Reconstruct Markov chain based on the sample paths Given a sample of small finite sequences of transitions between the states of a Markov chain with unknown probabilities how to find the probabilities of transitions between states? Would computing the naive probabilities be a right (best) method? By the naive method I mean to compute the fraction of transitions that went from a given state to one of the possible states? 
 A: Your intuition is correct, the empirical transition probabilities are in fact the Maximum Likelihood Estimator for the transition matrix.  Note that this is not the only right/best way, but it's probably the most convenient.  For example, you could cook up a prior over transition matrices and then use the likelihood given in the reference to calculate a posterior distribution over transition matrices; from there you would use the posterior mean as your "best estimate", rather than the MLE.
EDIT/ADDENDUM An example of the MLE approach is as follows: say we have a markov chain with states $S_i$.  Let $p_{ij}$ denote the probability of transition between $S_i$ and $S_j$.  We would like to attain the MLE estimate for $\hat{p}_{ij}$.  Following the reference above it can be show that this is just $\hat{p}_{ij} = \frac{n_{ij}}{\sum\limits_{j=1}^3 n_{ij}}$ where $n_{ij}$ is the total number of observed transitions from $S_i$ to $S_j$.  Note that the sum in the denominator is taken over the $j$ indices only.  This sum is calculates the total number of state transitions that have departed from $S_i$.
