Path: utzoo!mnetor!uunet!mcvax!ukc!its63b!hwcs!tom From: tom@cs.hw.ac.uk (Tom Kane) Newsgroups: comp.ai Subject: Probability Bounds from Bayes Theory: (A Problem). Message-ID: <1578@brahma.cs.hw.ac.uk> Date: 11 Dec 87 14:10:21 GMT Organization: Computer Science, Heriot-Watt U., Scotland Lines: 65 Keywords: Bayes Theorem, Probability, Expert Systems, Uncertainty I am sending this letter out to the network to ask for solutions to a particular problem of Bayesian Inference. Below is the text of the problem, and at the end is the mathematical statement of the information given. Simply, I am asking the questions: 1) Can you find bounds on the final result. If so, how? 2) If not, why is it not possible to do so? What is missing in the specification of the problem? 3) If you get nowhere with this problem, would you be able to solve it if you were given the information: p(pv|t or l)=0.9? I am interested in the problem of providing probability bounds for events specified in a Bayesian setting when not all the necessary conditional probabilities are provided in setting up the problem. PROBLEM ~~~~~~~ (A problem relevant to the handling of Uncertainty in Expert Systems.) We want to know the probability of a patient having both lung cancer and tuberculosis based on the fact that this person has had a positive reading in a chest X-ray. We are given the following pieces of information: 1. The probability that a person with lung cancer will have a positive chest X-ray is 0.9. 2. The probability that a person with tuberculosis will have a positive chest X-ray is 0.95. 3. The probability that a person with neither lung cancer nor tuberculosis will have a positive chest X-ray is 0.07. 4. In the town of interest, 4 percent of the population have lung cancer, and three percent have tuberculosis. EVENTS ~~~~~~ l = lung cancer; t = tuberculosis; pv = positive chest X-ray SETUP ~~~~~ In the statement of the problem below:- ~l means 'not l'. ~l, ~t means 'not l and not t'. t or l means 't or l' where 'not', 'and' , and 'or' are logical operators. so that: p(~l, ~t) means probability( not l and not t). Also, p(pv|l) means the conditional probability of event pv, given event l. PRIORS ~~~~~~ p(l) = 0.04; p(t) = 0.03; p(~l, ~t) = 0.95 CONDITIONALS ~~~~~~~~~~~~ p(pv|l) = 0.9; p(pv|t) = 0.95; p(pv| ~t,~l) = 0.07 (You are not given p(pv| t or l) ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Please mail all solutions or comments to me, and I will let interested parties know what the results are. (I will specially treasure attempts which don't use independence assumptions.) Thanks in advance to anyone who will spend time on this problem... Regards, Tom Kane.