I am currently working with Markov chains and calculated the Maximum Likelihood Estimate using transition probabilities as suggested by several sources (i.e., number of transitions from a to b divided by number of overall transitions from a to other nodes).
I now want to calculate the log-likelihood of the MLE.
Best Answer
Let $ \{ X_i \}_{i=1}^{T}$ be a path of the markov chain and let $P_{\theta}(X_1, ..., X_T)$ be the probability of observing the path when $\theta$ is the true parameter value (a.k.a. the likelihood function for $\theta$). Using the definition of conditional probability, we know
$$ P_{\theta}(X_1, ..., X_T) = P_{\theta}(X_T | X_{T-1}, ..., X_1) \cdot P_{\theta}(X_1, ..., X_{T-1})$$
Since this is a markov chain, we know that $P_{\theta}(X_T | X_{T-1}, ..., X_1) = P_{\theta}(X_T | X_{T-1} )$, so this simplifies this to
$$ P_{\theta}(X_1, ..., X_T) = P_{\theta}(X_T | X_{T-1}) \cdot P_{\theta}(X_1, ..., X_{T-1})$$
Now if you repeat this same logic $T$ times, you get
$$ P_{\theta}(X_1, ..., X_T) = \prod_{i=1}^{T} P_{\theta}(X_i | X_{i-1} ) $$
where $X_0$ is to be interpreted as the initial state of the process. The terms on the right hand side are just elements of the transition matrix. Since it was the log-likelihood you requested, the final answer is:
$$ {\bf L}(\theta) = \sum_{i=1}^{T} \log \Big( P_{\theta}(X_i | X_{i-1} ) \Big) $$
This is the likelihood of a single markov chain - if your data set includes several (independent) markov chains then the full likelihood will be a sum of terms of this form.