Department of Veterinary and Animal Sciences Introduction to Bayesian Networks Anders Ringgaard Kristensen
Department of Veterinary and Animal Sciences Outline Causal networks Bayesian Networks • Evidence • Conditional Independence and d-separation Compilation • The moral graph • The triangulated graph • The junction tree Slide 2
Department of Veterinary and Animal Sciences 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. Slide 3
A quiz – let’s try! 1 2 3
Department of Veterinary and Animal Sciences Can we model 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} Slide 5
Department of Veterinary and Animal Sciences Identify relations Chosen initially at random Choice 1 Causal Decided by the player Opened Choice 2 Causal Causal Gain True Chosen initially at random Slide 6
Department of Veterinary and Animal Sciences 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. Slide 7
Department of Veterinary and Animal Sciences Baysian networks 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, there is a structure and a set of parameters. All parameters are probabilities. Slide 8
Department of Veterinary and Animal Sciences 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. Slide 9
Department of Veterinary and Animal Sciences 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. Slide 10
Department of Veterinary and Animal Sciences 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 Slide 11
Department of Veterinary and Animal Sciences A small Bayesian net Pregnant = ”yes” Pregnant = ”no” Pregnant c = 0.5 d = 0.5 Heat = ”yes” Heat = ”no” Pregnant = ”yes” a p+ = 0.02 b p+ = 0.98 Heat Pregnant = ”no” a p- = 0.60 b p- = 0.40 Let us build the net! Slide 12
Department of Veterinary and Animal Sciences Experience with the net: Evidence 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 : Slide 13
Department of Veterinary and Animal Sciences Baye’s Theorem for our net Slide 14
Let us extend the example A sow model: • Insemination • Several heat observations • Pregnancy test Consistent combination of information from different sources
Department of Veterinary and Animal Sciences Why build a Bayesian network 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 Slide 16
Department of Veterinary and Animal Sciences Diagnostics/troubleshooting Risk 1 Risk 2 Risk 3 State Symp 1 Symp 2 Symp 3 Symp 4 Slide 17
Department of Veterinary and Animal Sciences The sow pregnancy model Risk factor Insem. Hypothesis variable Pregn. Heat 1 Heat 2 Heat 3 Test Symptoms Slide 18
Transmission 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”
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”
Converging connections Yes/No Yes/No Mastitis Heat 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.
Department of Veterinary and Animal Sciences Example: Mastitis detection Previous case Milk yield Mastitis index Heat Mastitis Temperature Conductivity Slide 22
Department of Veterinary and Animal Sciences Compilation of Bayesian networks Compilation: • Create a moral graph • Add edges between all pairs of nodes having a common child. • Remove all directions • Triangulate the moral graph • Add 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. The software system does it automatically (and can show all intermediate stages). Slide 23
Department of Veterinary and Animal Sciences Why use Bayesian networks? A consistent framework for • Representation and dealing with uncertainty • Combination of information from different sources. • Combination of numerical knowledge with structural expert knowledge. Slide 24
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