Hidden Markov Models: The Three Assumptions | Machine Learning and Real-World Data

Exam Context

This sub-question appears every year. You must state all three assumptions explicitly, give their formal mathematical form, and evaluate each in the context of the specific application given in the question. Generic statements without evaluation lose marks. Each assumption typically earns 1 mark for stating it plus 1 mark for contextual evaluation.

The Three Assumptions

1. First-Order Markov Assumption

$P(q_t \mid q_1, q_2, \ldots, q_{t-1}) = P(q_t \mid q_{t-1})$

Only the immediately preceding state matters. The entire history before $t-1$ is irrelevant once $q_{t-1}$ is known. This means the transition matrix $A$ needs only $|S| \times |S|$ entries rather than growing exponentially with the sequence length.

Fails when: trends depend on longer history. Multi-day weather patterns (a cold front lasting a week), economic cycles (a recession spanning quarters), or infection waves where the path from low to high typically passes through medium over multiple timesteps. A first-order model cannot distinguish “the first cold day of an autumn cold snap” from “the tenth consecutive freezing day of a deep winter freeze” — both cases have the same $P(F \mid F)$ transition, but the probability of continuing freezing is very different.

2. Output Independence Assumption

$P(o_t \mid q_1, \ldots, q_t, o_1, \ldots, o_{t-1}) = P(o_t \mid q_t)$

The observation at time $t$ depends solely on the current hidden state $q_t$ . All previous states, future states, and all other observations are irrelevant once $q_t$ is known. This means $b_i(v_k)$ captures everything about $o_t$ .

Fails when: consecutive observations are correlated beyond what the hidden state explains. If it rained today, rain is more likely tomorrow — regardless of whether the hidden temperature state has changed. In speech recognition, neighbouring phonemes influence each other acoustically (coarticulation): the sound of /t/ differs depending on whether the next phoneme is /i/ or /u/. In part-of-speech tagging, an adjective is more likely to be followed by a noun regardless of any hidden syntactic state.

3. Stationarity Assumption

$P(q_t = s_j \mid q_{t-1} = s_i) \text{ and } P(o_t = v_k \mid q_t = s_i) \text{ are constant for all } t$

Transition probabilities $A$ and emission probabilities $B$ do not change over time. The same matrices apply whether $t = 1$ , $t = 100$ , or $t = 1000$ .

Fails when: behaviour is seasonal (summer vs. winter transitions differ), structural (policy changes alter relationships between states and observations), or evolutionary (language changes over decades, market regimes shift gradually). For the 2024 weather HMM: $P(NF \to F)$ in October (heading into winter) is far higher than $P(NF \to F)$ in March (heading into spring). The stationary model averages these into a single number that overestimates freezing probability in summer and underestimates it in winter.

Evaluating in Context: COVID HMM (2022 Q9)

For an infection-level HMM with hidden states $\{L, M, H\}$ :

First-order Markov: infection levels follow multi-week trends (a wave building over a month). The probability of staying at H may depend on how many consecutive H weeks have already elapsed — the model cannot capture this duration dependence. First-order is probably too short a memory.
Output independence: consecutive test positivity rates are likely correlated. A testing centre with high positivity one week tends to have high positivity the next, independent of whether the true infection level changed, due to consistent testing demographics and reporting lags (the same people retested, the same labs processing). This violates output independence.
Stationarity: COVID behaviour changed dramatically between variants (Alpha, Delta, Omicron). Each variant had different transmission dynamics, different incubation periods, and different relationships between infection prevalence and test positivity. A model estimated from early-2020 data would produce wrong predictions on late-2021 data because the underlying process had changed.

Summary

Assumption	Formal Statement	Typical Failure Mode
First-order Markov	$P(q_t \mid q_{t-1})$ only; history before $t-1$ irrelevant	Trends persist over multiple timesteps (multi-week fronts)
Output independence	$P(o_t \mid q_t)$ only; all other observations irrelevant	Consecutive observations are correlated (rain clustering)
Stationarity	$A, B$ invariant for all $t$	Seasons, policy shifts, variant emergence, regime changes

Past paper questions: y2022p3q9, y2024p3q8, y2025p3q9