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A Quick Romp Through Probabilistic Relational Models

A Quick Romp Through Probabilistic Relational Models. Teg Grenager NLP Lunch February 13, 2003. Agenda. Bayesian Networks Probabilistic Relational Models Learning PRMs Expressivity Applications to NLP. Agenda. Bayesian Networks Probabilistic Relational Models Learning PRMs

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A Quick Romp Through Probabilistic Relational Models

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  1. A Quick Romp ThroughProbabilistic Relational Models Teg Grenager NLP Lunch February 13, 2003

  2. Agenda • Bayesian Networks • Probabilistic Relational Models • Learning PRMs • Expressivity • Applications to NLP

  3. Agenda • Bayesian Networks • Probabilistic Relational Models • Learning PRMs • Expressivity • Applications to NLP

  4. Propositional Logic • Ontological commitment: the world consists of propositions, or facts, which are either true or false: HighPaperRating • Set of 2n possible worlds – one for each truth assignment to the n propositions • Propositional logic allows us to compactly represent restrictions on possible worlds: • If HighPublicationRating then HighPaperRating • Means that we have eliminated the possible worlds where HighPublicationRating is true but HighPaperRating is false.

  5. Propositional Uncertainty • To model uncertainty we would like to represent a probability distribution over possible worlds. • To represent the full joint distribution we would need 2n-1parameters (infeasible) • Insight: the value of most propositions isn't affected by the value of most other propositions! • More formally, some propositions are conditionally independent of each other given the value of other propositions

  6. Bayesian Networks • We use a directed acyclic graph to encode these independence assumptions • This model encodes the assumption that each variable is independent of its non-descendents given its parents AuthorInstitution JournalRating AuthorRating PaperRating PaperCited

  7. Factorization • If a BN encodes the true independence assumptions of a distribution, we can use a factored representation for the distribution: • To specify the full joint we need only the conditional probabilities of a variable given its parents

  8. Bayesian Networks • The full joint over these five binary variables would need 25-1=31 parameters, but this factored representation only needs 10! AuthorInstitution JournalRating AuthorRating PaperRating PaperCited

  9. Inference in Bayes Nets • Query types (given evidence z): • Conditional probability query: what is the probability distribution over the values of subset y? • Most probable explanation query: what is the most likely assignment of values to all remaining variables x-z? • Maximum a posteriori query: what is the most likely assignment of values to subset y? • Worst case, inference is NP-hard • In practice, much easier

  10. Variable Elimination AuthorInstitution JournalRating AuthorRating PaperRating PaperCited

  11. Learning Bayes Nets • We want to learn a BN from a dataset D that consists of m tuples, each of the form x(m), specifying the value of all variables xi • Two problems: • Given a graphical model G, estimate the the conditional probability distribution at each node (parameter estimation) • Select the best graphical model (structure learning)

  12. Parameter Estimation • Note that we can decompose this and estimate the parameters separately: • Can also take a Bayesian approach

  13. Structure Learning • Hypothesis space: • Exponential number of possible structures over the variables • Scoring function (minimum description length or Bayesian) includes: • Likelihood of the structure given the data and the maximum likelihood parameters  • Description length of the graph and CPDs • Search algorithm: • Operators: add, delete, and reverse an edge • Greedy hill-climbing with random restart

  14. Agenda • Bayesian Networks • Probabilistic Relational Models • Learning PRMs • Expressivity • Applications to NLP

  15. Bayes Net Shortcomings • BNs lack the concept of an object • Cannot represent general rules about the relations between multiple similar objects • For example, if we wanted to represent the probabilities over multiple papers, authors, and journals: • We would need an explicit random variable for each paper/author/journal • The distributions would be separate, so knowledge about one wouldn't impart any knowledge about the others

  16. Relational Models • Relational models make a stronger ontological commitment: the world consists of objects, and relations over them • There are many possible relational models (more on this later) • PRMs are based on a particular “relational logic” borrowed from databases:

  17. Relational Schema • We define a relational schema to consist of • A set of n classes X = {X1,…,Xn} • Given a class X, a set of attributes A(X) • Attribute A of class X is denoted X.A, and its space of values is denoted V(X.A) • Given a class X, a set of reference slots R(X) • Reference slot  of class X is denoted X., with domain type X and range type of some class Y • Each reference slot  has an inverse slot -1 • A slot chain  is a sequence of slots 1,… k such that for all i, Range(i)=Domain(I+1)

  18. Author Author Name Name D. Koller Institution Author Institution U. Maryland Rating Name L. Getoor Rating 3 Authorship Institution U. Maryland Authorship Author D. Koller Rating 5 Authorship Author Paper Learning in PRMs Author D. Koller Paper Authorship Paper Learning in PRMs Paper Author L. Getoor Paper Title OOBNs Paper Learning in PRMs Title Paper Publication Proc. IJCAI Publication Title Learning in PRMs Rating 5 Rating Publication Proc. IJCAI Rating 5 Citation Publication To OOBNs Name Publication From Learning in PRMs Rating Name Proc. UAI Publication Rating 3 Citation Name Proc. IJCAI To Rating 3 From Relational Model Example

  19. Modeling Uncertainty • Given a schema, a possible world specifies: • A set of objects in each class • An assignment of objects to reference slots • An assignment of values to attributes • The set of possible worlds is infinite, hard to define a distribution over • Thus a PRM only specifies a distribution over the possible assignment of values to attributes given a set of ground objects and the relations between them

  20. Modeling Uncertainty • More formally, we define: • A relational skeleton, , to be a set of objects and relations between them (defined as reference slot values) • An instance, , to be an assignment of values to attributes • A PRM defines a probability distribution over possible completions  of a skeleton  • Let x.A be the value of x.A in instance 

  21. Relational Model Example Author Author Name Name D. Koller Institution Author Institution U. Maryland Rating Name L. Getoor Rating 3 Authorship Institution Authorship Author D. Koller Rating Authorship Author Paper Learning in PRMs Author D. Koller Paper Authorship Paper Learning in PRMs Paper Author L. Getoor Paper Title OOBNs Paper Learning in PRMs Title Paper Publication Proc. IJCAI Publication Title Learning in PRMs Rating 5 Rating Publication Proc. IJCAI Rating Citation Publication To OOBNs Name Publication From Learning in PRMs Rating Name Proc. UAI Publication Rating 3 Citation Name Proc. IJCAI To Rating From

  22. PRM Dependency Structure • PRMs assume that the attribute values of objects are each influenced by only a few other attribute values (as in a BN) • Thus we associate with each attribute X.A a set of parents Pa(X.A) • These are formal parents; they will be instantiated differently for different objects • These sets of parents (one for each attribute) define the dependency structure S of the PRM

  23. Types of Parents • We define two types of parents for X.A: • Another attribute X.B of the same class X • E.g., Author.Rating could depend on Author.Institution • An attribute of a related object X..B where  is a slot chain • E.g., Paper.Rating could depend on Paper.Publication.Rating

  24. Author Author Name Name D. Koller Institution Author Institution U. Maryland Rating Name L. Getoor Rating 3 Authorship Institution Authorship Author D. Koller Rating Authorship Author Paper Learning in PRMs Author D. Koller Paper Authorship Paper Learning in PRMs Paper Author L. Getoor Paper Title OOBNs Paper Learning in PRMs Title Paper Publication Proc. IJCAI Publication Title Learning in PRMs Rating 5 Rating Publication Proc. IJCAI Rating Citation Publication To OOBNs Name Publication From Learning in PRMs Rating Name Proc. UAI Publication Rating 3 Citation Name Proc. IJCAI To Rating From Relational Model Example

  25. Multisets as Parents • But what if X..B points to more than one value? • E.g., Paper.Authorship.Author.Rating points to the ratings of all coauthors of the paper • We define an aggregate function, , to map from a multiset of attributes to a summary value (e.g., sum, mean, max, cardinality) • We allow X.A to have as a parent (X..B) • E.g., Paper.Rating depends on mean(Paper.Authorship.Author.Rating)

  26. Author Author Name D. Koller Name Institution Institution Rating Rating Authorship Author Authorship Author D. Koller Name L. Getoor Authorship Author Paper Learning in PRMs Institution Author D. Koller Paper Rating Paper Learning in PRMs Paper mean Title Publication Rating Paper Title OOBNs Publication Publication Proc. IJCAI Name Rating Rating Publication Citation Name Proc. UAI To Rating From Relational Model Example

  27. PRM Parameters • As in a BN, for each attribute we define a conditional probability distribution (CPD) over the values of the attribute given the values of the parents • More precisely, let U = Pa(X.A) be the set of parents, V(U) be the possible values of U, and uV(U) be some tuple of of values • Then we can define a distribution P(X.A|u) parameterized by X.A|u • Let S be the union of all X.A|u

  28. Joint Probability Distribution • Now we can use the following factored representation for the joint probability distribution over possible instances  consistent with skeleton  : • Where O(X) denotes the set of objects in skeleton  whose class is X

  29. Acyclicity • Problem: Distribution is not coherent when dependency structure S has cycles • Naïve approach:Require acyclic class dependency graph • This would prohibit a dependency of the genotype of a person (child) on the genotype of a person (parent), even though it is clearly acyclic • Better: require certain guaranteed acyclic slots • The parent slot above is guaranteed acyclic • Graph coloring algorithm for checking legality of dependency structures

  30. Structural Uncertainty • Reference Uncertainty • Uncertainty over values of reference slots R(X) for each object in class X (but range is given) • Naïve approach: if range of X. has range Y, use distribution over all objects O(Y) • Too large, not general • Better: use a partition over objects O(Y) • Base partition on a subset of attributes Partition[]  A(Y) • Create a selector attribute S which takes a value in the space of values V(Partition[]) • Define parents and a CPD for S

  31. Structural Uncertainty (2) • Existence Uncertainty • Uncertainty over number of objects in a class • Distinguish between determined and undetermined classes • Undetermined classes are typically those representing many-many relations like Citation • Place distribution over objects in undetermined classes • Consider all “possible” objects in class • Create special existence attribute X.E whose values V(E) = {true, false} • Define parents and a CPD for X.E

  32. Inference in PRMs • Given a skeleton (set of objects and relations), a PRM defines a distribution over possible instances (assignments of values to attributes) • Same query types as in BNs • To answer queries, we compile the PRM to its associated BN, and use BN inference as described earlier (that’s it!)

  33. Agenda • Bayesian Networks • Probabilistic Relational Models • Learning PRMs • Expressivity • Applications to NLP

  34. Learning PRMs • Training set consists of a fully specified instance: a set of objects, the relations between them, and the values of all attributes • In other words, a database! • As in BNs, we split into two problems: • Given a dependency structure S, estimate the the conditional probability distribution at each node (parameter estimation) • Select the best dependency structure (structure learning)

  35. Parameter Estimation • We do maximum likelihood estimation for the BN induced by the structure given the skeleton • Parameters are tied for nodes of same class • As in BNs, the likelihood function can be decomposed and learned separately • As in BNs, can also take a Bayesian approach

  36. Structure Learning • We need a hypothesis space: • A hypothesis is a dependency structure, specifying parents for each attribute X.A. • Hypothesis space is infinite because the length of the slot chain leading to the parent attribute is unbounded • No problem, we let the search algorithm decide how long to make slot chains • We must ensure that the dependency structures we learn is acyclic • No problem, just test each dependency structure before we consider it

  37. Structure Learning (2) • We need a scoring function to evaluate the "goodness" of each candidate hypothesis: • We use Bayesian model selection, where the score of a structure S is defined as the posterior probability of the structure given the data : • The second component of the score P(S|) = P(S) is a prior over structures • To penalize length of slot chains we set logP(S) to be proportional to the total length of the chains in S

  38. Structure Learning (3) • The first component of the score is the marginal likelihood: • If we use a parameter independent Dirichlet prior (over parameters), this integral decomposes into a product of integrals each of which has a simple closed form solution • E.g., uniform Dirichlet prior over parameters • Asymptotically equivalent to explicit penalization, such as the MDL score

  39. Structure Learning (4) • Finally we need a search algorithm: • Greedy hill-climbing with random restart? Problems: • Infinitely many possible structures • Process of computing sufficient statistics at each search step requires expensive database joins • Heuristic search: • Series of phases. At phase k we identify a set of potential parents Potk(X.A) for each attribute X.A • We do standard structure search restricted to space of structures in which each attribute X.A has parents in Potk(X.A) • Allows us to precompute database views

  40. Agenda • Bayesian Networks • Probabilistic Relational Models • Learning PRMs • Expressivity • Applications to NLP

  41. Relational Models • The most general relational model: the world consists of objects and relations over them • First order logic is perhaps the most basic relational setting: • Syntax • Constants and quantified variables (representing objects) • Predicates (representing relations), stated in terms of constants and variables, composed with logical connectives • Semantics: • Set of possible worlds, one for each possible extent of each relation

  42. Relational Models • Different relational paradigms: • First-order logic (from math, phil) • Relational algebra (from DB) • Entity/Relation models (from DB) • Frame-relational systems (from KR) • How do they relate to each other? • FOL makes the fewest assumptions • E/R models add classes (of objects), attributes (of objects), and typed relations • FRS add class hierarchies • FOL can represent E/R, FRS (of course) • E/R, FRS can represent FOL (awkwardly)

  43. Relational Models • PRMs as currently defined cannot represent uncertainty in general FOL • Cannot represent uncertainty about whether or not a relation exists between a given tuple of objects • Even when we add “structural uncertainty” as proposed PRMs are too specialized • The probability of a relation between objects would conditioned on the values of some of their attributes, not on their participation in other relations

  44. Alternative Models • Taskar, Abbeel, Koller. Discriminative Probabilistic Models for Relational Data. UAI 2002. • Muggleton, S. Stochastic logic programs. Journal of Logic Programming. 2001. • Kersting, K. and De Raedt, L. Bayesian logic programs. Work-in-progress Track at the 10th International Conference on ILP. 2000. • Poole, D. Probabilistic Horn abduction and Bayesian networks. • Halpern, J. An analysis of first-order logics of probability

  45. Agenda • Bayesian Networks • Probabilistic Relational Models • Learning PRMs • Expressivity • Applications to NLP

  46. Applications to NLP • Can use it anywhere you want to have parameter tying • E.g., Naïve Bayes can be represented by PRM • Two classes: Document and Feature • A(Document) = {Document.Class} • A(Feature) = {Feature.Value} • R(Feature) = {Feature.Document} • Dependency structure: • Feature.Value depends on Feature.Document.Class

  47. Relational Classification • Hypertext classification, using text and links • Previously, iterative labeling algorithms that reassigns labels based on neighbors labels • PRMs with “existence uncertainty” provide unified framework for links and text • Classes: Page, Anchor, Link (undetermined) • A(Page) = {Category, Hub, Word1,…Wordn} • A(Link) = {Exists}, V(Link) = {From-Page, To-Page} • A(Anchor) = {Word}, V(Anchor) = {In-Link} • Learn PRM using Bayesian parameter est. • Predict using belief propagation

  48. Relational Clustering • Clustering in a relational domain: e.g., Papers and Citations • They use multi-valued reference slots • Classes: Paper, Author • A(Paper) = {Class, Word1,…,Wordn} • V(Author) = {Cited, Wrote} • A(Author) = {Class} • Dependency structure has aggregation ops • Pa(Paper.Class) = {Stoch-Mode(Wrote.Class), Stoch-Mode(Cited.Class)} • Pa(Paper.Wordi) = {Paper.Class} • Learn with EM, using belief prop for E-step

  49. Database Optimization • Estimating the size of complex queries • Previous approaches assume: • Attribute independence • Join uniformity (prob of two tuples joining is uniform • But PRMs provide compact representation of joint probability distribution over attributes • Represent skew in the join probabilities between tuples • Represent correlation between attributes of tuples joined via foreign-key.

  50. Other Applications • Modeling gene expression data…

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