Bayesian networks II. Model building. Anders Ringgaard Kristensen

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1 Bayesian networks II. Model building Anders Ringgaard Kristensen

2 Outline Determining the graphical structure Milk test Mastitis diagnosis Pregnancy Determining the conditional probabilities Modeling methods and tricks Object oriented Bayesian networks

3 Milk test Infected? { Yes, No } Test? { Positive, Negative } Sensitivity/Specificity determines the conditional probabilities Direction of edge! Causal direction Against the reasoning direction

4 Daily measurements Inf 1 Inf 2 Inf 3 Inf 4 Inf 5 Inf 6 Inf 7 Test 1 Test 2 Test 3 Test 4 Test 5 Test 6 Test 7 Are the infection states of different days independent? Probably not! Markov property Duration of disease

5 Dependence between test results Inf 1 Inf 3 Inf 5 Inf 7 Inf 2 Inf 4 Inf 6 Test 1 Test 3 Test 5 Test 7 Test 2 Test 4 Test 6 Correctness of test depends on whether it was correct yesterday. To determine whether it was correct yesterday: The true infection state yesterday The test result yesterday

6 Dependence between test results Inf 1 Inf 3 Inf 5 Inf 7 Inf 2 Inf 4 Inf 6 Cor 1 Cor 2 Cor 3 Cor 4 Cor 5 Cor 6 Test 1 Test 3 Test 5 Test 7 Test 2 Test 4 Test 6 A simplifying intermediate variable Cor i { yes, no } indicating whether the test was correct.

7 Mastitis diagnosis, AMS No Subclinical Clinical Mastitis Heat No Yes Conductivity Temperature Separate variables (don t pool mastitis & heat) Check conditional independence Are Conductivity and Temperature independent, given Mastitis?

8 If not conditional independent No Subclinical Clinical Mastitis Heat No Yes Conductivity Temperature If conductivity influences temperature.

9 If not conditional independent No Subclinical Clinical Mastitis Heat No Yes Conductivity Temperature If temperature influences conductivity. The causal direction may be difficult to determine

10 If not conditional independent No Subclinical Clinical Mastitis Heat No Yes Conductivity Temperature If the direction of an edge cannot be determined, a variable is often missing! Other disease

11 Pregnancy (again, again ) A goat is mated, and six weeks later we want to test it for pregnancy. We have three tests available: Blood test Urine test Scanning The variables of our problem are: Pregnant { yes, no } Blood { positive, negative } Urine { positive, negative } Scan { positive, negative }

12 Pregnancy test BN Pregnant Blood Urine Scan Check for conditional independence

13 Pregnancy test BN, revised Pregnant The blood test and the urine test both measure a hormone level. The scanning does something completely different. Hormone Blood Urine Scan

14 Outline Determining the graphical structure Milk test Mastitis diagnosis Pregnancy Determining the conditional probabilities Modeling methods and tricks

15 Determining the probabilities Statistical model with parameters estimated from data. Law of nature. Experts of the domain.

16 Milk test example Inf 1 Inf 3 Inf 5 Inf 7 Inf 2 Inf 4 Inf 6 Cor 1 Cor 2 Cor 3 Cor 4 Cor 5 Cor 6 Test 1 Test 3 Test 5 Test 7 Test 2 Test 4 Test 6 The P(Test i Inf i ) conditional probability is supplied by the test retailer.

17 P(Test i Inf i ) P(Test i = yes Inf i ) P(Test i = yes Inf i ) Inf i = yes Inf i = no Defined by the sensitivity and specificity of the test.

18 Milk test example Inf 1 Inf 3 Inf 5 Inf 7 Inf 2 Inf 4 Inf 6 Cor 1 Cor 2 Cor 3 Cor 4 Cor 5 Cor 6 Test 1 Test 3 Test 5 Test 7 Test 2 Test 4 Test 6 The P(Cor i Inf i, Test i ) conditional probability is trivial.

19 P(Cor i Inf i, Test i ) Inf i Test i P(Cor i =y Inf i,test i ) P(Cor i =n Inf i,test i ) yes yes 1 0 yes no 0 1 no yes 0 1 no no 1 0 If Inf i and Test i agree, the test is correct!

20 Milk test example Inf 1 Inf 3 Inf 5 Inf 7 Inf 2 Inf 4 Inf 6 Cor 1 Cor 2 Cor 3 Cor 4 Cor 5 Cor 6 Test 1 Test 3 Test 5 Test 7 Test 2 Test 4 Test 6 The P(Test i Inf i, Cor i-1 ) conditional probabilities needs some assumptions.

21 P(Test i Inf i, Cor i-1 ) Assumptions: A correct test has 99.9% chance of being correct next time. An incorrect test has 30% chance of being incorrect next time: Thus, it is still most likely to be correct. In agreement with the example file provided from the homepage. In disagreement with the textbook.

22 P(Test i Inf i, Cor i-1 ) Inf i Cor i-1 P(Test i =y Inf i,cor i-1 ) P(Test i =n Inf i,cor i-1 ) yes yes yes no no yes no no

23 Milk test example Inf 1 Inf 3 Inf 5 Inf 7 Inf 2 Inf 4 Inf 6 Cor 1 Cor 2 Cor 3 Cor 4 Cor 5 Cor 6 Test 1 Test 3 Test 5 Test 7 Test 2 Test 4 Test 6 The P(Inf 1 ) probabilities must be modeled.

24 P(Inf 1 ) Assume that the milk test is made on single cow level at the farm. We need the probability λ that the milk from a particular cow is infected on an arbitrary day (i.e. P(Inf i = yes ) = λ). The farmer has no knowledge about λ, but The dairy performs a very precise bulk tank test: If the milk from just one cow is infected, the bulk tank test will be positive. On average, the bulk tank test is positive once a month

25 P(Inf 1 ) Further assumptions: λ is the same for all cows Cows are infected independently. Under those assumptions: (1 - λ) 50 = 29/30 λ = 1 (29/30)

26 Milk test example Inf 1 Inf 3 Inf 5 Inf 7 Inf 2 Inf 4 Inf 6 Cor 1 Cor 2 Cor 3 Cor 4 Cor 5 Cor 6 Test 1 Test 3 Test 5 Test 7 Test 2 Test 4 Test 6 The P(Inf i Inf i-1, Inf i-2 ) conditional probabilities must be modeled.

27 P(Inf i Inf i-1, Inf i-2 ) Assume the following properties of the infection: A not-infected cow has probability q of becoming infected. An infection always lasts for at least 2 days After 2 days, the probability of recovery is π Define a state space model: s i {nn, ny, yn, yy} where e.g. ny means: not-infected day i-1 but infected day i

28 P(Inf i Inf i-2, Inf i-1 ) Transition propabilities: Day i Day i+1 nn ny yn yy P(yes) nn (1-q) q 0 0 q ny yn (1-q) q 0 0 q yy 0 0 π (1-π) (1-π) Only assumptions on min. duration, q and π

29 A procedure There are basically 3 parameters: Duration minimum 2 days Probability of becoming infected q Daily probability of recovery π (after 2 days) The 3 parameters should be estimated from data. If data is not available, we may have to rely on experts. Experts guesses may be calibrated to the overall probability of infection at a given day.

30 Limiting state distribution

31 Choice of parameters Let the probability of recovery (after 2 days) be π = 0.4 Let the probability of becoming diseased be q = The steady state distribution a is calculated in the Limit.R program

32 Milk test example Inf 1 Inf 3 Inf 5 Inf 7 Inf 2 Inf 4 Inf 6 Cor 1 Cor 2 Cor 3 Cor 4 Cor 5 Cor 6 Test 1 Test 3 Test 5 Test 7 Test 2 Test 4 Test 6 The P(Inf i Inf i-1 ) conditional probabilities must be modeled.

33 P(Inf i Inf i-1 ) Some assumption compensating for the fact that we don t know the infection state two days ago

34 A stud farm: Genealogical tree Ann Brian Cecily Fred Dorothy Eric Gwenn Henry Irene John is suffering from a serious hereditary disease caused by a recessive gene. State space for a horse: aa, aa or AA The genotype aa is diseased. The genotype aa is carrier. We want to cull all carriers! John

35 Contitional probabilities from genetics Mother Father aa aa AA aa (1, 0, 0) (0.5, 0.5, 0) (0, 1, 0) aa (0.5, 0.5, 0) (0.25, 0.5, 0.25) (0, 0.5, 0.5) AA (0, 1, 0) (0, 0.5, 0.5) (0, 0, 1)

36 Unknown parents Two unknown parents: Assume that the distribution reflects the population probabilities of being healthy or carrier (if they had been diseased they would not have survived until breeding age ). One unknown parent: Introduce a dummy parent reflecting the population distribution.

37 The diseased state aa Impossible for all other horses than John Two options: Delete the state and adjust all probabilities accordingly. Keep the state and enter the evidence that the horses are either healthy or carriers

38 Obtaining the probabilities Sources: Pure data estimation (frequency counts) Model and parameter estimation from data Provided by nature Subjective expert assessments

39 Outline Determining the graphical structure Milk test Mastitis diagnosis Pregnancy Determining the conditional probabilities Modeling methods and tricks

40 Logical constraints: SIR model We observe the spread of a contagious disease in a population of, say 50 animals. Each animal is either Susceptible Infective Removed (recovered/dead) Let S i, I i and R i be the number of susceptible, infective and removed, respectively, at time i

41 The SIR model S 1 S 2 S 3 I 1 R 1 I 2 R 2 I 3 R 3 Basic problem: S i, I i and R i are not independent Cannot be solved by directed (causal) edges. Even though the conditional probabilities of the model may be correct, it may happen that S i + I i + R i 50

42 The SIR model S 1 S 2 S 3 I 1 R 1 I 2 R 2 I 3 R 3 C 1 C 1 C 1 { Valid, Invalid } P(C i = Valid S i, I i, R i ) = 1 if S i + I i + R i = 50 P(C i = Valid S i, I i, R i ) = 0 if S i + I i + R i 50 Enter the evidence P(C i = Valid) and propagate

43 Logical constraints Refer also to the sock-sorting problem in the textbook.

44 Object oriented Bayesian networks For large networks with many more or less identical items, an object oriented approach may be relevant. Object Oriented Networks are like all other Bayesian network, but the object oriented approach makes the construction phase easier. Example: We wish to construct a model of a cow herd with a number of cows: We construct a herd model of which the variables are defined at herd level: Conception rate Heat detection rate We construct a cow model describing variables relating to individual cows. This cow model is used as a template for construction of separate cow objects having the same structure: Pregnancy status Heat detection(s) Pregnancy diagnosis Thus, we can easily create as many cows as we want and afterwards link each cow to the herd model.

45 Object oriented Bayesian networks: Cow class A model of a dairy cow. Used as a template for cow objects. The cow has been inseminated, and we observe it for heat We also perform a pregnancy diagnosis The value of the variable Pregnant depends on a herd level conception rate (CR). The outcomes of the Heat variables depends on the pregnancy state and the herd level heat detection rate (HDR).

46 Object oriented Bayesian networks: Herd class A model of a dairy herd. It contains the herd level variables CR and HDR and a number of cow objects. Each cow object is created from the cow class (template). The model is shown with the objects collapsed.

47 Object oriented Bayesian networks: Herd class expanded With the expanded view, the structure of the objects is visible The expanded model is just an ordinary Bayesian network

Bayesian Networks and Decision Graphs

Bayesian Networks and Decision Graphs Chapter 3 Chapter 3 p. 1/47 Building models Milk from a cow may be infected. To detect whether or not the milk is infected, you can apply a test which may either give