Prof. Carlos Bana e Costa

Transcription

1 Redes Bayesianas e Análise de Risco Prof. Carlos Bana e Costa Referências: Jensen, F.V. (1996), An Introduction to Bayesian Networks, UCL Press, London. (Ch. 2) Breese J. (Microsoft Corp.), Koller, D. (Stanford University): Norsys Software Corp. (NETICA soft.): Fred, A. (2008) Redes Bayesianas 1 Incerteza Modelos de AD Problema dominado por Objectivos múltiplos REVISÃO DE OPINIÃO Redes Bayesianas AVALIAR OPÇÕES Análise Multicritério ESCOLHA Árvores de Decisão Diagramas de Influência SEPARAÇÃO EM COMPONENTES Análise de Risco ALOCAÇÃO DE RECURSOS E NEGOCIAÇÃO Análise Equity 1

2 Remember: Rules of Probability Regra de Bayes (1763): P(Oil\Favourable) = P(Favourable\Oil).P(Oil) P(Favourable) Product rule : P(Favourable, Oil) = P(Favourable\Oil).P(Oil) = P(Oil\Favourable).P(Favourable) ( ) Marginalisation : P(Favourable) = P(Favourable\Oil).P(Oil) + P(Favourable\Dry).P(Dry) P(Favourable\Oil).P(Oil) P(Oil\Favourable) = P(Favourable\Oil).P(Oil)+P(Favourable\Dry).P(Dry) 3 The cornerstone of Bayesian probability theory is the inversion formula: P(H\E) = P(E\H).P(H) P(E) where P(H\E) denotes the probability of hypothesis H conditioned on the evidence E. Bayes rule provides an explicit relation for the degree of believe we accord a hypothesis H, in light of evidence E. Bayes Rule is useful in contexts where probabilities are more easily obtained in one inferential direction than another. 4 2

3 By providing the flexibility to reason probabilistically in either the causal or the diagnostic directions, Bayes Rule allows agents to assert beliefs in forms that are compatible with the way they actually reason about the process(es) or phenomena of interest. A bayesian network A particular type of Influence Diagram. It contains only chance (and deterministic) nodes. 5 Holmes & Watson example Police Inspector Smith is impatiently waiting the arrival of Mr Holmes and Dr Watson; they are late and Inspector Smith has another important appointment (lunch). Looking out of the window he wonders whether the roads are icy. Both are notoriously bad drivers, so if the roads are icy they may crash His secretary enters and tells him that Dr Watson has had a car accident, Watson? OK. It could be worse icy roads! Then Holmes has most probably crashed too. I ll go for lunch now. Icy roads?, the secretary replies, It is far from being that cold, and furthermore all the roads are salted. Inspector Smith is relieved. Bad luck for Watson. Let us give Holmes ten minutes more. 6 3

4 Police Inspector Smith is impatiently waiting the arrival of Mr Holmes and Dr Watson; they are late and Inspector Smith has another important appointment (lunch). Looking out of the window he wonders whether the roads are icy. Both are notoriously bad drivers, so if the roads are icy they may crash. Ice Holmes Bayesian net Watson Both are notoriously bad drivers roads are icy they may crash 7 Ice Holmes Watson if the roads are icy they may crash 8 4

5 if the roads are icy they may crash His secretary enters and tells him that Dr Watson has had a car accident, Watson? OK. It could be worse icy roads! Then Holmes has most probably crashed too. I ll go for lunch now. 9 Then Holmes has most probably crashed too Icy roads?, the secretary replies, It is far from being that cold, and furthermore all the roads are salted. Inspector Smith is relieved. Bad luck for Watson. Let us give Holmes ten minutes more. 10 5

6 Alec Morton OR425 Seminar 8: Bayes Nets and Influence Diagrams Qualitative properties of Bayes nets Inference can follow arrow direction Knowing T tells you something about H via I Inference can run against arrow direction Knowing H tells you something about T via I Both patterns of inference are broken if I know I Temperature below freezing (T) Roads icy (I) Holmes crashes car (H) Inference can also work up along an arrow and then back down Knowing H can tell me something about W, supposing I don t know I Alec Morton OR425 Seminar 8: Bayes Nets and Influence Diagrams Holmes crashes car (H) This is known as conditional independence Roads icy (I) H and W (or H and T) are independent if I know the state of I (otherwise they are dependent) Watson crashes car (W) 6

7 What about working down and then up? If I don t know the stateofl L, RandSare are independent But if I know L is the case, R and S become dependent Alec Morton OR425 Seminar 8: Bayes Nets and Influence Diagrams It rained (R) Holmes lawn is wet (L) Sprinkler was left on (S) Knowing S can explain away L and thus make R less likely This pattern of inference is enabled rather than broken by knowledge of intermediate variable 14 7

8 Bayesian (or Belief, or Probabilistic Causal) Networks Visit To Asia Sm ok i ng Tuberculosis Lung Cancer Bronchitis XRay Result Tuberculosis or Cancer Dyspnea Chest Clinic A BN is composed of a set of nodes representing variables of interest, connected by links to indicate dependencies, and containing information about the relationships between the nodes (often in the form of conditional probabilities). Its uses include prediction and diagnosis. A BN provides a complete probabilistic description of a particular system, i.e., completely specifies a joint probability distribution on the kinds of distinctions represented by the network. Remember: Conditional P(X = x \ Y = y) Probability that X=x given we know that Y=y. Joint P(x, y) P(X = x ^Y = y) Probability that both X = x and Y = y. 15 (FORMAL) DEFINITION OF A BN (Jansen, 1996) Visit To Asia Sm ok i ng A BN consists of the following: Tuberculosis Lung Cancer Bronchitis A set of variables and a set of directed edges between variables. Tuberculosis or Cancer Each variable has a finite set of Chest Clinic i states. XRay Result Dyspnea The variables together with the directed edges form a directed acyclic graph. To each variable A with parents B 1,, B n there is attached a conditional probability table P(A \ B 1,, B n ). Visit To Asia Visit 1.0 No Visit 99.0 Sm ok i ng Sm ok er 50.0 NonSmoker 50.0 Tuberculosis Pr esent 1.04 Absent 99.0 Lung Cancer Pr esent 5.50 Absent 94.5 Bronchitis Pr esent 45.0 Absent 55.0 A bnormal Normal XRay Result T uberculosis or Cancer True False Dyspnea Pr esent 43.6 Absent 56.4 Chest Clinic 16 8

9 Chest Clinic Copyright 1998 Norsys Software Corp. This belief network is also known as "Asia. It is a toy medical diagnosis example from: Lauritzen, Steffen L. and David J. Spiegelhalter (1988), Local computations with probabilities on graphical structures and their application to expert systems, J. Royal Statistics Society B, 50(2), It is a simplified version of a network that could be used to diagnose patients arriving at a clinic. Each node in the network corresponds to some condition of the patient, for example, "Visit to Asia" indicates whether the patient recently visited Asia. To diagnose a patient, values are entered for nodes when they are known. Netica then automatically re-calculates the probabilities for all the other nodes, based on the relationships between them. The links between the nodes indicate how the relationships between the nodes are structured. 18 9

10 The two top nodes are for predispositions which influence the likelihood of the diseases. Those diseases appear in the row below them. At the bottom are symptoms of the diseases. To a large degree, the links of the network correspond to causation. This is a common structure for diagnostic networks: predisposition nodes at the top, with links to nodes representing internal conditions and failure states, which in turn have links to nodes for observables. Often there are many layers of nodes representing internal conditions, with links between them representing their complex inter-relationships. 19 Probabilistic relation of "Lung Cancer with Smoking Functional dependence of "Tuberculosis or Cancer on Tuberculosis and Lung Cancer

11 Probabilities of each state of fthe node Bronchitis Suppose we want to "diagnose" a new patient. When she first enters the clinic, without having any information about her, we believe she has lung cancer with a probability of 5.5% (the number may be higher than that for the general population, because something has led her to the chest clinic). 21 Finding - She has an abnormal x-ray All the probability numbers and bars changed to take into account the finding. Now the probability that she has lung cancer has increased from 5.5% to 48.9%

12 New Finding: She has made a visit to Asia recently Visit No Visit Visit To Asia The probability of lung cancer decreases from 48.9% to to 37.1%, because the abnormal XRay is partially explained away by a greater chance of Tuberculosis (which she could catch in Asia) Sm ok er NonSmoker Sm ok i ng Pr esent Absent Tuberculosis Pr esent Absent Lung Cancer Pr esent Absent Bronchitis Tuberculosis or Cancer True False Chest Clinic Abnormal Normal XRay Result Pr esent Absent Dyspnea A new patient has just walked in: remove all the findings 24 12

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14 Evidence on A (or G) will influence the certainty of S. Similarly, evidence on C will influence the certainty on A (and G) through S. However, if the state of S is known, then A (G) and C become independent. We say that A (G) and C are d-separated given S. Serial connection. If S is instantiated (i.e., its state is known) it blocks communication between its parents and children 27 When nothing is known about the state of Cancer: However, when we know the state of Cancer: Diverging connection. If C is instantiated, it blocks communication between its children 28 14

15 29 When nothing is known about the state of Cancer: However, when we have information about Cancer: Converging connection. If C is instantiated, it opens communication between its parents 30 15

16 31 The joint probability distribution is the product of all conditional probabilities where: 32 16

17 Revisão de opinião Modelo Bayesiano A Regra de Bayes (1763) permite rever a probabilidade associada a uma hipótese H com base numa evidência E. Redes Bayesianas P(E H). P(H) P(H E) = P(E) Descoberta de novas classes de estrelas infra-vermelhas AutoClass Project (NASA) P( X 1,X2, K,X ) = P( X X 1, K,X n n i= 1 i i 1 ) A Taxonomy of Decision Models (In Decision Analysis in the 1990s - L.D. Phillips) Uncertainty Problem dominated by Multiple Objectives EXTEND conversation Event tree Fault tree Influence diagram REVISE opinion Bayesian nets SEPARATE into components Credence decomposition Risk analysis CHOOSE option Payoff matrix Decision tree EVALUATE options Multi-criteria decision analysis ALLOCATE resources Multi-criteria commons dilemma NEGOTIATE Multi-criteria bargaining analysis 17

18 Separate into components Example 1: Total duration of work required to covert city gas infrastructure to natural gas in a sector of flisbon s public gas network Objective: To define the probability distribution of the total duration 1) Decompose the work required into a set of related activities whose probability distributions of the respective durations are known or easily determined. 2) Use a simulation procedure (e.g. Monte Carlo) in order to estimate the probability distribution of the total duration Total duration (days) Risk analysis Probability distribution Duração da OMG para um Sector (dias) pi pi Distribuição de probabilidades de duração da OMG para um Sector Mínimo 95 mean Máximo Media Desvio Padrão Variância Moda 138 Probability that the total duration is less than 170 days 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

19 (PALISADE) Example 2: MODELLING RISK AS AN INDEPENDENT CRITERION Multicriteria evaluation of bids for the construction of anewstationofthe of the Lisbon Metro 19

20 WHAT IS THE PROBLEM? The construction of the new station at Terreiro do Paço is included in the expansion plan of the Lisbon Metro network In 1997, the Lisbon Metro Company launched an international call for tenders for the construction of the new station. Five bids were introduced by four bidders 2 COSTS TOTAL COST Initial value tree COST CREDIBILITY DEADLINES PROPOSED DEADLINES DEADLINES CREDIBILITY TECHNICAL WORTH AND FEASIBILITY QUALITY RESTRICTIONS LAID DOWN BY THE BIDDER ENVIRONMENTAL DISRUPTIONS AND IMPACT ESTABLISHED UNDERTAKINGS Question: If two bids turn out to be indifferent in all the criteria above, is there any reason to consider one bid better than the other? Answer: Yes, if we knew that the estimated costs and completion times proposed by one bidder were more credible than those proposed by other bidder. 20

21 Requisite value tree COSTS TOTAL COST COST CREDIBILITY DEADLINES PROPOSED DEADLINES DEADLINES CREDIBILITY TECHNICAL WORTH AND FEASIBILITY QUALITY RESTRICTIONS LAID DOWN BY THE BIDDER ENVIRONMENTAL DISRUPTIONS AND IMPACT ESTABLISHED UNDERTAKINGS Credibility of Total Cost: Bidsl Proposed Total cost Mean Standard Total cost not exceeded Deviation with 90% probability , , , , , , , , , , , , , , , , , , , ,36 21

22 Credibility of completion time: Simulation procedure 1. A random number that corresponds to the duration of each activity is generated according to a log-normal probability distribution. 2. The minimum total duration is calculated based on the generated durations d i of the activities and on the established precedence relations between them. 3. The process is repeated N times (sufficient number of times in order to guarantee desired confidence level). 4. The credibility p associated with the proposed total duration is equal to the relative frequency after N simulations. requency Fr Frequency Histogram and Cumulative Frequency curve for Completion time % % % % % 00% % % Classes umulative requency Cu Fr Percentile 10% Evolution of 10% percentile for Completion time Nº of Simulations 10 % percentile = weeks 22