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

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

Elimination in Chains Forward pass Using definition of probability, we have A B C E D - Inference (Variable Elimination)

Elimination in Chains By chain decomposition, we get A B C E D - Inference (Variable Elimination)

Elimination in Chains Rearranging terms ... A B C E D - Inference (Variable Elimination)

X Elimination in Chains Now we can perform innermost summation A B C E A has been eliminated A B C E D Now we can perform innermost summation - Inference (Variable Elimination)

X X … Elimination in Chains Rearranging and then summing again, we get B has been eliminated A B C E D Rearranging and then summing again, we get - Inference (Variable Elimination) …

Elimination in Chains with Evidence Similar due to Bayes’ Rule We write the query in explicit form A B C E D - Inference (Variable Elimination)

Variable Elimination Write query in the form Iteratively Move all irrelevant terms outside of innermost sum Perform innermost sum, getting a new term Insert the new term into the product Eliminate irrelevant variables Semantic Bayesian Network - Inference (Variable Elimination)

Asia - A More Complex Example Visit to Asia Smoking Lung Cancer Tuberculosis Abnormality in Chest Bronchitis X-Ray Dyspnoea - Inference (Variable Elimination)

We want to compute P(d) Need to eliminate: v,s,x,t,l,a,b Initial factors - Inference (Variable Elimination)

We want to compute P(d) Need to eliminate: v,s,x,t,l,a,b Initial factors Compute: Eliminate: v - Inference (Variable Elimination) but in general the result of eliminating a variable is not necessarily a probability term !

We want to compute P(d) Need to eliminate: s,x,t,l,a,b V S L T A B X D We want to compute P(d) Need to eliminate: s,x,t,l,a,b Initial factors Compute: Eliminate: s - Inference (Variable Elimination) Summing on s results in a factor with two arguments fs(b,l). In general, result of elimination may be a function of several variables

We want to compute P(d) Need to eliminate: x,t,l,a,b V S L T A B X D We want to compute P(d) Need to eliminate: x,t,l,a,b Initial factors - Inference (Variable Elimination) Compute: Eliminate: x Note that fx(a)=1 for all values of a

We want to compute P(d) Need to eliminate: t,l,a,b V S L T A B X D We want to compute P(d) Need to eliminate: t,l,a,b Initial factors - Inference (Variable Elimination) Compute: Eliminate: t

We want to compute P(d) Need to eliminate: l,a,b V S L T A B X D We want to compute P(d) Need to eliminate: l,a,b Initial factors - Inference (Variable Elimination) Compute: Eliminate: l

We want to compute P(d) Need to eliminate: a,b V S L T A B X D We want to compute P(d) Need to eliminate: a,b Initial factors - Inference (Variable Elimination) Compute: Eliminate: a,b

Variable Elimination (VE) We now understand VE as a sequence of rewriting operations Actual computation is done in elimination step Exactly the same procedure applies to Markov networks Computation depends on order of elimination - Inference (Variable Elimination)

Dealing with Evidence How do we deal with evidence? S L T A B X D Dealing with Evidence How do we deal with evidence? Suppose get evidence V = t, S = f, D = t We want to compute P(L, V = t, S = f, D = t) - Inference (Variable Elimination)

Dealing with Evidence We start by writing the factors: V S L T A B X D - Inference (Variable Elimination)

Dealing with Evidence We start by writing the factors: X D Dealing with Evidence We start by writing the factors: Since we know that V = t, we don’t need to eliminate V Instead, we can replace the factors P(V) and P(T|V) with - Inference (Variable Elimination)

Dealing with Evidence We start by writing the factors: X D Dealing with Evidence We start by writing the factors: Since we know that V = t, we don’t need to eliminate V Instead, we can replace the factors P(V) and P(T|V) with This “selects” the appropriate parts of the original factors given the evidence Note that fP(v) is a constant, and thus does not appear in elimination of other variables - Inference (Variable Elimination)

Dealing with Evidence Initial factors, after setting evidence Given evidence V = t, S = f, D = t, compute P(L, V = t, S = f, D = t ) V S L T A B X D Dealing with Evidence Initial factors, after setting evidence - Inference (Variable Elimination)

Dealing with Evidence Initial factors, after setting evidence Given evidence V = t, S = f, D = t, compute P(L, V = t, S = f, D = t ) V S L T A B X D Dealing with Evidence Initial factors, after setting evidence Eliminating x, we get - Inference (Variable Elimination)

Dealing with Evidence Initial factors, after setting evidence Given evidence V = t, S = f, D = t, compute P(L, V = t, S = f, D = t ) V S L T A B X D Dealing with Evidence Initial factors, after setting evidence Eliminating x, we get Eliminating t, we get - Inference (Variable Elimination)

Dealing with Evidence Initial factors, after setting evidence Given evidence V = t, S = f, D = t, compute P(L, V = t, S = f, D = t ) V S L T A B X D Dealing with Evidence Initial factors, after setting evidence Eliminating x, we get Eliminating t, we get Eliminating a, we get - Inference (Variable Elimination)

Dealing with Evidence Initial factors, after setting evidence Given evidence V = t, S = f, D = t, compute P(L, V = t, S = f, D = t ) V S L T A B X D Dealing with Evidence Initial factors, after setting evidence Eliminating x, t, a, we get Finally, when eliminating b, we get - Inference (Variable Elimination)

Dealing with Evidence Initial factors, after setting evidence Given evidence V = t, S = f, D = t, compute P(L, V = t, S = f, D = t ) V S L T A B X D Dealing with Evidence Initial factors, after setting evidence Eliminating x, t, a, we get Finally, when eliminating b, we get - Inference (Variable Elimination)

Reminder: Probability theory Basics of Bayesian Networks Outline Introduction Reminder: Probability theory Basics of Bayesian Networks Modeling Bayesian networks Inference (VE, Junction Tree) Excourse: Markov Networks Learning Bayesian networks Relational Models