Hidden Markov Models: Common Shortcomings | Machine Learning and Real-World Data

Overview

Identifying HMM limitations is worth 2–4 marks every year. You must name each shortcoming, explain why it matters, and tie it to a concrete failure example from the specific application in the question. Generic criticisms without specific examples earn partial marks at best.

1. Stationarity Violated

The problem: transition and emission probabilities are assumed constant over time. In real systems, these probabilities shift with seasons, policy changes, economic cycles, technological progress, or evolutionary processes.

Weather example: $P(NF \to F)$ is far higher in October–November (autumn, heading into winter) than in March–April (spring, heading into summer). A year-round average fails to predict either season: it overestimates the chance of freezing in summer and underestimates it in winter. The model is wrong in both seasons.

COVID example: new variants (Alpha, Delta, Omicron) changed transition dynamics (how quickly infection levels rose and fell) and emission probabilities (the relationship between true infection level and test positivity shifted as testing policy evolved). A model fitted to the Alpha wave fails on the Omicron wave.

2. First-Order Markov Assumption Violated

The problem: $P(q_t \mid q_{t-1})$ only looks back one step. Real trends persist over longer windows: weather fronts lasting a week, infection waves building over a month, economic recessions spanning quarters.

Example: a weather model using only yesterday’s temperature cannot distinguish “the first cold day of an autumn snap” from “the tenth consecutive freezing day in a deep January freeze.” Both have the same $P(F \mid F)$ transition. But the probability of continuing freezing is much higher after ten consecutive freezing days than after one. The model has no way to express this because it discards all history beyond $t-1$ .

3. Small Training Set

The problem: with only 7 data points (as in the COVID HMM), parameter estimates are unreliable. Many transitions and emissions have zero or single-digit counts, giving extreme probability estimates (0 or 1) that are artefacts of the small sample, not of the true underlying process.

Example: from the COVID training data, $a_{L \to M} = 1$ because we observed L exactly once and it transitioned to M. The model says it is impossible for L to stay L or transition elsewhere — but with more data, we would almost certainly observe some $L \to L$ and possibly $L \to H$ transitions. A single observation cannot produce a reliable probability.

Smoothing mitigates this (add-one makes $a_{L \to L} = 1/4$ instead of 0) but cannot fully compensate for a fundamentally sparse dataset. With more data, the smoothed estimates would converge toward the true probabilities; with only 7 points, the smoothed values are heavily influenced by the prior.

4. Discrete Hidden States Too Coarse

The problem: forcing a continuous reality into a small set of discrete bins loses information. Temperature is continuous; compressing it into $\{F, NF\}$ discards all fine-grained distinctions.

Example: 0.1°C and 40°C are both “NotFreezing.” The model treats them identically, yet a day at 0.1°C is vastly more likely to precede freezing tomorrow than a day at 40°C. The transition $P(NF \to F)$ is not a single number — it depends on how much above zero the temperature is. The discrete model cannot express this because both values map to the same state NF.

5. Output Independence Violated

The problem: $P(o_t \mid q_t)$ assumes observations depend only on the current hidden state. Consecutive observations are often correlated beyond what the hidden state explains.

Example: if it rained today, rain is more likely tomorrow — regardless of whether the hidden air temperature changed. This is because weather systems (fronts, pressure systems) have spatial and temporal coherence that the hidden state (a single temperature category) does not fully capture. Rain events cluster in time; the HMM’s output independence assumption flattens this clustering.

6. Supervised Training Requires Dual-Tape Data

The problem: to estimate $A$ and $B$ , the training data must have both the hidden state and the observation known at every timestep. This is called dual-tape data. Obtaining it is expensive and often impossible.

Example: we can trivially observe daily weather, but labelling the “true” atmospheric state (a meteorologically meaningful category) requires expert analysis. We can observe stock prices continuously, but labelling the “true” market regime (bull, bear, sideways) is subjective. For part-of-speech tagging, human annotators must label thousands of sentences — very time-consuming and expensive. The requirement for fully labelled sequences limits HMM applicability.

Summary Table

Shortcoming	Why It Matters	Concrete Failure
Stationarity	Transitions change with seasons/policy/variants	$P(NF \to F)$ differs in Oct vs Apr
First-order Markov	Real trends persist over longer windows	Cannot distinguish first vs tenth consecutive freezing day
Small training set	Unreliable parameters (zero counts from sparse data)	$a_{L \to M} = 1$ from a single observation
Coarse discrete states	Continuous reality forced into too-few bins	0.1°C and 40°C both “NF” — same transition probability
Output independence	Consecutive observations correlated beyond state	Rain clusters in time; rain today ⇒ rain tomorrow
Supervised training	Requires expensive dual-tape labelled data	Expert annotation for market regimes or POS tags

Past paper questions: y2022p3q9, y2024p3q8