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Hidden Markov Models

A Markov chain assumes you can see the state. Often you cannot: you see something the state produces.

Markets are the standard example. There is arguably a "regime" (calm, stressed, trending) that is never announced. What you observe is returns and volumes, which the regime generates.

The pieces

Hidden states {S1,…,Sn}\{S_1,\ldots,S_n\}, unobserved.

Transition matrix Aij=P(St+1=j∣St=i)A_{ij} = P(S_{t+1}=j \mid S_t=i), how the regime evolves.

Emission matrix Bik=P(Ot=k∣St=i)B_{ik} = P(O_t = k \mid S_t = i), what each state tends to produce.

Initial distribution π\pi.

A two-state market model might have a calm state emitting small returns and a volatile state emitting large ones, with the transition matrix encoding that regimes persist.

The three questions

Filtering: where am I now? Given observations so far, what is the probability distribution over current states? Solved by the forward algorithm, and it is the question a live trading system asks.

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