KVL Introduction to Bayesian Networks Anders Ringgaard Kristensen Advanced Herd Management 2006 1
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Outline Causal networks � Bayesian Networks � � Evidence � Conditional Independence and d-separation Compilation � � The moral graph � The triangulated graph � The junction tree Advanced Herd Management 2006 2
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Causal networks A quiz You have signed up for a quiz in a TV-show � The rules are as follows: � � The host of the show will show you 3 doors � Behind one of the doors a treasure is hidden � You just have to choose the right door and the treasure is yours. � You have two choices: � Initially you choose a door and tell the host which one you have chosen. � The host will open one of the other doors. He always opens a door where the treasure is not hidden. � You can now choose Either to keep your initial choice and the host will open the door � you first mentioned. Or you can change your choice and the host will open the new � door you have chosen. Advanced Herd Management 2006 3
4 3 2 A quiz – let’s try! Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Advanced Herd Management 2006 1 Causal networks
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Causal networks Can w e m odel the quiz? Identify the variables: � � True placement, ”True” ∈ { 1, 2, 3} � First choice, ”Choice 1” ∈ { 1, 2, 3} � Door opened, ”Opened” ∈ { 1, 2, 3} � Second choice, ”Choice 2” ∈ { Keep, Change} � Reward, ”Gain” ∈ { 0, 1000} Advanced Herd Management 2006 5
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Causal networks Chosen initially at random I dentify relations Choice 1 Causal Decided by the player Opened Choice 2 Causal Causal Gain True Advanced Herd Management 2006 6 Chosen initially at random
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks Notation Random variable, Chance node C � Edges into a chance node Parent 1 Parent 2 (yellow circle) correspond to a set of conditional probabilities. They express the influence of the values Child of the parents on the value of the child. Advanced Herd Management 2006 7
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks Baysian netw orks Basically a static method � A static version of data filtering � Like dynamic linear models we may: � Model observed phenomena by underlying unobservable � variables. Combine with our knowledge on animal production. � Like Markov decision processes (Wednesday), there is a � structure and a set of parameters. All parameters are probabilities. � Advanced Herd Management 2006 8
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks The textbook A general textbook on Bayesian networks and decision graphs. � Written by professor Finn Verner Jensen from Ålborg � University – one of the leading research centers for Bayesian networks. � Many agricultural examples due to close collaboration with KVL and DJF through the Dina network, Danish Informatics Network in the Agricultural Sciences. Advanced Herd Management 2006 9
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks Probabilities What is the probability that a farmer observes a particular � cow in heat during a 3-week period? P( Heat = ”yes”) = a � P( Heat = ”no”) = b � a + b = 1 (no other options) � The value of Heat (”yes” or ”no”) is observable. � What is the probability that the cow is pregnant? � P( Pregnant = ”yes”) = c � P( Pregnant = ”no”) = d � � c + d = 1 (no other options) The value of Pregnant (”yes” or ”no”) is not observable. � Advanced Herd Management 2006 10
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks Conditional probabilities Now, assume that the cow is pregnant. What is the � conditional probability that the farmer observes it in heat? � P( Heat = ”yes” | Pregnant = ”yes”) = a p+ � P( Heat = ”no” | Pregnant = ”yes”) = b p+ � Again, a p+ + b p+ = 1 Now, assume that the cow is not pregnant. Accordingly: � � P( Heat = ”yes” | Pregnant = ”no”) = a p- � P( Heat = ”no” | Pregnant = ”no”) = b p- � Again, a p- + b p- = 1 Each value of Pregnant defines a full probability � distribution for Heat . Such a distribution is called conditional Advanced Herd Management 2006 11
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks A sm all Bayesian net Pregnant = ”yes” Pregnant = ”no” Pregnant c = 0.5 d = 0.5 Heat = ”yes” Heat = ”no” Heat Pregnant = ”yes” a p+ = 0.02 b p+ = 0.98 Pregnant = ”no” a p- = 0.60 b p- = 0.40 Let us build the net! � Advanced Herd Management 2006 12
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks: Evidence Experience w ith the net: Evindence � By entering information on an observed value of Heat we can revise our belief in the value of the unobservable variable Pregnant. � The observed value of a variable is called evidence. � The revision of beliefs is done by use of Baye’s Theorem : Advanced Herd Management 2006 13
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks: Evidence Baye’s Theorem for our net Advanced Herd Management 2006 14
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks: Evidence Let us extend the exam ple � A sow model: � Insemination Several heat observations � Pregnancy test � Consistent combination of information from different sources � Advanced Herd Management 2006 15
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks W hy build a Bayesian netw ork Because you wish to estimate certainties for the values of � variables that are not observable (or only observable at an unacceptable cost). Such variables are called “hypothesis variables”. The estimates are obtained by observing “information variables” � that either Influence the value of the hypothesis variable (“risk factors”), or � Depend on the hypothesis variable (“symptoms”) � Diagnostics/ Trouble shooting � Advanced Herd Management 2006 16
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks Diagnostics/ troubleshooting Risk 1 Risk 2 Risk 3 State Symp 1 Symp 2 Symp 3 Symp 4 Advanced Herd Management 2006 17
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Bayesian networks The sow pregnancy m odel Risk factor Insem. Hypothesis variable Pregn. Heat 1 Heat 2 Heat 3 Test Symptoms Advanced Herd Management 2006 18
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Conditional independence and d-separation Transm ission of evidence Age Calved Lact. Num. Yes/No Yes/No Age of a heifer/ cow influences the probability that it has calved. � Information on the “Calved” variable influences the probability that � the animal is lactating. Thus, information on “Age” will influence our belief in the state of � “Lact.” � If, however, “Calved” is observed, there will be no influence of “Age” on “Lact.”! Evidence may be transmitted through a serial connection, unless the � state of the intermediate variable is known. “Age” and “Lact” are d-separated given “Calved”. � They are conditionally independent given observation of “Calved” � Advanced Herd Management 2006 19
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Conditional independence and d-separation Diverging connections Landrace/Yorkshire/Duroc… Breed Litter Num. Color White/Black/Brown… size The breed of a sow influences litter size as well as color. � Observing the value of “Color” will tell us something about the � “Breed” and, thus, indirectly about the “Litter size”. If, however, “Breed” is observed, there will be no influence of “Color” � on “Litter size”! � Evidence may be transmitted through a diverging connection, unless the state of the intermediate variable is known. “Litter size” and “Color” are d-separated given “Breed”. � They are conditionally independent given observation of “Breed” � Advanced Herd Management 2006 20
Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Conditional independence and d-separation Converging connections Yes/No Mastitis Heat Yes/No Temp. Num. If nothing is known about “Temp.”, the values of “Mastitis” and “Heat” are � independent. If, however, “Temp.” is observed at a high level, the supplementary information � that the cow is in heat will decrease our believe in the state “Yes” for “Mastitis”. � “Explaining away” effect. Evidence may only be transmitted through a converging connection if the � connecting variable (or a descendant) is observed. Advanced Herd Management 2006 21
Recommend
More recommend
