The Hidden Markov Model

Consider a sensor which tells you whether it is cloudy or clear, but is wrong with some probability. Now, the weather *is* cloudy or clear, we could go and see which it was, so there is a “true” state, but we only have noisy observations on which to attempt to infer it.  

We might model this process (with the assumption of sufficiently precious weather), and attempt to make inferences about the true state of the weather over time, the rate of change of the weather and how noisy our sensor is by using a Hidden Markov Model. 

The Hidden Markov Model describes a hidden Markov Chain which at each step emits an observation with a probability that depends on the current state. In general both the hidden state and the observations may be discrete or continuous.

But for simplicity’s sake let’s consider the case where both the hidden and observed spaces are discrete. Then, the Hidden Markov Model is parameterised by two matrices: 

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