Hidden Markov Models

Markets seem to switch personalities. For weeks they drift calmly, small moves, low volatility, and then something flips and every day is a big, jumpy swing. A hidden Markov model (HMM) is the natural way to capture that. It says the world is always in one of a few hidden regimes, it hops between them at random according to fixed odds, and each regime spits out data with its own characteristic look. You never get told which regime you're in; you have to infer it from the observations, and that inference is the whole game.

It's the discrete-state member of the State-Space Models family. Where the The Kalman Filter tracks a continuous hidden number, an HMM tracks a hidden category, calm or stormy, bull or bear, trending or mean-reverting.

The three ingredients

An HMM is fully described by three things:

1. Transition probabilities, the odds of hopping between regimes from one step to the next. This is a Markov chain on the hidden states: where you go next depends only on where you are now.
2. Emission distributions, for each regime, the distribution of the data it produces. The calm regime might emit small, tight returns; the stormy regime, wide, fat-tailed ones.
3. Initial probabilities, your belief about which regime you start in.

Worked example: did the regime just flip?

Use the two regimes in the diagram. Suppose the persistence is high: a calm day stays calm with probability $0.9$ and turns stormy with $0.1$; a stormy day stays stormy with $0.8$. Emissions: a "big move" (say a daily return past $\pm 2\%$) happens with probability $0.1$ in the Calm regime but $0.5$ in the Storm regime.

Yesterday you were fairly sure it was Calm: $P(\text{Calm}) = 0.8$, $P(\text{Storm}) = 0.2$. Today a big move prints. What should you now believe? This is one step of the forward algorithm, just Bayes' rule with the emission as evidence:

$$P(\text{Storm}\mid \text{big move}) = \frac{P(\text{Storm})\,P(\text{big}\mid \text{Storm})}{P(\text{Calm})\,P(\text{big}\mid \text{Calm}) + P(\text{Storm})\,P(\text{big}\mid \text{Storm})}.$$

Plug in the numbers:

$$= \frac{0.2 \times 0.5}{0.8 \times 0.1 + 0.2 \times 0.5} = \frac{0.10}{0.08 + 0.10} = \frac{0.10}{0.18} \approx 0.56.$$

A single big move drags your belief from $20\%$ stormy to $56\%$ stormy, now more likely than not that the regime has flipped. The model quantifies exactly how much one surprising observation should shift your read on the hidden state, and the persistence in the transition matrix keeps it from over-reacting to every wiggle. Run this update forward day after day and you get a live, probabilistic regime signal.

The three questions HMMs answer

Practitioners lean on three standard algorithms, each answering a different question:

• "What regime am I in now?" The forward-backward algorithm gives the probability of each hidden state at each time, filtering and smoothing.
• "What's the single most likely sequence of regimes?" The Viterbi algorithm finds the best path through the hidden states.
• "What are the model's parameters?" The Baum-Welch algorithm (an EM procedure) learns the transition and emission probabilities from data when you don't know them up front.

Where it misleads

• Picking the number of regimes is an art. Two states? Three? Too few and you blur distinct regimes together; too many and you overfit, inventing regimes that are just noise. Use out-of-sample checks, not in-sample fit, to choose.
• Label switching and instability. The fitting can converge to different local optima, and "regime 1" versus "regime 2" is arbitrary, so parameters can be hard to compare across runs. Multiple random starts are essential.
• Regimes are detected with a lag. By construction, you only recognise a regime shift after the tell-tale observations arrive. An HMM will confirm a crash is underway, but it won't front-run it, don't mistake its smooth in-sample regime plot for real-time foresight.
• Markov and stationarity assumptions. Real transition odds drift, and the "memoryless" assumption (next regime depends only on the current one) is a simplification. Long, structured regimes can violate it.

The core intuition to keep: an HMM assumes a small cast of hidden "moods," each with its own fingerprint in the data, and it uses every new observation to update the odds of which mood you're in. That makes it the standard formal engine behind Regime Detection, and the discrete-state sibling of the The Kalman Filter within the broader State-Space Models world.