How is the posterior probability of Steve being a librarian calculated using the representative sample grid in the two-occupation example?
Conditions
- Population consists only of librarians and farmers with a 1:20 ratio
- Likelihoods are stipulated as 40% for librarians and 10% for farmers
- Conditioning requires
Reasoning, step by step
- Construct a hypothetical population of 210 individuals based on the base rate assumption.
- Calculate the number of librarians matching the evidence: .
- Calculate the number of farmers matching the evidence: .
- Sum these counts to find the total number of people exhibiting the evidence: .
- Divide the count of matching librarians by the total matching count: .
- Simplify the fraction to get the final probability: .
Example
"Imagine a population of 210 people: 10 librarians and 200 farmers... This yields 4 matching librarians and 20 matching farmers. Out of the 24 total people who fit the description, only 4 are librarians. Thus, the probability Steve is a librarian given the evidence is approximately 16.7%."
Common misconceptions
- Assuming the prior reflects actual current occupational statistics rather than the specific illustrative setup.
- Confusing the likelihood of the evidence given the hypothesis with the posterior probability of the hypothesis given the evidence.
- Ignoring the base rates entirely and judging solely by how well Steve's profile resembles a stereotype.
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In this two-category example, the prior is the probability of the hypothesis 'Steve is a librarian' before seeing evidence, calculated as based on the stipulated population. The likelihood is the probability of the evidence 'fits the description' given the hypothesis is true, stipulated as 0.4.
Conditions: Hypothesis H: 'Steve is a librarian'; Evidence E: 'Fits the description'; Population assumption: 10 librarians, 200 farmers; Likelihood assumption: 40% for librarians, 10% for farmers
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