Suppose that a new study shows that the proportion of pregnant women who have gestational diabetes is actually larger than the previous estimate - rather than 5%, approximately 10% of pregnant women have gestational diabetes. Compared to your previous estimate of the probability that a woman has gestational diabetes given that she tests positive, the new estimate for the rate of a true positive using the new prior is:

(Note: Try doing this without recomputing the exact probability! Just use your intuition about the role of the prior to update your previous response.)

Question

Suppose that a new study shows that the proportion of pregnant women who have gestational diabetes is actually larger than the previous estimate - rather than 5%, approximately 10% of pregnant women have gestational diabetes.  Compared to your previous estimate of the probability that a woman has gestational diabetes given that she tests positive, the new estimate for the rate of a true positive using the new prior is:

(Note:  Try doing this without recomputing the exact probability!  Just use your intuition about the role of the prior to update your previous response.)

BlackTom AI · Accepted Answer

This question asks you to reason about how updating the prior probability for gestational diabetes (GD) from 5% to 10% affects your posterior probability P(D|+), i.e., the probability a woman has GD given a positive test. The key idea is to understand how priors influence Bayes updating: when the base rate (prior) of the disease increases, a positive test result tends to be more informative about a true positive, raising the posterior probability.

First, consider what a positive test tells us. The posterior P(D|+) combines two pieces: how common the disease is in the population (the prior P(D)) and how reliable the test is (its sensitivity and specificity). If the prior probability of the disease doubles from 0.05 to 0.10, you’re starting from a higher baseline likelihood that a person with a positive result actually has the disease. All else being equal, increasing the prior makes P(D|+) larger because the numerator P(D)×P(+|D) grows and the denominator P(+) = P(D)×P(+|D) + P(~D)×P(+|~D) also shifts in a way that still supports a higher ratio in favor of D given a positive test.

Option analysis:
- The statement that the new estimate is larger aligns with the intuitive Bayesian update: more disease in the population makes a positive result more likely to be a true positive. Therefore, P(D|+) increases when the prior P(D) increases from 5% to 10%.
- If we were to consider alternative implications, such as test accuracy meaning the same positive result would become less reliable, that would require changes in the test’s sensitivity or specificity. Since the question asks you to reason about the role of the prior and not to recompute exact probabilities, we maintain the intuition that a higher prior raises the posterior for a positive result.

In short, increasing the prior GD rate makes a positive test more likely to reflect true disease, so the posterior probability given a positive test should be larger. This preserves coherence with Bayesian updating: a higher base rate strengthens the claim that a positive result indicates true disease.

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