Free lesson · Time series

Hidden Markov models: predict, observe, update

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A hidden state is a model category you cannot directly read from the data. An emission is the observation distribution conditional on that state. For a first example, use the observed category “large move” instead of a complicated multivariate Gaussian density. Neither the category nor the state should be called profitable by definition.

Perform two operations in order. First use the transition model to predict the state probabilities before seeing today’s observation. Then weight each predicted state probability by the likelihood of the observed evidence under that state, and normalize the weights to sum to one. This is a Markov prediction followed by Bayes’ rule.

Filtering estimates the current state using observations available through now. Smoothing revises past state probabilities using later observations. Viterbi finds a most likely whole path under a fitted model, which differs from choosing the most probable state separately at every date. Full-sample smoothing or decoding is useful for retrospective interpretation but cannot be a live feature for an earlier decision.

Learning an HMM commonly alternates estimates of hidden-state responsibility with parameter updates, called expectation maximization or Baum–Welch. Keep that fitting inside each training window. During refits the numeric state labels may swap: state 0 today need not describe state 0 next month. Align economic meaning using only prior information and measure the turnover induced by refits.

Apply it in a strategy

• Start with the two-state discrete example and verify every probability.
• Fit transition and emission models on past data, then filter sequentially.
• Test probability stability, label alignment, turnover and net outcomes against a simple regime baseline.

Research deliverable

Produce a table with prior, likelihood, unnormalized weight and posterior for each state at each step.

Sources & evidence · reviewed 12 September 2026

Reviewed 12 September 2026: the supplied URL served the draft dated 19 August 2026. Markov chains, emissions, forward inference, decoding and parameter-learning sections reviewed. This is a textbook treatment in language processing, not a finance backtest. Our trading analogy uses causal filtering; the simple discrete filter is not a full Gaussian-HMM fitter.

Further reading: Jurafsky & Martin · Speech and Language Processing, Appendix A: Hidden Markov Models ↗

The allocation lesson records the supplied 2026 regime-aware investing preprint and explains its research limitations.

Further reading: Regime-aware investing: implementation checkpoint ↗

Research sources, review dates and limitations

Extend the research question

Annotate observation, release, revision and decision timestamps. A lagged series can still contain hindsight information if its values were revised later.

Continue with the connected research module →

Connect the ideas: Information and decision time

Retrieve: Use only information available when the decision is made.

Check the change: Observation dates, release delays, revisions and label maturity require different availability checks.

Statistics → Research & backtests → Point-in-time data → Research & robust tuning → Financial machine learning → Putting it all together

Connect the ideas: Dependence and diversification

Retrieve: Joint behavior matters when combining uncertain outcomes.

Check the change: Correlation, cointegration, covariance and event dependence answer different questions.

Statistics → Linear algebra → Portfolio construction → Portfolio management theory → Prediction foundations

from math import isfinite

def hmm_update(previous,transition,likelihood):
    n=len(previous)
    if not n or len(likelihood)!=n or len(transition)!=n or any(len(row)!=n for row in transition): raise ValueError("State dimensions must match")
    for row in [previous]+list(transition):
        if not all(isfinite(x) and 0<=x<=1 for x in row) or abs(sum(row)-1)>1e-9: raise ValueError("Invalid probability row")
    if not all(isfinite(x) and x>=0 for x in likelihood): raise ValueError("Nonnegative finite likelihoods required")
    prior=[sum(previous[i]*transition[i][j] for i in range(n)) for j in range(n)]
    # Rescale likelihoods first; relative weights and posterior are unchanged.
    scale=max(likelihood)
    if scale==0: raise ValueError("Impossible evidence")
    weights=[p*(b/scale) for p,b in zip(prior,likelihood)]
    total=sum(weights)
    if total==0: raise ValueError("Evidence has zero probability under the prior")
    return prior,[x/total for x in weights]

print(hmm_update([.8,.2],[[.9,.1],[.3,.7]],[.1,.6]))