Hidden states and emissions

The state is hidden, the move is seen

In a market you never see whether the day is Calm or Stormy. You only see the size of the move: say Small or Large. A hidden Markov model (HMM) has three parts. The initial probabilities $\pi$ say how likely each state is on day one. The transition matrix $A$ says how the hidden state moves. The emission probabilities $B_j(o)$ say how likely each observation $o$ is when the state is $j$. Here a calm day gives a small move with probability $0.9$ and a stormy day gives a large move with probability $0.7$, and $\pi=(0.6,\,0.4)$.

The formula multiplies the start, then for each later day the chance of the state move and the chance of the observed move. The catch is that we only see the $o$'s. A large move is more likely on a stormy day, but it can also happen on a calm one, so one observation is only evidence about the state, never proof. The rest of this skill is about turning a sequence of observations into beliefs about the hidden states: how likely is the data, which state is each day in, and what are the model's numbers.

• An HMM is three things: start probabilities $\pi$, transitions $A$ and emission probabilities $B$.
• Hidden versus seen: the states are never observed, only the emissions are.
• One observation is evidence, not proof, because every state can produce every outcome.

Lab · what's graded

• A path and its observations15 pts
• A small move on day one15 pts
• Was day one calm?15 pts
• A large move tomorrow15 pts
• Generating data from an HMM10 pts
• Match the parts of an HMM15 pts
• Why one observation is not proof15 pts