What is the Hidden Markov Model 🌱

The Hidden Markov Model (HMM) is a time series model that assumes the ground truth is made up of observed variables that we see $o_1,\dots ,o_t$ and hidden variables $s_1,\dots ,s_t$ that we never see but determine the values we observe. There are two main assumptions with the HMM:

1. $P(s_t\mid s_{t-1},s_{t-2},\dots ,s_1)=P(s_t\mid s_{t-1})$
2. $P(o_t\mid o_{t-1},o_{t-2},\dots ,o_1,s_t,s_{t-1},\dots ,s_1)=P(s_t\mid s_{t-1})=P(o_t\mid s_t)$

• The first assumption means that the next hidden state is only dependent on the directly previous hidden state and none of the other past hidden states.
• The second assumption means that the observation at a certain time point is only dependent on the hidden state at that time point.

Here is a visualization of what the HMM looks like where $x_t$ are the hidden states and $y_t$ are the observed values.

There are 3 main problems we can solve with the HMM model, where $A$ is a $n\times n$ matrix of transition probabilities like $P(s_t\mid s_{t-1})$, $B$ is a $n\times v$ matrix of emission probabilities like $P(o_t\mid s_t)$, $O$ is the sequence of observed values, and $S$ is the sequence of states. This model assumes that transition and emission probabilities are constant over time (but this can be generalized by having $A$ as a $T\times n\times n$ tensor).

1. Compute $P(O\mid A,B)$
2. Compute $S^{\ast }={\text{argmax}}_{S}P(S,O\mid A,B)$
3. Compute $A^,B^ = \text{argmax}_{A,B} P(O \mid A,B)$