What is it all about? Introduction to Bayesian Networks. Method to reasoning under uncertainty. Where we reason using probabilities

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1 What is it all about? Introduction to ayesian Networks Method to reasoning under uncertainty dvanced Herd Management 28th of september 2009 Where we reason using probabilities Tina irk Jensen Reasoning under uncertainty: n example Reasoning under uncertainty: n example Reasoning under uncertainty: n example Reasoning under uncertainty: n example 1

2 Where is ayesian networks placed in HM? Text books/literature 1. ayesian Networks and Decision Graphs general textbook on ayesian networks and decision graphs. Written by professor Finn Verner Jensen from Ålborg University one of the leading research centers for ayesian networks. 2. ayesian Networks without Tears rticle written by Eugene Charniak Software Esthauge LIMID Software System You can download the software from the course homepage Outline Today (28th of september) General introduction to ayesian networks: What is a ayesian network? Exercise 11.1 Transmission of evidence Exercise 11.2 Tuesday (29th of september) uilding ayesian network models Friday (2nd of October) Case example ayesian networks (in general) Graphical model with some restrictions (next slide) asically a static method ( here and now imagine) static version of data filtering ll parameters are probabilities ayesian networks - definition ayesian networks consist of: set of variables and a set of directed edges between variables Each variable has a finite set of mutually exclusive states 2

3 Each variable has a definte set of mutually excusive states ayesian networks - definition diarrhea Some diarrhoea lot of diarrhoea 0-100g/day coughing g/day Some coughing >200 g/day lot of coughing ayesian networks consist of: set of variables and a set of directed edges between variables Each variable has a finite set of mutually exclusive states The variables and directed edges form a directed acyclic graph The variables and directed edges form a directed acyclic graph The variables and directed edges form a directed acyclic graph C D C D F E G F E G The variables and directed edges form a directed acyclic graph ayesian networks - definition ayesian networks consist of: set of variables and a set of directed edges between variables Each variable has a finite set of mutually exclusive states The variables and directed edges form a directed acyclic graph C D To each variables with parents 1, 2 to n there is attached the probability table 1, 2. n ) F E G 3

4 aye s Theorem w an example!! 1, 2,., n small ayesian network: Pregnancy and heat detection in cows 1, 2,., m t observable Pregnant i) i) i ) 1) p( 1) + 2) 2) n) n) Observable ) n k 1 k) k) Pregnancy and heat detection in cows Conditional probabilities What is the probability that a farmer observes a particular cow in heat during a 3-week period? yes ) a no ) b a + b 1 (no other options) What is the probability that the cow is pregnant? Pregnant yes ) c Pregnant no ) d c + d 1 (no other options) w, assume that the cow is pregnant. What is the conditional probability that the farmer observes it in heat? yes Pregnant yes ) a p+ no Pregnant yes ) b p+ gain, a p+ + b p+ 1 w, assume that the cow is not pregnant. ccordingly: yes Pregnant no ) a p- no Pregnant no ) b p- gain, a p- + b p- 1 Each value of Pregnant defines a full probability distribution for. Such a distribution is called conditional small ayesian net Experience with the net: Evidence Pregnant Pregnant yes c 0.5 Pregnant no d 0.5 y entering information on an observed value of we can revise our belief in the value of the unobservable variable Pregnant. The observed value of a variable is called evidence. Pregnant yes yes a p no b p The revision of beliefs is done by use of aye s Theorem: Let us build the net! Pregnant no a p b p i) i) i ) 1) p( 1) + 2) 2) n) n) 4

5 aye s Theorem for our net aye s Theorem for our net How do we use ayes formula to calculate: Pregnant yes yes ) How do we use ayes formula to calculate: Pregnant yes no ) i) i) i ) 1) p( 1) + 2) 2) n) n) Extension of the net w time for exercise 11.1! Info. variables Insem. Prior probability Hypothesis variable Pregnant Info. variables Test dvantages of ayesian networks Herd diagnostics Consistent combination of information from various sources Risk factors Herd size SPF status Purchase policy Can estimate certainties for the values of variables that are not observable (or very costly to observe). These variables are called hypothesis variables. These estimates are obtained by entering evidence in information variables that Hypothesis variable Mycoplasma pneumonia Influence the hypothesis variable Depend on the hypothesis variable Symptoms DWG Temp 5

6 Transmission of evidence Serial connections Transmission of evidence Diverging connection Feed Colic Death reed If Colic is observed, there will be no connection between Feed and Death Feed and Death are d-separated given Colic Evidence may be transmitted through a serial connection unless, the state of the intermediate variable is known Litter size Color If reed is observed, there will be no influence of Color on Litter size Litter size and Color are d-separated given reed Evidence may be transmitted through a diverging connection unless, the state of the intermediate variable is known Transmission of evidence The previous example d-separation Converging connection Mastitis Temp If Temp is observed, the information that a cow is not in heat will influence the belief that the cow has mastitis Evidence may only be transmitted through a converging connection if a connecting variable (or descendant is observed) Diarrhea The previous example d-separation The previous example d-separation ge Season ge Season Diarrhea Diarrhea 6

7 Exercise: Mastitis detection Previous case Milk yield Mastitis index Mastitis Compilation of ayesian networks Cursory Compilation: Create a moral graph dd edges between all pairs of nodes having a common child. Remove all directions Triangulate the moral graph dd edges until all cycles of more than 3 nodes have a chord Identify the cliques of the triangulated graph and organize them into a junction tree. Conductivity Temperature The software system does it automatically (and can show all intermediate stages). Sum up Next time (29th of september) ayesian networks Reasoning under uncertainty Graphical model with some restrictions Variables and nodes form a DG ll interpendencies are descibed using conditional probabilty distributions Can reason against the causal direction uilding ayesian networks Determining the graphical structure Determining the conditional probabilities Modeling tricks and tips Consistent combination of information from various sources Can estimates certainties for hypothesis variables 7