概率论札记 - 2 - 用贝叶斯定理来讨论“医疗诊断的可靠性到底有多少”

医学诊断的贝叶斯定理分析：误诊率与真实患病概率

最新推荐文章于 2023-09-24 18:04:03 发布

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最新推荐文章于 2023-09-24 18:04:03 发布 · 8.3k 阅读

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本文通过一道概率论习题探讨了在艾滋病筛查中，即使测试准确率高达99%，若艾滋病患病率仅为万分之一，当测试结果为阳性时，实际患病概率仅为约千分之二。贝叶斯定理揭示了误诊率与发病率对诊断可靠性的双重影响，强调了理解医学诊断概率的重要性。

只有愚蠢的人才会相信眼睛看到的。
——安·兰德

故事要从一道贝叶斯定理的简单习题讲起。大意是艾滋病患病率为万分之一，误诊率为5%，患有艾滋病者被诊断出来的概率为99%，请问在这样的设定下如果你被诊断为艾滋病阳性，那么你患艾滋病的概率是多少，原题如下——

Problem Denoted blood is screened for AIDS. Suppose the test has 99% accuracy, and that one in ten thousand people in your age group are HIV positive. The test has a 5% false positive rating, as well. Suppose the test screens you as positive. What is the probability you have AIDS? Is it 99%?

Solution: E_1=”test positive”, E_2=”test negative”. A_1=”You have AIDS”, A_2=”You don’t have AIDS”. Now we know $P(E_1|A_1)=99\%$ , we need to find $P(A_1|E_1)$ . Since “one in ten thousand people in your age group are HIV positive”, $P(A_1)=1/10000$ .”5% false positive rating” means $P(E_1|A_2)=5\%$ . By Bayes’ Theorem

$P (A 1 | E 1) = = \approx P ( E 1 | A$