2.3 Bayesian Risk Calculations

2.3 Bayesian Risk Calculations

Quick Answer: Build a two-hypothesis table (carrier vs not): list prior, multiply by conditional probability of the data under each hypothesis to get joint probabilities, then normalize joints to posterior probabilities. Negative tests reduce—but do not eliminate—carrier risk when detection rate < 100%.

Domain 2A Bayesian items reward orderly tables more than mental shortcuts. If your posterior does not sit between 0 and the prior after negative evidence, restart the arithmetic.

Why Bayes Appears on the CGC Exam

Real counseling almost never stops at 1/2 or 2/3. Clients bring negative panels, normal CK values, multiple healthy sons, or ethnicity-specific screens with published detection rates. Bayesian analysis is how you turn those facts into a residual carrier probability.

The Four-Column Habit

For hypothesis H1 = carrier and H2 = not carrier:

Posterior odds = joint(H1):joint(H2). Posterior probability carrier = joint(H1)/(joint1+joint2).

Worked Example A — X-Linked Carrier Risk with Negative Information

Stem (teaching numbers): A woman has one brother with Duchenne muscular dystrophy (DMD). Assume for this item that her mother is an obligate carrier (for example, the stem also notes an affected maternal uncle), so the woman's prior carrier risk = 1/2. She has two unaffected sons and no molecular testing yet. Approximate the probability an affected son would have been recognized as nearly 1 if she were a carrier (teaching simplification: each son of a carrier has 1/2 chance of being affected, so probability both sons unaffected given she is a carrier = (1/2)×(1/2) = 1/4). Probability both sons unaffected if she is not a carrier ≈ 1.

Arithmetic check: Joints 1/8 + 4/8 = 5/8. Posterior carrier risk = 1/5 (20%), down from 50%.

If the stem instead gives a CK-based conditional (for example, normal CK detects 2/3 of adult carriers—illustrative), replace the conditional column with P(normal CK | carrier) and P(normal CK | not carrier)≈1, then renormalize. Never mix CK and molecular conditionals unless the stem stacks independent tests carefully.

Variant: Isolated Case and Maternal Prior < 1

If the affected male is an isolated case, the mother's carrier prior may be less than 1 because of de novo variants in the son or maternal mosaicism. Exam stems usually supply the prior to use (for example, 2/3). Use the given prior; do not invent a different empiric number unless provided.

Worked Example B — Autosomal Recessive Carrier Screening with Detection Rate

Stem (teaching numbers): An Ashkenazi Jewish consultand has a population prior carrier risk of 1/25 for a condition. She has a negative targeted carrier screen with detection rate 95% (sensitivity 0.95 for carriers). Specificity ≈ 1 for this teaching item (true non-carriers almost never screen positive).

Conditionals for a negative result:

• P(neg | carrier) = 1 − 0.95 = 0.05
• P(neg | not carrier) ≈ 1

Priors:

• P(carrier) = 1/25
• P(not) = 24/25

Posterior carrier risk ≈ 0.00208 ≈ 1/481 (about 1/480).

Counseling translation: Her residual carrier risk after a negative 95% detection screen is roughly 1/480, not zero. Partner testing and residual-risk couple calculations (Section 2.5) still matter for some indications.

Same Math with Fractions Only

Joint carrier = (1/25)×(1/20) if you express 0.05 as 1/20 → 1/500. Joint not = 24/25. Posterior = (1/500) / (1/500 + 24/25) = (1/500) / (1/500 + 480/500) = 1/481. Same result.

Worked Example C — Combining Mendelian 2/3 with a Negative Screen

Unaffected sib of an AR proband: prior carrier = 2/3. Negative screen detection rate 90%.

Posterior ≈ 1/6. The 2/3 prior falls substantially but remains clinically meaningful if the partner is a carrier or untested.

Stacking Independent Conditionals

If two independent negative tests apply, multiply conditionals under the carrier row: P(data | carrier) = P(test1 neg | carrier) × P(test2 neg | carrier). Only multiply when independence is reasonable (exam stems usually imply this). Dependent tests (overlapping variant coverage) should not be multiplied as if independent.

Exam Traps in Bayesian Items

• Forgetting to normalize joints (reporting the joint as if it were the posterior)
• Using P(pos | carrier) when the person tested negative
• Setting residual risk to 0 after any negative screen
• Using 1/2 for an unaffected AR sib prior instead of 2/3 before Bayes
• Ignoring that unaffected sons modify XL carrier risk only under the carrier hypothesis
• Mixing population prior with Mendelian prior incorrectly (choose the prior that matches pedigree information)

Communication After the Math

Present posteriors as absolute risks and, when helpful, as reductions from the prior ("from 50% to about 20% after two unaffected sons"). Pair the number with the next medical action: partner testing, expanded sequencing, prenatal options, or reassurance with caveats. Bayesian output without a follow-up plan is incomplete counseling—and incomplete Domain 2A performance.