The Markov Assumption: Memory Limits in Language Models

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When you talk to someone, you don’t need to remember everything they’ve ever said to follow the conversation—you just need to pay attention to what they said recently. The same idea holds in computational language modeling, where a foundational principle known as the Markov assumption says: “The next word depends only on a limited number of previous words.”

In this post, we’ll explore what the Markov assumption is, why it's useful in building language models, and how it helps simplify an otherwise overwhelming task.

1. The Challenge: Predicting What Comes Next

Language is full of possibilities. If you’re given a sentence beginning with:

“The cat sat on the…”

You might expect:

• “mat”

• “couch”

• “sofa”

But what if the sentence was longer?

“The little black cat from the neighbor’s house sat quietly and stretched its legs on the…”

Now predicting the next word becomes harder. There are more dependencies, more context, more ambiguity. Ideally, you would want to consider the entire sentence before making a prediction.

But there’s a problem: computational complexity.

2. Enter the Markov Assumption

To make prediction feasible, we apply the Markov assumption, which simplifies the problem by limiting the model’s memory to a fixed window of prior words.

Formally:

An n-gram model assumes that the probability of a word depends only on the previous n−1 words.

This is what the Markov assumption looks like:

Instead of calculating:

P(w_n | w_1, w_2, ..., w_{n-1})
(the probability of the next word given the entire history)

We simplify it to:

P(w_n | w_{n−1}) (for bigrams)
P(w_n | w_{n−2}, w_{n−1}) (for trigrams)

This makes the model tractable and trainable using real-world data.

3. A Real Example: From Full Context to Simplified Context

Suppose we have this sentence:

“The water of Walden Pond is so beautifully…”

Let’s say we want to predict the next word.

Full-context (ideal):

P(blue | The water of Walden Pond is so beautifully)

But computing this directly is impossible unless we’ve seen this exact phrase many times in a huge corpus (which is unlikely).

Bigram approximation (Markov assumption):

P(blue | beautifully)

Now we only need to know how often “beautifully blue” appears. That’s much more manageable.

4. Visualizing the Difference

Chain Rule (Full History)

P(w1 w2 w3 w4 w5) = 
P(w1) * P(w2|w1) * P(w3|w1 w2) * P(w4|w1 w2 w3) * ...

This gets very complicated fast.

Bigram Model (Markov, order-1)

P(w1 w2 w3 w4 w5) ≈ 
P(w1) * P(w2|w1) * P(w3|w2) * P(w4|w3) * P(w5|w4)

Here’s a visual comparing the two:

Full History Dependency (Exponential Growth)
   w1 --> w2 --> w3 --> w4 --> w5
   ↑      ↑      ↑      ↑
 (all before) (all before) ...

Bigram (Markov Assumption)
   w1 → w2 → w3 → w4 → w5
     (1 step each)

This drastically reduces the number of conditional probabilities we need to estimate.

5. Trigrams: A Slightly Longer Memory

If bigrams are too limited, we can go one step further with trigrams.

Here we assume:

P(w_n | w_{n−2}, w_{n−1})

So instead of just looking at the last word, we look at the last two words.

Example:

“He drank a cup of hot ___”

Bigram: P(tea | hot)
Trigram: P(tea | of hot) — more context, better prediction

Trigram models often perform better than bigrams (lower perplexity) because they remember more context, while still keeping computation manageable.

6. Why the Markov Assumption Matters

Without this assumption, language modeling would require:

• Storing and computing probabilities for billions of word sequences

• Massive corpora to estimate even moderately long contexts

• Handling extreme sparsity (most word combinations are rare or unseen)

With the Markov assumption, we:

• Reduce computational load

• Simplify probability estimation

• Make language modeling possible with limited data

7. But What About Long-Range Dependencies?

Yes, n-grams ignore long-term structure. This is a key limitation:

• Bigrams can’t track subject-verb agreement.

• Trigrams can’t detect sarcasm or metaphor.

• 5-grams might memorize but not generalize.

That’s why modern models like transformers (e.g., GPT, BERT) have moved beyond fixed-length memory windows. Still, n-grams remain important for understanding the foundations of probabilistic language modeling.

Conclusion

The Markov assumption is a clever simplification that makes language modeling computationally possible. By assuming that only the last few words matter, we trade depth for speed and practicality.

Though newer models now capture richer context, n-grams built on the Markov assumption continue to be used in fast, interpretable, and resource-efficient systems — and are a must-know concept for anyone entering NLP.

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