Advanced Probability
h and g can’t both be true at the same time. Probability calculus 1 ≥ Pr(h) ≥ 0 If e deductively implies h, then Pr(h|e) = 1. (disjunction rule) If h and g are mutually exclusive, then Pr(h or g) = Pr(h) + Pr(g). (disjunction rule) If h and g are mutually exclusive, then Pr(h or g) = Pr(h) + Pr(g) (conditional probability) Pr(h|g) = Pr(h&g)/Pr(g) h and g can’t both be true at the same time. 5. (conjunction rule) Pr(h&g) = Pr(h) x Pr(g|h)
Probability calculus 6. (conjunction rule) Pr(h&g) = Pr(h) x Pr(g|h) Independent: the occurrence of one doesn’t affect the probability of the other. Pr(g) = Pr(g|h) 6. (conjunction rule) Pr(h&g) = Pr(h) x Pr(g|h) 7. (conjunction rule) If h and g are independent, then Pr(h&g) = Pr(h) x Pr(g).
Probability calculus 1 ≥ Pr(h) ≥ 0 If e deductively implies h, then Pr(h|e) = 1. (disjunction rule) If h and g are mutually exclusive, then Pr(h or g) = Pr(h) + Pr(g). (conditional probability) Pr(h|g) = Pr(h&g)/Pr(g) 5. (conjunction rule) Pr(h&g) = Pr(h) x Pr(g|h)
Bayes’s theorem Pr(e|h) x Pr(h) Pr(h|e) = Pr(e)
Bayes’s theorem probability of evidence given hypothesis “prior probability” of the hypothesis Pr(e|h) x Pr(h) Pr(h|e) = Pr(e) probability of the hypothesis given the evidence “posterior probability” “expectedness” of the evidence “likelihood” How strongly does the hypothesis lead us to expect this evidence? How plausible was the hypothesis before the new evidence? What is the probability of the hypothesis, given this new newevidence? How unsurprising is the new evidence? base rate [Pr(e|h) x Pr(h)] + [Pr(e|not-h) x Pr(not-h)]
[Pr(w|not-r) x Pr(not-r)] Suppose there is a 40% chance of rain today and a 90% chance that you get wet if it rains. What is the chance that you get wet? Pr(r) = .4 Pr(not-r) = .6 Pr(w|r) = .9 so, Pr(w) = Pr(w|r) x Pr(r) [ ] .9 x .4 + .36 + .36 + .12 + [Pr(w|not-r) x Pr(not-r)] Pr(w|not-r) .2 x .6 0 x .6 ? .48 .36
Bayes’s theorem Pr(e|h) x Pr(h) Pr(h|e) = Pr(e) Pr(e|h) x Pr(h) [Pr(e|h) x Pr(h)] + [Pr(e|not-h) x Pr(not-h)]
Bayes and Frequency Trees
The probability that a woman in this group has breast cancer is. 8% The probability that a woman in this group has breast cancer is .8%. If a woman has breast cancer, the probability is 90% that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 7% that she will still have a positive mammogram. x has a positive mammogram. What is the probability that x has breast cancer?
Base rate (prior probability) The probability that a woman in this group has breast cancer is .8%. If a woman has breast cancer, the probability is 90% that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 7% that she will still have a positive mammogram. x has a positive mammogram. What is the probability that x has breast cancer?
Base rate (prior probability) Sensitivity inverse of false negatives likelihood The probability that a woman in this group has breast cancer is .8%. If a woman has breast cancer, the probability is 90% that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 7% that she will still have a positive mammogram. x has a positive mammogram. What is the probability that x has breast cancer?
Base rate (prior probability) Sensitivity False positive The probability that a woman in this group has breast cancer is .8%. If a woman has breast cancer, the probability is 90% that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 7% that she will still have a positive mammogram. x has a positive mammogram. What is the probability that x has breast cancer?
Odds are 10:1 you don’t have cancer! Probability of having cancer given positive screen is 7/77, approximately 9%. base rate = .8% 1,000 women sensitivity = 90% 8 have cancer 992 don’t false positives = 7% ~1 negative ~7 positive ~7 positive ~70 positive ~70 positive ~922 negative Odds are 10:1 you don’t have cancer!
base rate = .8% base rate = .8% Women in group sensitivity = 90% sensitivity = 90% false positives = 7% false positives = 7% tested positive women with cancer
Don’t confuse sensitivity with posterior probability! Pr(e|h) x Pr(h) Pr(h|e) = Pr(e) likelihood = posterior sensitivity Don’t confuse sensitivity with posterior probability!