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.