Definition. Transition-observation kernel [ftip-001Z]
Definition. Transition-observation kernel [ftip-001Z]
A transition-observation kernel at time \(t\) is a probability kernel \[ K_t(ds_{t+1},do_{t+1}\mid s_t,a_t) \] from the current hidden state and action to the next hidden state and public observation. After \(A_t=a_t\), the environment draws \((S_{t+1},O_{t+1})\) from this kernel.
The factorization \(K_t(ds',do'\mid s,a)=P_t(ds'\mid s,a)\,Q_t(do'\mid s',a)\) for declared kernels \(P_t\) and \(Q_t\) recovers the usual separate state-transition and observation kernels when such a factorization is declared. The joint-kernel form does not assume it.