Sets, Logic, and Probability As Used in Decision Support Systems

Transcription

1 Sets, Logic, and Probability As Used in Decision Support Systems HAP 752 Advanced Health Informa6on Systems Janusz Wojtusiak, PhD George Mason University Spring 2014

2 Part of the inhumanity of the computer is that, once it is competently programmed and working smoothly, it is completely honest. Isaac Asimov

3 Why? Three areas that give mathema6cal bases of decision support systems Set theory describes objects to deal with Logic describes how to reason Probability describes how to handle uncertainty

4 Sets

5 Set Theory A = {a, b, c, d} B = {a, b, d} C = {a, c, e} D = {a,b,{a,b}} Set size A = 4 B = 3 C = 3 D =? 3

6 Set Theory A = {a, b, c, d} B = {a, b, d} C = {a, c, e} Intersec6on: A B = {a, b, d} Union: A C = {a,b,c,d,e} Difference: A \ B = {c}, A \ C = {b,d}

7 Set Theory A = {a, b, c, d} B = {a, b, d} C = {a, c, e} Empty set is subset of every set (including itself) B A C A a A

8 S = {a, b, c, {a, b}, {d}} Set Theory a S? d S? {a,b} S? {b,a} S? S? S? yes no yes yes no yes

9 Examples Symptoms = {high BP, high fas6ng glucose, large waist circumference, low HDL, high triglycerides} Treatments = {surgery, radia6on, chemotherapy, brachytherapy, watchful wai6ng}

10 Examples MS Symptoms = {high BP, high fas6ng glucose, large waist circumference, low HDL, high triglycerides} Pa6ent Symptoms = {high BP, high fas6ng glucose, large waist circumference, chest pain} MS Symptoms Pa6ent symptoms = {high BP, high fas6ng glucose, large waist circumference}

11 Logic

12 Boolean Logic Conjunc6on: a AND b, a b, a & b, a && b True only if both a and b are true Disjunc6on: a OR b, a b, a b, False only if both a and b are false Nega6on: NOT a True only if a is false Implica6on: a IMPLIES b False only when a true and b false Equivalence: a IS EQUIVALENT TO b, a b, a == b, True when a and b are the same

13 Order of Opera6ons Opera6ons need to be done in specific order NOT, AND, OR For example a AND NOT b OR c Is the same as (a AND (NOT b) ) OR c It is the same as in arithme6c.

14 True or False? Many ways of encoding true/false True, 1, T, Y,. False, 0, F, N,.

15 When these are true? x AND (y OR x) a OR (b OR (c OR d)) (a à b) OR a (x = y x = z y = z )

16 Rules

17 If.. Then.. Else Rules The most common and important way of crea6ng logic expressions Used in programming languages Used in decision support systems IF <condi6on> THEN <ac6on 1> ELSE <ac6on 2> or simplified IF <condi6on> THEN <ac6on1> This statement comes in may different variants

18 Example Rule Example rule when ordering CT with contrast. IF THEN last_creat > 1.5 OR last_bun > 30 alert Possible impaired kidney func6on for contrast studies

19 Example Rule Example rule when ordering CT with contrast. IF last_creat is null AND last_bun in null THEN alert Consider crea6nine tes6ng ELSE IF last_creat > 1.5 OR last_bun > 30 THEN alert Possible impaired kidney func6on for contrast studies

20 Not true? Not false? In real life not everything is true or false. people oten answer maybe There are many ways to extend Boolean logic to do so Probability Fuzzy logic Rough Sets

21 Probability The following slides are based on those by Dr. Farrohk Alemi for HAP 730 class.

22 Probability Most common way of presen6ng uncertain situa6ons Provides strict methods of calcula6on Many methods, formulas, theorems, etc. available

23 What is probability? In the Figure, where are the events that are not A?

24 How to Calculate Probability? A P(A)=

25 Calculus of Probabili6es Helps Us Keep Track of Uncertainty of Mul6ple Events Joint probability, probability of either event occurring, revising probability ater knew knowledge is available, etc.

26 Probability of One or Other Event Occurring P(A or B) = P(A) + P(B) - P(A & B)

27 Probability of One or Other Event Occurring P( diabetes OR hypertension ) = P( diabetes ) + P( hypertension ) P( diabetes AND hypertension )

28 Effect of New Knowledge

29 Condi6onal Probability P( B A ) = p( A & B ) / p( A )

30 Independence A and B are independent if P( A ) = P( A B ) When A and B are independent, then P( A & B ) = P( A ) x P( B )

31 Example P( pneumonia AND abnormal CXR ) = P( pneumonia ) x P ( abnormal CXR ) WRONG! P( pneumonia AND abnormal CXR ) = P( pneumonia ) x P ( abnormal CXR pneumonia )

32 Breast Cancer Example 1 in 100 pa6ents in a clinic have breast cancer For the ini6al screening test a false posi6ve rate is 0.2 (20% of women without cancer will test posi6ve) For the ini6al screening test a false nega6ve rate is 0.1 (10% of women with cancer will test nega6ve) If a pa6ent has cancer there is 90% chance that will test posi6ve.

33 Breast Cancer Suppose a pa6ent s test is posi6ve. What is chance that the pa6ent has cancer? 90% 80% Something else? From Bayes formula you can calculate that the probability is about 4.3%

34 People v. Collins Suspects of robbery Robbers were reported to be (6 clues) The probability that stopped pair was innocent was calculated as 1/ Convicted! Using Bayes theorem the probability of innocence is 99.8%

35 Summary Set theory, logic and probabili6es are basis for knowledge representa6on methods in used healthcare