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Relational Markov Models and their Application to Adaptive Web Navigation Corin Anderson, Pedro Domingos, Daniel Weld
Review: Hidden Markov Models • Sequence of States • Transition distribution to next state dependant on evidence variable and current state only (in first-order case)
DBNs: A Step Up • Multiple nodes at each state in the sequence • “Object-oriented” view
Complaint 1: Too General! • Absolute restriction imposed on structure of state • Could individual state structure be exploited? • Adopt “Polymorphic” view
Complaint 2: Not General Enough! • Accurate probability estimation difficult with sparse data • P(Mac_instock|mainpage) = 0 • P(apple|mainpage) = 0.375 Web log for pages linked from main_page.html
Solution: RMM • Represent sets of states as a relation (predicate) with instantiations of the relation defining specific states • E.g. – product_page(mac, in_stock) main_page() • Transitions can now occur from sets to sets and sets to states
Going further . . . • Using predicates defines a basic hierarchy over the state space • Why not impose further structure within each predicate argument? • Tree leaves correspond to unique arguements
Sets of sets • Define the set of abstractions of a given state q = R( . . .) to be a set of states such that each argument to R in the abstraction is an ancestor to the corresponding argument in q. • More Formally (scientists love fancy math): R – The relation, Q – set of all possible instantiations of R d – possible arguments, D – the tree of a particular type of argument, delta – the arguments to the predicate for the given state q.
Learning with Abstractions • Transitions between a state and an abstraction: • In practice, we can just count instances in the data. Where a is a transition matrix, q is a ground state (individual state), and alpha and beta are abstracted arguments.
Defining a mixture model • Estimating the transition probability between ground states qs and qd: • Note that choosing lambda properly is crucial Where lambda is a mixing coefficient based on alpha and beta in the range [0,1] such that the sum of all lambdas is 1.
Choosing Lambda: • There is a bias-variance tradeoff • Possibly high variance at lowest abstraction level • High bias at highest abstraction level • A good choice will have two properities: • Gets higher as abstraction depth increases • Gets lower as available training data decreases Where n = possible transitions from alpha to beta in the data, k is a design parameter to penalize lack of data, and rank(a) is where each d is an argument to the relation defining a.
Probability Estimation Tree • With deep hierarchies and lots of arguments, considering every possible abstraction may not be feasible • Learn a decision tree over possible abstractions • To the left is the tree for the page Product_page(mac, in_stock)
Empirical Evaluation • How well do the various flavors of RMM (RMM-uniform, RMM-Rank, RMM-PET) compare to traditional MMs? • Analyzed several web logs in various domains. • Tried to predict transition probabilities between pages using varied amounts of training data.
Related Work: The Cube • Exploiting structure within RMM states (DPRM)