pr.probability - What is hidden in Hidden Markov Models?

The unobserved state.

Let's consider a hidden Markov model for my cat's behavior. Bella can be in five states: hungry, tired, playful, cuddly, bored. She can respond to these states with six behaviors: whining, scratching, cuddling, pouncing, sleeping and stalking.

A hidden Markov model would consist of two matrices, one 5x5 and the other 5x6. The 5x5 matrix gives the probabilities that, if she is hungry at time $t$, she will be tired at time $t+1$, and so forth. So we can compute the probability that she is in different emotional states by taking powers of this matrix.

However, we can't observe her emotions -- they are hidden. The 5x6 matrix gives the probability that, if she is hungry at time $t$, she will whine at time $t+1$. (Very close to $1$.) These are the behaviors we observe.

In an ordinary Markov model, there would just be a single 6x6 matrix, which directly described the probability of transitions like whining ---> clawing. As you can see, an ordinary Markov model is less able to reflect the complexity of my cat's inner life.

See the wikipedia article for much more information.