Should I use Markov or Hidden Markov model when there are only 2 possible states? I am just learning Markov Models and HMM. I have a problem on which I am trying to implement them but am not sure which one is appropriate. Following is the problem.
The data is time-series. There are 2 states - 1 & 0. When at state 1, observations can take a range of values usually between say 1 to 10000. But when in state 0 the observation always takes the value of 0. There are other independent variables which are correlated to the observations i.e. they can be used to predict the observation values but they are not available in the data. 
I have data till today. I don't what the state or observations will be for the next week. I also have to predict the probability of the occurrence of the states/observation from the previous step.
I read the paper on Markov by Rabiner. While since the states in my case are not Hidden I initially thought it is direct Markov Chain Model but in the paper Rabiner uses the Coin-toss as an example for HMM even though the states in a coin-toss are not really hidden.
 A: The question can be generalized to how to select the right model to certain data. To some degree the question is equivalent to flowing question: if I have some 1 dimensional data, should I fit it with Gaussian or mixture of Gaussian. I have a related answer here. Markov chains vs. HMM Also for choosing the right model, we can do "knowledge driven" or "data driven". And a related discussion can be found here. How to chose the order for polynomial regression?
If we want to do "data driven", Using MM or HMM is depending the nature of the data. In general, HMM is more "complex" (check the bias variance trade off) than MM. Therefore, if we have sufficient data, and can make sure we are not over fitting, HMM is better.
Note, choosing between MM and HMM have little thing to do with Number of Observed Stages. We can have a super complicated binary string data, that requires a HMM with many hidden stages. On the other hand, we can also have a simple data with multiple possible outcomes, that using MM is sufficient.
