Definition

Given $N+1$ RVs $X_{0},\ldots,X_{N}$ from $\mbox{Markov}(\lambda,P)$ the likelihood $L$ is defined as $$L=P(X_{0}=x_{0},\ldots,X_{N}=x_{N})=\lambda_{x_{0}}p_{x_{0},x_{1}}\cdots p_{x_{N-1},x_{N}}$$