Suppose I have some data on transitions between states of a Discrete Time Markov Chain. Let's say that transitions between some events are observed more frequently from others. For example, in a 3 state Markov Chain with states A, B, C ... suppose we observe lots of activity between (A,B) and (B, C) - but we observe little activity between (A,C), i.e. P_a,c and P_c,a.
Intuitively, for states that observe fewer transitions, this would result in their transition probabilities (to and from) being closer to 0 or closer to 1.
I am wondering what can be done to correct for these biases. For example, suppose we are confident that neither state A or C are absorbing states, is there something we can do to correct for these extreme estimations?
The idea that comes to mind is Bayesian Estimation.
For example, in a regular Markov Chain, we estimate transition probabilities using Maximum Likelihood based on the Multinomial Distribution.
In the example I described, could we use a Bayesian approach with a Dirichlet Prior and choose parameters of the Dirichlet Distribution that discourage very large and very small probability estimates?
Has anyone ever dealt with this kind of problem before?
