What is the difference between markov chain models and hidden markov model? I've read in Wikipedia, but couldn't understand the differences.
The difference between markov chains and hidden markov model
hidden-markov-modelsmarkovmarkov-chains
Related Solutions
The Forward-Backward algorithm combines the forward step and the backward step to get the probability of being at each state at a specific time. Doing this for all time steps can therefore give us a sequence of individually most likely states at each time (although not guaranteed to be valid sequence, since it considers individual state at each step, and it can happen that the probability p(q_i -> q_j)=0 in the transition model), in other words:
, where 
On the other hand, Viterbi algorithm finds the most likely state sequence given an observation sequence, by maximizing a different optimality criterion:
I suggest you refer to this well-known paper for a detailed explanation (see Problem #2):
LAWRENCE R. RABINER, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition
Markov chains can be represented by finite state machines. The idea is that a Markov chain describes a process in which the transition to a state at time t+1 depends only on the state at time t. The main thing to keep in mind is that the transitions in a Markov chain are probabilistic rather than deterministic, which means that you can't always say with perfect certainty what will happen at time t+1.
The Wikipedia articles on Finite-state machines has a subsection on Finite Markov-chain processes, I'd recommend reading that for more information. Also, the Wikipedia article on Markov chains has a brief sentence describing the use of finite state machines in representing a Markov chain. That states:
A finite state machine can be used as a representation of a Markov chain. Assuming a sequence of independent and identically distributed input signals (for example, symbols from a binary alphabet chosen by coin tosses), if the machine is in state y at time n, then the probability that it moves to state x at time n + 1 depends only on the current state.
Best Answer
To explain by example, I'll use an example from natural language processing. Imagine you want to know the probability of this sentence:
I enjoy coffee
In a Markov model, you could estimate its probability by calculating:
Now, imagine we wanted to know the parts-of-speech tags of this sentence, that is, if a word is a past tense verb, a noun, etc.
We did not observe any parts-of-speech tags in that sentence, but we assume they are there. Thus, we calculate what's the probability of the parts-of-speech tag sequence. In our case, the actual sequence is:
(where PRP=“Personal Pronoun”, VBP=“Verb, non-3rd person singular present”, NN=“Noun, singular or mass”. See https://cs.nyu.edu/grishman/jet/guide/PennPOS.html for complete notation of Penn POS tagging)
But wait! This is a sequence that we can apply a Markov model to. But we call it hidden, since the parts-of-speech sequence is never directly observed. Of course in practice, we will calculate many such sequences and we'd like to find the hidden sequence that best explains our observation (e.g. we are more likely to see words such as 'the', 'this', generated from the determiner (DET) tag)
The best explanation I have ever encountered is in a paper from 1989 by Lawrence R. Rabiner: http://www.cs.ubc.ca/~murphyk/Bayes/rabiner.pdf