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Markov Chains
A Markov chain models movement among states when the probability of the next state depends on the current state, given the model, rather than the full earlier path.
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Résumé
A transition matrix records those probabilities and can be used to calculate multi-step behavior. The model is useful only when its chosen states and transition assumptions fit the real process.
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The states of a Markov chain are the categories the model tracks at each step. The Markov property says that, conditional on the present state, the next-state distribution does not additionally depend on the earlier sequence of states. It is an assumption about the chosen state representation, not a claim that real life has no history. A state that omits important context, such as how long a machine has been failing, may not make the next transition adequately predictable. For a simple time-homogeneous two-state weather example, let the states be sunny and rainy. From sunny, suppose tomorrow is sunny with probability 0.8 and rainy with probability 0.2. From rainy, suppose tomorrow is sunny with probability 0.4 and rainy with probability 0.6. Put these in rows of a transition matrix, ordered sunny then rainy: the first row is 0.8, 0.2 and the second is 0.4, 0.6. Each row sums to one because the next day must be in one of the defined states. These numbers are invented for illustration, not a weather forecast. Starting from sunny, the chance of rain two days later is 0.8 × 0.2 plus 0.2 × 0.6, or 0.28. One path goes through sunny and the other through rainy. Matrix multiplication performs this path accounting for every state pair; the square of the one-step transition matrix gives two-step probabilities. A stationary distribution is a mixture of states unchanged by another transition. For this illustrative matrix, two-thirds sunny and one-third rainy is stationary: the next sunny share is (2/3 × 0.8) + (1/3 × 0.4) = 2/3. That is a long-run mathematical property of the model, not a promise that any particular day is sunny. Some chains have multiple stationary distributions or do not converge from every starting state, so do not assume every chain forgets its start. Evaluate the transition estimates on relevant data and revisit them when conditions change.
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The Future of Markov Chains
Markov models remain useful because their assumptions and calculations are inspectable. They support teaching, reliability analysis and some sequential simulations, while richer models can add hidden states, varying transition rates or more context. In text generation, a next-token rule based on only a short state can demonstrate sequence probabilities but cannot capture all long-range dependencies in language. Modern AI systems may use very different architectures even when they also predict sequences. Future applications should document state definitions, check whether transition patterns drift and compare the model with alternatives on held-out sequences. A convenient matrix is not evidence that the process is truly memoryless.
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A weather exercise uses sunny and rainy states to calculate the chance of rain tomorrow and two days from now.
A support team models movement among ticket states while checking whether customer history must be included in the state definition.
A reliability analyst estimates equipment transitions between working and broken states using observed operating periods.
A teacher contrasts a one-token text chain with a language model that can use much longer context.
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Document where Markov Chains helps and where simpler methods are better.
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What is Markov Chains?
A Markov chain models movement among states when the probability of the next state depends on the current state, given the model, rather than the full earlier path. A transition matrix records those probabilities and can be used to calculate multi-step behavior. The model is useful only when its chosen states and transition assumptions fit the real process.
In this guide, what does the Markov property say about predicting the next state?
The property is conditional on the chosen current state; it does not claim deterministic transitions or that real processes literally lack history.
Why must each row of the guide's transition matrix sum to one?
From one current state, the probabilities of all defined possible next states exhaust the outcomes and sum to one.
If today is sunny in the guide's illustrative matrix, what is the probability of rain tomorrow?
The sunny row is [0.8 sunny, 0.2 rainy], so the one-step sunny-to-rainy probability is 0.2.
Starting sunny, what is the guide's illustrative probability of rain two days later?
The two possible intermediate paths contribute 0.8 × 0.2 and 0.2 × 0.6, which sum to 0.28.
What makes a state distribution stationary for a transition matrix?
A stationary distribution satisfies πP = π; one step leaves the distribution the same.
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