier | Non-carrier | |
|---|---|---|
| Prior | 1/2 | 1/2 |
| Conditional: 2 unaffected sons | (1/2)^2 = 1/4 | 1 |
| Conditional: normal CK | 1/3 | 1 |
| Combined conditional | 1/4 x 1/3 = 1/12 | 1 x 1 = 1 |
| Joint | 1/2 x 1/12 = 1/24 | 1/2 x 1 = 1/2 |
| Posterior | (1/24) / (1/24 + 1/2) = 1/13 | 12/13 |
- Updated carrier risk: 1/13 (approximately 8%)
Example 3: Prenatal/child risk for late-onset conditions
Problem: A 30-year-old man has a parent with Huntington disease (AD, 100% penetrance by age 70). The man is currently unaffected. What is his risk of carrying the variant?
Solution:
| Carries HD variant | Does not carry | |
|---|---|---|
| Prior | 1/2 | 1/2 |
| Conditional (unaffected at 30) | ~0.85 (about 15% penetrant by 30) | 1 |
| Joint | 1/2 x 0.85 = 0.425 | 1/2 x 1 = 0.50 |
| Posterior | 0.425 / 0.925 = 46% | 54% |
- His risk is slightly reduced from 50% to 46% because he is still unaffected at 30
- As he ages without symptoms, his posterior risk continues to decrease
- Prior probability: What you know before the new evidence. Comes from pedigree analysis, population frequencies, or inheritance patterns.
- Conditional probability: How likely is the evidence under each hypothesis?
- Residual risk: The posterior probability of being a carrier after a negative test. Never zero unless the test has 100% sensitivity.
- Multiple pieces of evidence: Multiply the conditional probabilities together (they are independent observations)
- Forgetting to use conditional probability for the alternative hypothesis: The conditional for non-carrier is usually 1 (certain to have a negative test, certain to have unaffected sons) but not always
- Confusing sensitivity with detection rate: For carrier testing, the detection rate (proportion of variants detectable) is the relevant number, not analytical sensitivity
- Not recognizing when Bayesian analysis is needed: Any time you have a prior risk AND new information, you should consider updating with Bayes