Markov Chains: The Original Language Models

Markov chains represent the foundational approach to language modeling that predates modern LLMs by decades. Understanding their mechanics reveals both the evolution of text generation and the mathematical principles underlying today’s transformer architectures.

What Makes Markov Chains Work

A Markov chain generates text by predicting the next word based solely on the current state—typically the previous word or sequence of words. The key insight is the Markov property: future states depend only on the present state, not the entire history.

Here’s how temperature control works in a simple Markov implementation:

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from math import log, exp
from random import choices

def next_word(current_word, transitions, temp=1.0):
    if current_word not in transitions:
        return random.choice(list(transitions.keys()))
    
    probabilities = transitions[current_word]
    next_words = list(probabilities.keys())
    pvals = list(probabilities.values())
    
    # Apply temperature scaling
    logits = [log(p) for p in pvals]
    scaled_logits = [logit/temp for logit in logits]
    max_logit = max(scaled_logits)
    exps = [exp(s - max_logit) for s in scaled_logits]
    sum_exps = sum(exps)
    softmax_probs = [exp_val/sum_exps for exp_val in exps]
    
    return choices(next_words, weights=softmax_probs)[0]

Temperature demonstrates the same creativity control found in modern LLMs. Low temperatures (0.1) produce repetitive, predictable text. Higher temperatures (1.0+) increase randomness and creativity.

The Fundamental Limitation

Markov chains excel at local patterns but struggle with long-range dependencies. Consider a 2D bitmap with vertical patterns—a left-to-right Markov chain misses these completely because it processes data linearly.

This limitation stems from exponential state explosion. To capture patterns separated by random data, you need states for every possible intermediate sequence. A pattern with 32 random bits between meaningful elements requires 2^32 states—computationally intractable.

The Bridge to Modern LLMs

Modern transformers are mathematically equivalent to very high-order Markov chains where the entire context window forms the “state.” The breakthrough wasn’t abandoning Markov chains but finding efficient ways to represent massive state spaces.

Attention mechanisms solve the exponential blowup problem by learning which parts of the context matter for each prediction. Instead of explicitly storing every possible state, transformers parameterize these relationships through learned weights.

Practical Applications Today

Despite their limitations, Markov chains remain valuable for:

Teaching language modeling concepts: They demonstrate core principles without complexity
Rapid prototyping: Simple to implement and understand
Specialized domains: Effective when context requirements are limited
Baseline comparisons: Useful benchmarks for more complex models

Implementation Considerations

When building Markov chains:

Use n-grams (2-5 words) instead of single words for better context
Apply smoothing techniques for unseen transitions
Consider character-level models for different applications
Implement proper normalization for probability distributions

The Evolution Continues

Understanding Markov chains provides crucial insight into why modern LLMs work. They’re not fundamentally different approaches—they’re sophisticated solutions to the same core problem of predicting sequential data while managing computational constraints.

The journey from simple word-to-word transitions to transformer attention illustrates how mathematical principles evolve through engineering innovation, creating more powerful tools while preserving foundational concepts.