16 terms

3. Bayes' Theorem and its Uses

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Bayes' Theorem (3.1)
Pr(H|O) = Pr(O|H)Pr(H)/Pr(O)
Posterior Probability of Bayes' Theorem (3.2)
Pr(H|O)
Likelihood of Bayes' Theorem (3.2)
Pr(O|H)
Prior Probability of Bayes' Theorem (3.2)
Pr(H)
Unconditional Probabilty of the Observation of Bayes' Theorem (3.2)
Pr(O)
Reliable Definition 1 (3.4)
Pr(X has tuberculosis | X's test is positive) is high
Pr(X does not have tuberculosis | X's test is negative) is high

This deals with Posterior Probabilities [Pr(H|O)]
Reliable Definition 2 (3.4)
Pr(S's test is positive | S has tuberculosis) is high
Pr(S's test is negative | S does not have tuberculosis) is high

This deals with Likelihoods [Pr(O|H)]
Type 1 Error (3.4)
A false negative error.
Type 2 Error (3.4)
false positive error.
Partition (3.5)
A set of mutually exclusive and exhaustive hypotheses. The sum of their probabilities = 1 but the sum of the likelihoods does not necessarily = 1 (can be greater or less than 1).
Repeated usage (3.6)
We have priors along with new evidence A, using Bayes' theorems we find posterior probabilities. Then we obtain more evidence B, we can uses the previous posterior probability as priors to find the new posterior.
Odds formulation of Bayes's theorem (3.6)
the purpose is to eliminate the unconditional probability of the observation by comparing two different hypotheses
features of the odd formula (3.6)
It tells you what needs to be true if the ratio of posterior probabilities is to differ from the ratio of prior probabilities. What needs to be true then?
Likelihood ratio (3.6)
Pr (O | H₁) / Pr (O | H₂)
Law of Likelihood (3.7)
In the odds formulation, changes in the posterior probability change due solely to the likelihoods ratio
Bayesianism (3.8)
Determine how probable a theory (hypothesis) is given certain observations.
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