Definition

Given $N+1$ RVs $X_{0},\ldots,X_{N}$ from $\mbox{Markov}(\lambda,P)$ the log-likelihood $L$ is defined as $$\log(L)=\log(\lambda_{X_{0}})+\sum\limits_{i,j\in S}\left(\sum\limits_{k=0}^{N-1}\mathbb{1}_{\{X_{k}=i,X_{k+1}=j\}} \right)\log(p_{ij})$$