条件付け、ベイズ推定およびサンプリング誤差
導入済み| English | 意味 |
|---|---|
| standard error/ˈstændəd ˈerə/ | 標準誤差 |
| posterior probability/pɒˈstɪərɪə ˌprɒbəˈbɪlɪti/ | 事後確率 |
A decision before an answer
- A positive screening result is not the same event as having the condition. When the condition is rare, false positives can outnumber true positives even with a fairly specific test.
- Your goal: Separate conditional, joint and independent-event probabilities.
Read the relationship
- For P(B)>0, P(A|B)=P(A∩B)/P(B): restrict the population to B before computing the fraction in A. Independence means P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A) when the denominator is nonzero. Mutually exclusive events with positive probabilities cannot be independent, because their joint probability is zero. Sampling without replacement usually changes later probabilities; a fixed denominator for successive draws would describe a different experiment.
- Calculate posterior probabilities with a complete base-rate table.
P(A)=0.4, P(B)=0.5 and P(A∩B)=0.3. What is P(A|B)?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Restrict to B: divide the joint probability 0.3 by P(B)=0.5, obtaining 0.6.
Use the defining rule
- Build joint probabilities by multiplying a prior category proportion by the relevant conditional probability. If D is the condition and + a positive result, P(D∩+)=P(D)P(+|D). The total positive probability is P(D)P(+|D)+P(not D)P(+|not D), because these are disjoint and cover every positive result. Bayes divides the first joint probability by this total. Sensitivity is P(+|D), while specificity is P(−|not D); the false-positive rate is one minus specificity.
- Use variance and sample size to determine a sample mean standard error.
An iid sample of size 25 has mean standard error 6. What is the mean standard error for size 100 from the same population?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Population standard deviation is 6√25=30. Dividing by √100=10 gives standard error 3.
Check the conditions
- For a hypothetical test with prevalence 2%, sensitivity 90% and specificity 95%, use a population of 10,000 for clarity. Of 200 people with the condition, 180 test positive. Of 9,800 without it, 490 test positive. Therefore 180 of 670 positive results have the condition: posterior 18/67, about 26.9%. Reversing the conditional would give 90%, answering a different question. This is a mathematical model, not a recommendation about clinical decisions.
- Use variance and sample size to determine a sample mean standard error.
Let P(A)=0.4, P(B)=0.5 and P(A∩B)=0.3. Then P(A|B)=0.6, so A and B are not independent, since 0.3≠0.4·0.5. If a variable has standard deviation 12, an independent sample of size 36 has mean standard error 2. Increasing the sample size to 144 gives 1; the individual-observation standard deviation stays 12.
For independent observations with population standard deviation 15 and n=25, the sample mean standard error is ____.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Divide 15 by √25=5. Independence and finite population variance are the stated conditions.
Apply the task format
- For independent identically distributed observations with finite variance σ², the sample mean has expectation μ and variance σ²/n; its standard error is σ/√n. This is spread of repeated sample means, not the spread of individual observations. Quadrupling n halves the standard error, rather than quartering it. Normal population data give an exactly normal mean; otherwise a central-limit approximation needs adequate conditions and sample size. Correlation invalidates the simple independent variance calculation.
- Use variance and sample size to determine a sample mean standard error.
Do not reverse P(+|D) into P(D|+). Include false positives in the posterior denominator. Standard deviation of individual observations and standard error of their mean have different sample-size behaviour.
このケースに当てはまる答えはどれか？
条件付き確率、同時確率、独立事象の確率を分離する
Two mutually exclusive events with positive probabilities are independent.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Their joint probability is zero, but the product of their positive probabilities is positive.
Keep the distinctions
- posterior probability 后验概率 — A probability conditional on the observed evidence after accounting for prior proportions.
- standard error 标准误 — The standard deviation of a statistic across repeated samples.
- Separate conditional, joint and independent-event probabilities.
- Calculate posterior probabilities with a complete base-rate table.
- Use variance and sample size to determine a sample mean standard error.
Match each term with its precise meaning in this lesson.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
難易度 3/5 · 200 コイン + 今日初回完了時に小ボーナス付与
The order follows the stated process; check each stage before the next.