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Bn Bn Presentation Transcript

  • Learning the Structure of Dynamic Probabilistic Networks Matt Hink March 27, 2012
  • Overview• Definitions and Introduction to DPNs• Learning from complete data• Experimental Results• Applications
  • Regular probabilistic networks (Bayesian networks) are well established for representing probabilistic relationships among many random variables. Dynamic Probabilistic Networks (DPNs), however, extend this representation to the modeling ofstochastic evolution of a set of random variables over time. (Think “probabilistic state machines”)
  • Notation• Capital letters (X,Y,Z)- sets of variables• X - Random variable of set X i• Val(X ) - Finite set of values of X i i• |X | - Size of Val(X ) i i• Lowercase italic (x,y,z)- set instantiations
  • DPNs are an extension to the common Bayesian network representation where the probabilitydistribution changes with respect to time according to some stochastic process. Assume that X is a set of variables in a PN which vary according to time. Then Xi[t] is the value of the attribute Xi at time t, and X[t] is the collection of such variables.
  • For simplicity’s sake, we assume that the stochastic process governing transitions is Markovian: P(X[t+1] | X[0...t]) = P(X[t+1] | X[t])That is, the probability of a certain instantiation isdependent only upon its immediate predecessor.
  • We also assume the process is stationary, i.e., P(X[t+1] | X[t]) is independent of t.
  • Given these two assumptions, we can describe a DPN representing the joint distribution over all possible trajectories of a process using two parts:A prior network B0 that specifies a distribution over the initial states X[0]; and A transition network B-> over the variables X[0] ∪ X[1] which specifies the transition probability P(X[t+1] | X[t]) for all t.
  • A prior network (left) and transition network (right) for a dynamic probabilistic network
  • In light of this structure, the joint distribution over the entire history of the DPN at time T is given as PB(x[0...T]) = PB0(x[0]) ∏(t=0...T-1) PB->(x[t+1] | x[t]) in other words, the product of all previous distributions.
  • Learning fromComplete Data
  • Common traditional methods:search algorithims using scoring methods (BIC, BDe) (given a dataset D) DPN methods: search algorithms using scoring methods! (given a dataset D, consisting of Nseq observations)
  • So each entry in our dataset consists of an observation of a set of variables over time. The mth such sequence has length Nm and specifies values for the variable set Xm[0...Nm] We then have Nseq instances of the initial state, andN = ∑m Nm instances of transitions. We can use these to learn the structure of the prior network and the transition network, respectively.
  • BIC scores for DPNs• Let
  • BIC scores for DPNs• We find the log-likelihood using
  • BIC scores for DPNs• And can then find the BIC score using
  • Experimental Results
  • Application: Modeling driver behavior