Decision Theory III (MATH 3071) Lecture 1 2017/18 Useful - PowerPoint PPT Presentation

Decision Theory III (MATH 3071) Lecture 1 2017/18

Useful Information ◮ Lecturers: Dr Camila Caiado (Michaelmas), Prof Frank Coolen (Epiphany) ◮ Office: CM321 ◮ Email: c.c.d.s.caiado@durham.ac.uk ◮ Lecture Times: Wednesdays and Fridays 9h CG85 ◮ Problem Classes: Mondays 13h CG85, Weeks: 3, 5, 7, 9, 12, 14, 16, 18 ◮ DUO page: Course ID 73497

Resources DUO: Summaries, problem sheets, solutions, extras Recommended books: ◮ M. Peterson, An Introduction to Decision Theory , Cambridge University Press, 2009 ◮ J. Q. Smith, Decision Analysis - A Bayesian Approach , Chapman and Hall, 1988 ◮ D. V. Lindley, Making Decision , 2nd Ed, Wiley, 1985 ◮ S. French, Decision Theory: An Introduction to the Mathematics of Rationality , Ellis, Horwood, Chichester, 1986 ◮ M. H. DeGroot, Optiomal Statistical Decisions , McGraw-Hill, 2004 ◮ R. T. Clemen, Making Hard Decisions , 2nd Ed, Duxbury Press, 1995

Homework and Office Hours ◮ Homework: 4 per term ◮ Office Hours: Friday 10h-12h00 (tentative) ◮ Contact: Questions by e-mail are also welcome, include a picture/scan of your attempt when possible. Answers to the most common questions will be posted on DUO.

Revision - Probability Definition: Suppose Ω is a sample space. A probability distribution on Ω is a collection of numbers P ( A ), one for each event A ⊆ Ω, satisfying the following axioms: ◮ A1. P ( A ) ≥ 0 for every A ⊆ Ω ◮ A2. P (Ω) = 1 ◮ A3. if A and B are incompatible or mutually exclusive events then P ( A ∪ B ) = P ( A ) + P ( B ). Mutually Exclusive Black Red ♣ ♠ ♥ ♦

Revision - Probability Definition: Suppose Ω is a sample space. A probability distribution on Ω is a collection of numbers P ( A ), one for each event A ⊆ Ω, satisfying the following axioms: ◮ A1. P ( A ) ≥ 0 for every A ⊆ Ω ◮ A2. P (Ω) = 1 ◮ A3. if A and B are incompatible or mutually exclusive events then P ( A ∪ B ) = P ( A ) + P ( B ). From the axioms, we can derive: ◮ C1. For any events A and B , P ( B − A ) = P ( B ) − P ( A ∩ B ). ◮ C2. For any event A , P ( A C ) = 1 − P ( A ). ◮ C3. P ( ∅ ) = 0. ◮ C4. For any two events A and B , P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) ◮ C5. If A ⊆ B then P ( A ) ≤ P ( B ). ◮ C6. For any two events A and B , P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ).

Revision - Probability Conditional Probability: The conditional probability of an event A given B , P ( A | B ), is defined as: P ( A | B ) = P ( A ∩ B ) P ( B ) � = 0 . P ( B ) , Independent Events: Two events A and B are independent if and only if P ( A ∩ B ) = P ( A ) P ( B ) .

Revision - Probability (Partition Theorem) Let A 1 , . . . , A n be a partition of Ω, i.e. A 1 , . . . , A n are mutually disjoint and A 1 ∪ . . . ∪ A n = Ω. If P ( A j ) � = 0, j = 1 , . . . , n then, for an event B ⊆ Ω, we have n � P ( B ) = P ( B | A j ) P ( A j ) . j =1 (Bayes’ Theorem) Let A 1 , . . . , A n be a partition of Ω. For B ∈ Ω, P ( B ) � = 0, we have P ( B | A j ) P ( A j ) P ( A j | B ) = k =1 P ( B | A k ) P ( A k ) , j = 1 , . . . , n . � n

Example 1 In a survey it is found that 40% of the population like dogs, 60% like cats, and 70% of those who like dogs also like cats. 1. Find the probability that a randomly selected member of the population likes both dogs and cats. 2. Find the probability that a randomly selected member of the population likes dogs or cats but not both. 3. What proportion of those who like cats also like dogs?

Example 2 Suppose 0.001 of the population have a certain disease. A diagnostic test is carried out for the disease: the outcome of the test is either positive or negative . It is known from past experience that the test is 90% reliable, i.e. a person with the disease will test (correctly) positive with probability 0.9, while a person without the disease will test (incorrectly) positive with probability 0.1. 1. A person is tested for the disease. What is the probability that they test positive? 2. A person’s test result is positive. What is the probability that they have the disease?

Example 3 A plane is missing, and it is presumed that it was equally likely to have gone down in any of three possible regions. The probability that the plane will be found upon a search of region 1, when the plane actually is in that region, is judged to be 0.9. The corresponding figures for regions 2 and 3 are 0.95 and 0.85 respectively. What is the conditional probability that the plane is in region 3 given that searches in regions 1 and 2 have been unsuccessful?