Điều kiện, suy luận Bayesian và sai số lấy mẫu
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| standard error/ˈstændəd ˈerə/ | sai số chuẩn |
| sensitivity/ˌsensɪˈtɪvɪti/ | The probability that a test is positive when the condition is present. |
| specificity/ˌspesɪˈfɪsɪti/ | The probability that a test is negative when the condition is absent. |
A positive test, a rare condition
- A test is 90% sensitive and 95% specific; 2% of people have the condition. A random person tests positive. The chance they have it is only about 27%.
- GRE probability tests conditioning, independence, Bayes and the standard error 标准误 of a mean. (This is a mathematical model, not clinical advice.)
Conditioning and independence
- $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$: shrink the population to $B$ first.
- Independent: $P(A \cap B) = P(A)P(B)$. With $P(A) = 0.4$, $P(B) = 0.5$, $P(A \cap B) = 0.3$: $P(A \mid B) = 0.6 \neq 0.4$, not independent.
$P(A \cap B) = 0.12$ and $P(B) = 0.3$. Find $P(A \mid B)$.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
P(A | B) = P(A ∩ B) / P(B) = 0.12 / 0.3 = 0.4.
If $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cap B) = 0.2$, then $A$ and $B$ are independent.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. A and B are independent exactly when P(A ∩ B) = P(A)P(B), and 0.5 × 0.4 = 0.2.
Bayes with a table
- $P(D \mid +) = \dfrac{P(D)P(+\mid D)}{P(D)P(+\mid D) + P(\bar D)P(+\mid \bar D)}$. Sensitivity 灵敏度 $P(+\mid D)$; specificity 特异度 $P(-\mid \bar D)$.
Take 10,000 people. 200 have the condition, 180 of whom test positive. Of 9,800 without it, 5% — 490 — test positive. So $P(D \mid +) = \dfrac{180}{670} \approx 26.9\%$. The 90% is $P(+ \mid D)$, a different question.
Which probability is it?
Name the conditional probability a statement describes.
Prevalence 10%, sensitivity 80%, specificity 90%. Find $P(D \mid +)$ to three decimal places.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
0.08 / (0.08 + 0.09) = 8/17.
Standard error
- For iid observations with standard deviation $\sigma$, the sample mean has standard error $\sigma/\sqrt n$. $\sigma = 12$, $n = 36$: SE $= 2$; $n = 144$: SE $= 1$.
- Quadrupling $n$ halves the SE. The spread of individual observations stays $\sigma$.
Observations have $\sigma = 20$. What is the standard error of the mean of 100 independent observations?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The standard error of the mean is σ/√n = 20/√100 = 20/10 = 2.
Multiplying the sample size by 4 changes the standard error of the mean by a factor of
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
√4 = 2 in the denominator.
Common traps
- Reversing $P(+ \mid D)$ into $P(D \mid +)$.
- Leaving false positives out of the denominator.
- Confusing standard deviation with standard error.
Recap
- Condition by restricting the population; test independence by multiplying.
- Bayes: true positives over all positives — a table makes it concrete.
- SE $= \sigma/\sqrt n$.
- sensitivity: The probability that a test is positive when the condition is present.
- specificity: The probability that a test is negative when the condition is absent.