Template model for Bayesian network

1. Template variable $X(U_1, U_2,...,U_k)$ 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
   $$(X^{(t+1)}\perp X^{(0:t-1)}|X^{(t)})$$
   $$P(X^{(0:T)})=P(X^{(0)})\prod_{t=0}^{T-1}P(X^{(t+1)}|X^{(0:t)}) =P(X^{(0)})\prod_{t=0}^{T-1}P(X^{(t+1)}|X^{(t)})$$
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 $X_1,X_2,\dots, X_n$ is specified as a BN fragment such that:
     
     • The nodes include $X_1',X_2',\dots,X_n'$ and a subset of $X_1,X_2,\dots, X_n$
     • Only the nodes $X_1',X_2',\dots,X_n'$ have parents and a CPD
   • The 2TBN defines a conditional distribution
     $$P(X'|X)=\prod_{i=1}^n P(X_i'|\text{Pa}_{X_i'})$$
6. Dynamic Bayesian network
   
   • A dynamic Bayesian network over $X_1,\dots,X_n$ is defined by a
     
     • 2TBN $\text{BN}_{\rightarrow}$ over $X_1,\dots,X_n$
     • a Bayesian network $\text{BN}^{(0)}$ over $X_1^{(0)},\dots,X_n^{(0)}$
7. Ground Network
   For a trajectory over $0,\dots,T$ we define a ground (unrolled network) such that
   
   • The dependency model for $X_1^{(0)},\dots,X_n^{(0)}$ is copied from $\text{BN}^{(0)}$
   • The dependency model for $X_1^{(t)},\dots,X_n^{(t)}$ for all $t>0$ is copied from $\text{BN}_{\rightarrow}$