Bayes' Theorem

The Medical Testing Paradox & Conditional Probability

📚 Introduction

🏥 Medical Testing Paradox

假设有一种罕见疾病，患病率为1%。现在有一种检测方法，准确率高达99%（灵敏度99%，假阳性率5%）。 如果你的检测结果呈阳性，你真正患病的概率是多少？

Intuition says ~99%, but the real answer will surprise you...

📐 Bayes' Formula

P(A|B) = P(B|A) × P(A) / P(B)

Posterior = Likelihood × Prior / Marginal

🔑 Key Concepts

• Prior P(A): Probability before new evidence (e.g., disease prevalence)
• Likelihood P(B|A): Probability of evidence given event (e.g., sensitivity)
• Posterior P(A|B): Probability of event after observing evidence

💡 Why is it counterintuitive?

当疾病很罕见时，即使检测很准确，假阳性的绝对数量也可能远超真阳性。 这就是为什么不建议对低风险人群进行大规模筛查的原因之一。

⚙️ Adjust Parameters

Disease Rate (Prior): 1.0%

Sensitivity (True Positive Rate): 99%

False Positive Rate: 5.0%

🎯 Bayesian Calculation

If you test positive, the probability of actually having the disease is

16.7%

Not the intuitive 99%

🤯 Surprised?

即使检测准确率很高，当疾病罕见时，阳性结果中大部分可能是假阳性！ 这就是贝叶斯定理的威力——它告诉我们要考虑基础概率（先验）。

📊 Confusion Matrix

Actually Sick

Actually Healthy

Test Positive

True Positive
0.99%

False Positive
4.95%

Test Negative

False Negative
0.01%

True Negative
94.05%

👥 Population Visualization

Imagine 1000 people tested, see the distribution:

True Positive (10)

False Positive (50)

False Negative (0)

True Negative (940)

60

Total Positive

10

Actually Sick

50

False Alarm

16.7%

PPV