Let $\{X_i\}_{i=1}^{n}$ be the path of a Markov chain with 3 possible states $\{1,2,3\}$. Given the path, I know how to get the maximum likelihood estimator for the $3 \times 3$ transition probability matrix $P$.
However, I am missing some values of the chain. For example, I have $\{X_i\}_{i=1}^5$ and $\{X_i\}_{i=11}^{15}$ but am missing $\{X_i\}_{i=6}^{10}$. Given an initial probability distribution $p$, the likelihood function for this problem would look like this:
$$ \begin{align*} f_{\theta}(X_1, X_2, ..., X_5, X_{11}, ..., X_{15}) & = \sum_{X_i, i \in\{6,7,...,10\}} p_{X_1} \prod_{i=2}^{15} P_{X_{i-1}, X_i}\\ & = \sum_{X_6=1}^{3} \sum_{X_7=1}^{3} ... \sum_{X_{10}=1}^{3} \left[ p_{X_1} \prod_{i=2}^{15} P_{X_{i-1}, X_i} \right] \end{align*} $$
How should I go about solving for the maximum likelihood estimator for $P$ in this case?