Template model for Bayesian network

1. Template variable X(U1,U2,...,Uk) is instantiated (duplicated) multiple times.
2. Template models:
   
   • languages that specify how variables inherit dependency model from template
   • Dynamic Bayesian networks
   • Object-relational models:
     
     • Directed: plate models
     • Undirected
3. Markov assumption
   
   (X(t+1)⊥X(0:t−1)|X(t))
   
   
   P(X(0:T))=P(X(0))∏t=0T−1P(X(t+1)|X(0:t))=P(X(0))∏t=0T−1P(X(t+1)|X(t))
4. Time Invariance
   For all t , P(X(t+1)|X(t))=P(X′|X)
5. 2-time-slice Bayesian network
   
   • A tranisition model (2TBN) over X1,X2,…,Xn is specified as a BN fragment such that:
     
     • The nodes include X′1,X′2,…,X′n and a subset of X1,X2,…,Xn
     • Only the nodes X′1,X′2,…,X′n have parents and a CPD
   • The 2TBN defines a conditional distribution
     
     P(X′|X)=∏i=1nP(X′i|PaX′i)
6. Dynamic Bayesian network
   
   • A dynamic Bayesian network over X1,…,Xn is defined by a
     
     • 2TBN BN→ over X1,…,Xn
     • a Bayesian network BN(0) over X(0)1,…,X(0)n
7. Ground Network
   For a trajectory over 0,…,T we define a ground (unrolled network) such that
   
   • The dependency model for X(0)1,…,X(0)n is copied from BN(0)
   • The dependency model for X(t)1,…,X(t)n for all t>0 is copied from BN→