I am aware that Bayesian Networks and Dynamic Bayesian Networks do not allow cycles. However, there is something I can't figure out and which is simple: what is a cycle in a DBN?
Consider two nodes, $A$ and $B$. We note $A^T$ (resp. $B^T$) the outcome of node $A$ at time slice $T$. If you have the following inter-slice edges: $A^T \rightarrow B^{T+1}$ and $B^T \rightarrow A^{T+1}$, is it considered to form a cycle? Strictly speaking, if you unfold the $2-TBN$ over $N$ time slices, you will never found a circuit such that $A^{T} \rightarrow B^{T+1} \rightarrow \dots \rightarrow B^{T+N}\rightarrow A^T$ exists (for example).
However, if you assume that $A^T = A^{T+N}, \forall N \in \mathbb{N}$, then there is a circuit and the DBN is not valid. So my question is simple: is node $A^T$ the same as node $A^{T+N}$? If yes, why? I suppose that it is not the case and that as soon as you don't have cycles in the $2-TBN$, you can assume there will be no cycle also in an unfolded $2-TBN$, over $N$ time slices.
I am aware that Bayesian Networks and Dynamic Bayesian Networks do not allow cycles. However, there is something I can't figure out and which is simple: what is a cycle in a DBN?
Consider two nodes, $A$ and $B$. We note $A^T$ (resp. $B^T$) the outcome of node $A$ at time slice $T$. If you have the following inter-slice edges: $A^T \rightarrow B^{T+1}$ and $B^T \rightarrow A^{T+1}$, is it considered to form a cycle? Strictly speaking, if you unfold the $2-TBN$ over $N$ time slices, you will never found a circuit such that $A^{T} \rightarrow B^{T+1} \rightarrow \dots \rightarrow B^{T+N}\rightarrow A^T$ exists (for example).
However, if you assume that $A^T = A^{T+N}, \forall N \in \mathbb{N}$, then there is a circuit and the DBN is not valid. So my question is simple: is node $A^T$ the same as node $A^{T+N}$? If yes, why? I suppose that it is not the case and that as soon as you don't have cycles in the $2-TBN$, you can assume there will be no cycle also in an unfolded $2-TBN$, over $N$ time slices.
edit 18/03/2019: on the BayesServer tool, which is one of the few that allows the use of DBNs, it seems that the above example is not considered as a cycle. However, there is no source or reference to this. I believe it is true if $A^{T+1} \neq A^{T}$ (like the CPTs are different or the outcomes of both nodes are different). But what if $A^{T} = A^{T + 1}$?
I have found the answer to my question: Dynamic Bayesian Network could be represented under the form of a cyclic graph (compact representation of a DBN), but cycles are allowed only for the edges that are temporal. Edges that are not temporal should not create a cycle.
Once you unroll this compact representation, the DBN becomes acyclic.