Conditioning, Bayesian inference and sampling error
| English | 中文 | Pinyin |
|---|---|---|
| standard error/ˈstændəd ˈerə/ | 标准误 | biāo zhǔn wù |
| posterior probability/pɒˈstɪərɪə ˌprɒbəˈbɪlɪti/ | 后验概率 | hòu yàn gài lǜ |
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)?
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?
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 ____.
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.
Which answer fits this case?
Separate conditional, joint and independent-event probabilities
Two mutually exclusive events with positive probabilities are independent.
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.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.