Does a trigram use three previous words?

No. It uses the previous two words to predict the next, making a three-word grouping.

Can a Markov state contain a history summary?

Yes. The assumption concerns sufficiency of the chosen state, not its size.

Is a Markov model always stationary?

No. The Markov property, time-homogeneous transitions and a stationary distribution are distinct concepts.

AI Glossary / Natural language and speech / Intermediate

Markov assumption

Definition

The Markov assumption says that the chosen present state contains the information from the past needed to predict the next state. In a first-order model, the next state is conditionally independent of earlier states given the current one. In language modeling, an n-gram approximation retains only a fixed recent word history.

What it means

The important phrase is “given the current state.” The assumption does not say that consecutive events are independent. A transition can depend strongly on the present, while earlier history provides no additional predictive information once that present state is known.

Whether this is reasonable depends on what counts as a state. A machine's operating label alone may omit relevant information such as elapsed running time. An expanded state that includes that information may support a better approximation.

In a bigram language model, the preceding word supplies the retained context for predicting the next word. This simplifies estimation, but real language can depend on earlier topics, names and grammatical relationships. The assumption is a modeling choice, not a claim that language or human memory has no longer history.

How it works

A first-order Markov model uses P(Xt+1 | Xt) in place of conditioning on the complete state history. For histories with nonzero probability, the formal property equates those conditional distributions. Higher-order versions retain several recent states.

An n-gram language model predicts a word from the preceding n − 1 words. A trigram therefore uses two previous words. Grouping those retained words into a larger state can express the same bounded-history idea as a first-order transition over compound states.

A separate assumption, time homogeneity, says that the transition rule does not change with time. It is often used with Markov models but is not the same property. A Markov model can have transitions that vary with the time step.

The practical question is whether discarded history still helps prediction. Evaluation on held-out sequences can reveal limitations, while a better state representation or longer context may address them at additional complexity or data cost.

Three practical examples

A simple autocomplete model estimates the next word using only the immediately preceding word as a bigram baseline.
A workflow simulation predicts the next status from the current status, then checks whether time already spent there adds useful information.
A speech-labeling model uses a specified state-transition structure while keeping its assumptions about observations distinct from state dynamics.

Think of it this way

Think of receiving a handover note that is intended to summarize everything needed for the next decision. If the note is sufficient, rereading the full history adds no further predictive information.

The analogy depends on the note's contents. An incomplete summary may omit a detail that changes the next step. Likewise, a Markov assumption can be reasonable for one state definition and poor for another; it is not a universal license to discard records.

Worked example

Predicting two steps with a stated model

Consider an invented two-state process, Ready and Busy. Assume time-homogeneous first-order transitions: from Ready, the next state is Ready with probability 0.7 and Busy with probability 0.3. From Busy, the probabilities are 0.4 for Ready and 0.6 for Busy.

Starting in Ready, the probability of being Busy two steps later sums two paths: Ready → Ready → Busy has probability 0.7 × 0.3 = 0.21, and Ready → Busy → Busy has probability 0.3 × 0.6 = 0.18. The total is 0.39.

The calculation uses the model's specified transitions. If the chance of leaving Busy actually depends on how long the process has already been Busy, the two-state representation may violate the intended assumption. Adding duration to the state or choosing a different model would change the analysis. The numerical result is therefore conditional on the teaching model, not a measured operational forecast.

Illustrative case study

A fictional support team models ticket movement using only status labels. Tickets in Waiting receive the same next-step probabilities regardless of whether they have waited for one hour or two weeks.

Reviewers suspect that waiting duration and the type of missing information affect what happens next. They compare predictions with and without those fields on a separate evaluation period, ensuring that only information available at prediction time is used.

The lesson is that a compact state representation should be justified by predictive performance and task needs. A status diagram can be easy to explain while still omitting relevant history. The fictional team does not claim that a richer model necessarily wins before evaluation.

Common misconceptions and mistakes

Markov means every event is independent. The assumption is conditional independence given the retained state.
First-order means the world remembers only one raw observation. The chosen state can itself summarize many observations.
Markov automatically means unchanging transition probabilities. Time homogeneity is a separate condition.
A convenient assumption proves the data satisfy it. Model suitability requires evidence and attention to state definition.

Business use

The Markov assumption can make sequence models tractable and explainable. It is useful as a baseline and as a framework for asking what information a system needs to retain.

For business applications, examine cases where duration, previous interventions or long-range context may matter. Avoid simplifying away information that the decision depends on merely because a small transition table is easy to present. Document the state variables and prediction interval, then evaluate the model under the conditions where it will operate. Changes in process policy can affect transitions even if the state names stay the same.

Check your understanding

A model predicts the next ticket status from current status alone, but elapsed waiting time still improves prediction. What should be questioned?

Answer: Whether current status is a sufficient state for the assumed Markov approximation. Relevant history may need to be represented or modeled differently.

Related terms

Learn next

Define the state and prediction interval for a small sequence problem. Identify a piece of discarded history that might matter, then compare predictions with and without it using information available at the time of prediction.

Sources

Jurafsky and Martin, Speech and Language Processing, Chapter 3: https://web.stanford.edu/~jurafsky/slp3/3.pdf. N-gram Markov approximations. Marco Taboga, Markov chain: https://www.statlect.com/fundamentals-of-statistics/Markov-chains. Conditional independence, time homogeneity and stationary distributions. The two-state transition example is original and arithmetically checked.

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