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Probabilistic networks Inference and Other Problems

Probabilistic networks Inference and Other Problems. Hans L. Bodlaender Utrecht University. Overview. Probabilistic networks The inference problem Tree decompositions and an algorithm for probabilistic inference The Maximum Probable Assignment problem Monotonicity.

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Probabilistic networks Inference and Other Problems

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  1. Probabilistic networksInference and Other Problems Hans L. Bodlaender Utrecht University

  2. Overview • Probabilistic networks • The inference problem • Tree decompositions and an algorithm for probabilistic inference • The Maximum Probable Assignment problem • Monotonicity Probabilistic networks - IPA fall days

  3. Decision support systemsReasoning with uncertainty • Decision support systems (and/or expert systems) • Reasoning with uncertainty • Set of stochastic variables • Observations • Other variables • Variable(s) of interest • In 1980s the probabilistic network model was proposed. • Also called: Bayesian networks, belief networks, graphical models Probabilistic networks - IPA fall days

  4. Probabilistic networks Pr(x1)=0.7 Pr(¬x1)=0.3 • Directed acyclic graph • Each node is a (discrete) stochastic variable • E.g. Boolean variable • Given for each variable is its conditional probability distribution: • Conditional to values for the parents of the node x1 x2 x3 … … x5 x4 Pr(x5|x3)= 0.6 Pr(¬x5|x3)= 0.4 Pr(x5|¬x3)= 0.2 Pr(¬x5|¬x3)=0.8 Pr(x4| x2 and x3) = 0.12 etc. Probabilistic networks - IPA fall days

  5. Probabilistic networks - IPA fall days

  6. Probabilistic networks - IPA fall days

  7. A probabilistic networkconsists of • A directed acyclic graph G=(V,E) • For each vertex (variable) v, a table with conditional probabilities (conditional to values of parents of v) Pr(v | w and x) Pr(¬v | w and x)Pr(v | ¬w and x) Pr(¬v | ¬w and x)Pr(v | w and ¬x) Pr(¬v | w and ¬x)Pr(v | ¬w and ¬x) Pr(¬v | ¬w and ¬x) w x v Example Probabilistic networks - IPA fall days

  8. Configuration • A configurationc is an assignment of a value to each variable (node). • For set W of variables, or variable v, and configuration c, denote cWand cvfor the restrictions (partial configurations). • Probability of configuration c: Probabilistic networks - IPA fall days

  9. Topological sort of directed acyclic graph • Order of vertices such that edges go from left to right: • List vertices v1, …, vn such that for each arc (vi,vj): i < j. • Always exists for dag, and can be found in O(|V|+|E|) time. Probabilistic networks - IPA fall days

  10. Generating a random configuration Pr(v1)=0.7 Pr(¬v1)=0.3 x1 Make a topological sort of G For i= 1 to n dogenerate a value for viusing the probabilities dictated by values already generated for the parents of i x2 x3 … Pr(v2|v1) = 0.3Pr(¬v2|v1) = 0.7Pr(v2|¬v1) = 0.4Pr(¬v2|¬v1) = 0.6 x4 x5 … Pr(v5|v3)= 0.6 Pr(¬v5|v3)= 0.4 Pr(v5|¬v3)= 0.2 Pr(¬v5|¬v3)=0.8 Probabilistic networks - IPA fall days

  11. Inference problem • Given: values for some variables (observations) cO • Question: probability distribution on one variable conditional to observations, or: Probabilistic networks - IPA fall days

  12. Use of inference problem • Network models information from application domain (medical, agricultural, weather forecasting, …) • User gives values for some variables (symptoms of patient, observed values) and wants to know distribution for other variables (likeliness of success of treatment, diagnostic) • Used nowadays in many applications Probabilistic networks - IPA fall days

  13. Probabilistic networks - IPA fall days

  14. Inference problem is #P-complete • #P-completeness implies NP-hardness. • Proof of #P-hardness: • Number of satisfying truth assignments of 3CNF formula is #P-complete • E.g.: (x1 or x2 or ¬x4) and (x5 or ¬x1 or ¬x3) and … • Transform to probabilistic network Probabilistic networks - IPA fall days

  15. Example transformation (x1 or x2 or ¬ x4) and … Pr(xi) = 0.5 Pr(¬xi) = 0.5 x1 x2 x3 x4 x5 x6 T: Probability 1 when satisfied; otherwise 0 T: Probability 1 whenboth parents true Probability equals #sat / 2n Probabilistic networks - IPA fall days

  16. Using tree decompositions to solve the problem Fast when width of tree decomposition is small Tree decomposition of moralisation of G: Tree with each node a bag: set of variables For all v: Bags with v form connected subtree There is a bag containing v and its parents (bag coversv) Lauritzen-Spiegelhalter algorithm x1 x2 x3 x4 x5 x1 x2 x3 x3 x2 x4 x4 x5 Probabilistic networks - IPA fall days

  17. Here: description without observations. Take node with variable of interest in it as root. Compute for each node i with bag X of T a table For each assignment cX of values to the variables in X, with Y the variables covered in the subtree with root i, compute vi(cx): LS algorithm X Y extends Probabilistic networks - IPA fall days

  18. A table for a node can be computed when the tables of the children are known E.g., compute tables in postorder (bottom up) Computing tables bottom up Probabilistic networks - IPA fall days

  19. Example: node with two children with all bags identical X i X X j1 j2 Y1 Y2 Probabilistic networks - IPA fall days

  20. LS algorithm • For other types of nodes, similarly table can be computed. • Time for one table linear in size of table: bag size k gives time O(2k) for binary variables. • Linear time when bag size bounded by constant (bounded treewidth). Happens often in practice! • Table of root allows to compute distribution for variables in root bag • Similar scheme when observations are given; when variables are discrete but not all binary • Scheme with also moving downwards in tree to compute distribution for all variables: also linear time for bounded treewidth Probabilistic networks - IPA fall days

  21. MAP problem • Given: probabilistic network, some observations • Question: most likely configuration given the observations • Applications: most likely explanation, verification of design of probabilistic networks Probabilistic networks - IPA fall days

  22. MAP is NP-hard Pr(xi) = 0.5 Pr(¬xi) = 0.5 (x1 or x2 or ¬ x4) and … x1 x2 x3 x4 x5 x6 Shimoney, 1994 T: Probability 1 when satisfied or Y is T otherwise 1/2 Probabilistic networks - IPA fall days

  23. Similar algorithm as for inference can solve MAP in linear time when tree decomposition of moralisation with bounded bag size (treewidth) is given Compute for cX: MAP with tree decompositions X Y Probabilistic networks - IPA fall days

  24. Fixed parameter variant of MAP • MAP(p): • Given: probabilistic network • Question: is there a configuration with probability at least p? • Can be solved in O(f(p) n) time, i.e., linear for fixed p. • Joint work with van der Gaag and van den Eijkhof. • Similar result when there are observations (values for some variables), and we look to a configuration consistent with the observations Probabilistic networks - IPA fall days

  25. Look at variables in order of a topological sort Recursive process: Branch for assignment of value to next variable Plus … bounding mechanism Algorithm uses branch and bound Start here v1=T v1=F v1=T, v2=T v1=T, v2=F v1=T, v2=T v1=T, v2=F v1=T, v2=T, v3=T v1=T, v2=T, v3=T … Probabilistic networks - IPA fall days

  26. Bounding • Recall: • Parents of v are before v in topological sort • Compute for a node z in branch and bound tree with assigned values • P(z) can be computed from P(parent(z)) and choice for ith variable • Bound when P(z) < p: this can never be a solution Probabilistic networks - IPA fall days

  27. Recursive scheme • E-MPA-p(values for first i variables, p, pz) • If i=n (we have done all variables), then return true (output the sequence); stop. • Else: For each possible value x for vi+1: • Compute pznew = pz * Pr(vi+1=x | values for first i vertices) • If pznew ³ p, thenE-MPA-p( values for first i variables and then x, p, pznew) Probabilistic networks - IPA fall days

  28. Time analysis • If a node has at least two children in the tree, then • For each child, pznew³p, hence Pr(vi+1=x | values for first i vertices) ³p • Hence: pznew£ pz * (1-Pr(vi+1=x | values for first i vertices)) £pz * (1-p) • After a node in the tree with two children, value of pz is a factor at least (1-p) smaller • Tree has at mostlog p / log (1-p)leaves. (How often can you divide 1 by 1-p till you are smaller than p?) • Time is O(f(p) * n). Probabilistic networks - IPA fall days

  29. Partial MAP • Variant of MAP where we ask for values to subset of variables with maximum probability, given some observations • Park: NPPP-complete, and NP-complete when G is a polytree (underlying undirected graph is a tree) Probabilistic networks - IPA fall days

  30. Monotonicity • Joint work with Linda van der Gaag and Ad Feelders • Monotonicity is often a requested property of a probabilistic network • E.g.: if a patient has more severe symptons, one expects the diagnosis is more severe • Ordering on the values of variables • cX£c’X if for all x in X: cX(x) £c’X(x) • Two observations that are ordered should imply ordering of probabilities of values for variable of interest (formal definition follows). Probabilistic networks - IPA fall days

  31. Monotonicity in mode • Let z be the output variable. • The mode of z given values cX for some other variables X: T(z | cX) is that value for z such that Pr(z| cx) is maximal. (+ tie-breaking rule) • Take ordering on values of each variable. • The probabilistic network with observable nodes X and output variable z is isotone when each pair of value assignments to X, cX,c’X, one has: • cX£ c’X implies T(z | cX) £ T(z | c’X) • Antitone: cX£ c’X implies T(z | cX) ³ T(z | c’X) • Monotone: isotone or antitone • Monotone in distribution: similar, but looking to cumulative distribution. Identical to monotonicity in mode when all variables are binary. Probabilistic networks - IPA fall days

  32. Results • Testing if network is monotone (isotone, antitone) in mode (in distribution) is: • coNPPP complete • coNP-complete for polytrees Probabilistic networks - IPA fall days

  33. Transformation from variant of Partial MAP problem Can we set values for M, such that Pr(E=T|cM) > p ? Pr(A=T| E=T) = 1 Pr(A=T| E=F) = (1/2 –p)/(1-p) Pr(C=T| A, B) = 1 if A and B F, otherwise 0 Proof shows that the new network is monotone in mode, and monotone in distribution, if and only if there is a cM with Pr(E=T|cM) > p Hardness proof (sketch) G: instance of Partial MAP M E A B C M U B set of observable variables; C variable of interest Probabilistic networks - IPA fall days

  34. Conclusions • Probabilistic (belief, Bayesian) networks form mathematical precise model • Used in several decision support system • Use and design of networks pose interesting challenges, many algorithmic • Sometimes special structures help (tree decompositions), also in practice Probabilistic networks - IPA fall days

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