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乳腺癌筛查概率计算

用户:

这个题目，请用中文解答一下：

Consider mammographic screening for breast cancer. Using made up percentages for the sake of simplifying the numbers, let’s assume that 5 in 100 women have breast cancer. Suppose that if a person has breast cancer, then the mammograph test will find it 80 times out of 100. When the test comes out suggesting that breast cancer is present, we say that the result is positive, although of course there is nothing positive about this for the person being tested (a technical way of saying this is that the sensitivity of the test is 80%).

The test may also fail in the other direction, namely to indicate breast cancer when none exists. This is called a false positive finding. Suppose that if the person being tested actually doesn’t have breast cancer, the chances that the test nevertheless comes out positive are 10 in 100. (In technical terms, we would say that the specificity of the test is 90%.)

Based on the above probabilities, you are able to calculate the likelihood ratio. You’ll find use for it in the next exercise. If you forgot how the likelihood ratio is calculated, you may wish to check the terminology box earlier in this section and revisit the rain example.

Note: You can use the above diagram with stick figures to validate that your result is in the ballpark (about right) but note that diagram isn’t quite precise. Out of the 95 women who don’t have cancer (the gray figures in the top panel), about nine and a half are expected to get a (false) positive result. The remaining 85 and a half are expected to get a (true) negative result. We didn’t want to be so cruel as to cut people – even stick figures – in half, so we used 9 and 86 as an approximation.

Answered
Exercise 11: Bayes rule (part 2 of 2)
Consider the above breast cancer scenario. An average woman takes the mammograph test and gets a positive test result suggesting breast cancer. What do you think are the odds that she has breast cancer given the observation that the test is positive?

First, use your intuition without applying the Bayes rule, and write down on a piece of paper (not in the answer box below) what you think the chances of having breast cancer are after a positive test result. The intuitive answer will not be a part of your answer. It will be just for your own information.

Next, calculate the posterior odds for her having breast cancer using the Bayes rule. This will be your answer.

Hints:

Start by calculating the prior odds.
Determine the probability of the observation in case of the event (cancer).
Determine the probability of the observation in case of no event (no cancer).
Obtain the likelihood ratio as the ratio of the above two probabilities.
Finally, multiply the prior odds by the likelihood ratio.
Enter the posterior odds as your solution below. Give the answer in the form xx:yy where xx and yy are numbers, without simplifying the expression even if both sides have a common factor.

ChatGPT:

我们来一步步地用贝叶斯定理（Bayes’ Rule）计算在检测结果为阳性的条件下，一个平均女性实际患有乳腺癌的后验概率。

已知条件：

• 患病率（先验概率）：

  • 100 名女性中有 5 人患乳腺癌 ⇒
    $ P(\text{癌症}) = 0.05 $
    $ P(\text{无癌症}) = 0.95 $

• 检测准确性：

  • 如果有癌症，检测出阳性的概率是 80% ⇒
    $ P(\text{阳性}|\text{癌症}) = 0.80 $ （灵敏度）

  • 如果没有癌症，错误检测为阳性的概率是 10% ⇒
    $ P(\text{阳性}|\text{无癌症}) = 0.10 $ （假阳性率）

✅ 第一步：计算先验几率（Prior Odds）

先验几率 = 患病的概率 : 不患病的概率

$$\text{Prior Odds} = \frac{P(\text{癌症})}{P(\text{无癌症})} = \frac{0.05}{0.95} = 5 : 95$$

✅ 第二步：计算似然比（Likelihood Ratio）

似然比 = 有病时出现阳性结果的概率 ÷ 没病时出现阳性结果的概率

$$\text{Likelihood Ratio} = \frac{P(\text{阳性}|\text{癌症})}{P(\text{阳性}|\text{无癌症})} = \frac{0.80}{0.10} = 8 : 1$$

✅ 第三步：计算后验几率（Posterior Odds）

根据贝叶斯定理的几率形式：

$$\text{Posterior Odds} = \text{Prior Odds} \times \text{Likelihood Ratio}$$

代入数值：

$$\text{Posterior Odds} = (5 : 95) \times (8 : 1) = (5 \times 8) : (95 \times 1) = 40 : 95$$

✅ 最终答案：

$$\boxed{40:95}$$

这是在检测结果为阳性的条件下，该女性确实患癌与不患癌的后验几率。注意题中要求不要化简这个比例，所以答案保持为 40:95。

📌 补充说明（仅供理解）：

如果你愿意进一步将这个“几率”转化为“概率”，可以用公式：

$$P(\text{癌症}|\text{阳性}) = \frac{40}{40 + 95} ≈ 0.296 \Rightarrow 约 29.6\%$$

也就是说，即使检测结果是阳性，真正患癌的可能性也只有大约 30%，这可能和直觉相差很远，但这就是贝叶斯定理揭示的真相 —— 在低发病率下，假阳性可以大大影响结果。