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Graphical Models - Modeling -

MINVOLSET. KINKEDTUBE. PULMEMBOLUS. INTUBATION. VENTMACH. DISCONNECT. PAP. SHUNT. VENTLUNG. VENITUBE. PRESS. MINOVL. FIO2. VENTALV. PVSAT. ANAPHYLAXIS. ARTCO2. EXPCO2. SAO2. TPR. INSUFFANESTH. HYPOVOLEMIA. LVFAILURE. CATECHOL. LVEDVOLUME. STROEVOLUME. ERRCAUTER. HR.

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Graphical Models - Modeling -

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  1. MINVOLSET KINKEDTUBE PULMEMBOLUS INTUBATION VENTMACH DISCONNECT PAP SHUNT VENTLUNG VENITUBE PRESS MINOVL FIO2 VENTALV PVSAT ANAPHYLAXIS ARTCO2 EXPCO2 SAO2 TPR INSUFFANESTH HYPOVOLEMIA LVFAILURE CATECHOL LVEDVOLUME STROEVOLUME ERRCAUTER HR ERRBLOWOUTPUT HISTORY CO CVP PCWP HREKG HRSAT HRBP BP Mainly based on F. V. Jensen, „Bayesian Networks and Decision Graphs“, Springer-Verlag New York, 2001. Advanced IWS 06/07 Graphical Models- Modeling - Wolfram Burgard, Luc De Raedt, Kristian Kersting, Bernhard Nebel Albert-Ludwigs University Freiburg, Germany

  2. Outline • Introduction • Reminder: Probability theory • Basics of Bayesian Networks • Modeling Bayesian networks • Inference • Excourse: Markov Networks • Learning Bayesian networks • Relational Models

  3. (Discrete) Random Variables and Probability • Arandom variableXdescribes a value that is the result of a process which can not be determined in advance: the rank of the card we draw from a deck • The sample space S of a random variable is the set of all possible values of the variable: Ace, King, Queen, Jack, … • An event is a subset of S: {Ass, King}, {Queen}, … • The probability function P(X=x), short P(x),defines the probability that X yields a certain value x. - Reminder: Probability Theory

  4. (Discrete) Random Variables and Probability The world is described as a set of random variables and divided it into elementary events or states States are mutually exclusive A probability space has the following properties: - Reminder: Probability Theory

  5. Conditional probability and Bayes’ Rule If it is known that a certain event occurred this may change the probability that another event will occur P(x|y) denotes the conditional probability that x will happen given that y has happened. Bayes’ rule states that: - Reminder: Probability Theory

  6. Fundamental rule • The fundamental rule (sometimes called ‚chain rule‘) for probability calculus is - Reminder: Probability Theory

  7. Conditional probability and Bayes’ Rule The complete probability formulastates that Note: mirrors our intuition that the unconditional probability of an event is somewhere between its conditional probability based on two opposing assumptions. - Reminder: Probability Theory

  8. Joint Probability Distribution • „truth table“ of set of random variables • Any probability we are interested in can be computed from it - Reminder: Probability Theory

  9. Bayes´rule and Inference Bayes’ is fundamental in learning when viewed in terms of evidence and hypothesis: priorprobability posteriorprobability likelihood - Reminder: Probability Theory This allows us to update our belief in a hypothesis based on prior belief and in response to evidence.

  10. Independence among random variables Two random variables X and Y are independent if knowledge about X does not change the uncertainty about Y and vice versa. Formally for a probability distribution P: - Reminder: Probability Theory

  11. Independence among random variables - Reminder: Probability Theory

  12. Outline • Introduction • Reminder: Probability theory • Basics of Bayesian Networks • Modeling Bayesian networks • Inference • Excourse: Markov Networks • Learning Bayesian networks • Relational Models

  13. E B A J M Bayesian Networks • Finite, acyclic graph • Nodes: (discrete) random variables • Edges: direct influences • Associated with each node: a table representing a conditionalprobability distribution (CPD), quantifying the effect the parents have on the node - Bayesian Networks

  14. B E A Associated CPDs • naive representation • tables • other representations • decision trees • rules • neural networks • support vector machines • ... E B P(A | E,B) .9 .1 e b - Bayesian Networks e b .7 .3 .8 .2 e b e .99 .01 b

  15. Bayesian Networks X2 X1 X3 - Bayesian Networks

  16. Conditional Independence I Each node / random variable is (conditionally) independent of all its non-descendants given a joint state of its parents. - Bayesian Networks

  17. Example Visit to Asia? Smoker? Has tuberculosis Has lung cancer Has bronchitis - Bayesian Networks Tuberculosis or cancer Positive X-ray? Dyspnoea?

  18. Example Visit to Asia? Smoker? Has tuberculosis Has lung cancer Has bronchitis - Bayesian Networks Tuberculosis or cancer Positive X-ray? Dyspnoea?

  19. Example Visit to Asia? Smoker? Has tuberculosis Has lung cancer Has bronchitis - Bayesian Networks Tuberculosis or cancer Positive X-ray? Dyspnoea?

  20. Example Visit to Asia? Smoker? Has tuberculosis Has lung cancer Has bronchitis - Bayesian Networks Tuberculosis or cancer Positive X-ray? Dyspnoea?

  21. B T I X S Summary Semantics Compact & natural representation: nodes have  k parents   2k n instead of 2n params conditional independencies in BN structure local probability models full joint distribution over domain + = - Bayesian Networks

  22. Operations • Inference: • Most probable explanation: • Most probable configuration of given the evidence • Data Conflict (coherence of evidence) • positive conf(e) indicates a possible conflict • Sensitivity analysis • Which evidence is in favour of/against/irrelevant for H • Which evidence discriminates H from H´ - Bayesian Networks

  23. Conditional Independence II • One could graphically read off other conditional independencies based on • Serial connections • Diverging connections • Converging connections A B C A ... B E C - Bayesian Networks ... B C E A

  24. Serial Connections - Example Opponent´s hand after the first change of cards Initial opponent´s hand Final opponent´s hand • Poker: If we do not know the opponent´s hand after the first change of cards, knowing her initial hand will tell us more about her final hand. - Bayesian Networks

  25. Serial Connections - Example Opponent´s hand after the first change of cards Initial opponent´s hand Final opponent´s hand pair pair pair • Poker: If we do not know the opponent´s hand after the first change of cards, knowing her initial hand will tell us more about her final hand. - Bayesian Networks

  26. Serial Connections - Example Opponent´s hand after the first change of cards Initial opponent´s hand Final opponent´s hand • If we know the opponent´s hand after the first change of cards, knowing her initial hand will tell us nothing about her final hand. - Bayesian Networks

  27. Serial Connections - Example Opponent´s hand after the first change of cards Initial opponent´s hand Final opponent´s hand pair pair flush • If we know the opponent´s hand after the first change of cards, knowing her initial hand will tell us nothing about her final hand. - Bayesian Networks

  28. Serial Connections A B C • A has influence on B, and B has influence on C - Bayesian Networks

  29. Serial Connections A B C a • A has influence on B, and B has influence on C - Bayesian Networks

  30. Serial Connections A B C a a • A has influence on B, and B has influence on C - Bayesian Networks

  31. Serial Connections A B C b • A has influence on B, and B has influence on C - Bayesian Networks

  32. Serial Connections A B C b b • A has influence on B, and B has influence on C - Bayesian Networks

  33. Serial Connections A B C • A has influence on B, and B has influence on C • Evidence on A will influence the certainty of C (through B) and vice versa - Bayesian Networks

  34. Serial Connections A B C a • A has influence on B, and B has influence on C • Evidence on A will influence the certainty of C (through B) and vice versa - Bayesian Networks

  35. Serial Connections A B C a a • A has influence on B, and B has influence on C • Evidence on A will influence the certainty of C (through B) and vice versa - Bayesian Networks

  36. Serial Connections A B C a a a • A has influence on B, and B has influence on C • Evidence on A will influence the certainty of C (through B) and vice versa - Bayesian Networks

  37. Serial Connections A B C • A has influence on B, and B has influence on C • Evidence on A will influence the certainty of C (through B) and vice versa - Bayesian Networks • IF the state of B is known, THEN the channel through B is blocked, A and B become independent

  38. Serial Connections A B C b • A has influence on B, and B has influence on C • Evidence on A will influence the certainty of C (through B) and vice versa - Bayesian Networks • IF the state of B is known, THEN the channel through B is blocked, A and B become independent

  39. Serial Connections A B C a a b • A has influence on B, and B has influence on C • Evidence on A will influence the certainty of C (through B) and vice versa - Bayesian Networks • IF the state of B is known, THEN the channel through B is blocked, A and B become independent

  40. Conditional Independence II • One could graphically read off other conditional independencies based on • Serial connections • Diverging connections • Converging connections A B C A ... B E C - Bayesian Networks B C ... E A

  41. Diverging Connections - Example Sex Stature Hair length • If we do not know the sex of a person, seeing the length of her hair will tell us more about the sex, and this in turn will focus our belief on her stature. - Bayesian Networks

  42. Diverging Connections - Example Sex long long Stature Hair length long • If we do not know the sex of a person, seeing the length of her hair will tell us more about the sex, and this in turn will focus our belief on her stature. - Bayesian Networks

  43. Diverging Connections - Example Sex Stature Hair length • If we know that the person is a man, then the length of hair gives us no extra clue on his stature. - Bayesian Networks

  44. Diverging Connections - Example male Sex long Stature Hair length long • If we know that the person is a man, then the length of hair gives us no extra clue on his stature. - Bayesian Networks

  45. Diverging Connections A B E C ... - Bayesian Networks • Influence can pass between all the children • of A unless the state of A is known

  46. Diverging Connections A B E C ... b - Bayesian Networks • Influence can pass between all the children • of A unless the state of A is known

  47. Diverging Connections A b B E C ... b - Bayesian Networks • Influence can pass between all the children • of A unless the state of A is known

  48. Diverging Connections A b b B E C ... b - Bayesian Networks • Influence can pass between all the children • of A unless the state of A is known

  49. Diverging Connections A b b B E C ... b - Bayesian Networks • Influence can pass between all the children • of A unless the state of A is known

  50. Diverging Connections A a B E C ... - Bayesian Networks • Influence can pass between all the children • of A unless the state of A is known

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