Strong and weak sampling

Strong and weak sampling are two sampling approach^[1] in Statistics, and are popular in computational cognitive science and language learning.^[2] In strong sampling, it is assumed that the data are intentionally generated as positive examples of a concept,^[3] while in weak sampling, it is assumed that the data are generated without any restrictions.^[4]

Formal Definition

In strong sampling, we assume observation is randomly sampled from the true hypothesis:

${\displaystyle P(x|h)={\begin{cases}{\frac {1}{|h|}}&{\text{, if }}x\in h\\0&{\text{, otherwise}}\end{cases}}}$

In weak sampling, we assume observations randomly sampled and then classified:

${\displaystyle P(x|h)={\begin{cases}1&{\text{, if }}x\in h\\0&{\text{, otherwise}}\end{cases}}}$

Consequence: Posterior computation under Weak Sampling

${\displaystyle P(h|x)={\frac {P(x|h)P(h)}{\sum \limits _{h'}P(x|h')P(h')}}={\begin{cases}{\frac {P(h)}{\sum \limits _{h':x\in h'}P(h')}}&{\text{, if }}x\in h\\0&{\text{, otherwise}}\end{cases}}}$

Therefore the likelihood ${\displaystyle P(x|h')}$ for all hypotheses ${\displaystyle h'}$ will be "ignored".

References

1. Xu, Fei. "Bayesian word learning Sensitivity to sampling in Bayesian word learning" (PDF). MIT. Developmental Science.
2. Hsu, Anne. "Sampling assumptions in language learning 1 Running head: SAMPLING ASSUMPTIONS IN LANGUAGE LEARNING Sampling assumptions affect use of indirect negative evidence in language learning". ResearchGate.
3. Navarro, Danielle. "Lecture 20: Strong vs weak sampling" (PDF). Computational Cognitive Science.
4. Navarro, Daniel (2012). "Sampling assumptions in inductive generalization". Cognitive Science. 36 (2): 187–223. doi:10.1111/j.1551-6709.2011.01212.x. PMID 22141440.

External links

• Lecture 20: Strong vs weak sampling