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A stochastic process where the next state depends only on the current state, not the full history.
DESCRIPTION
A Markov chain models systems that transition between states with memoryless dynamics. The transition probabilities define movement between states and form a stochastic matrix. Chains may be discrete or continuous-time and can have absorbing, transient, or recurrent states. Long-run behavior often converges to a stationary distribution under mild conditions. Markov chains underpin HMMs, PageRank, queueing models, and MCMC methods. Analyzing ergodicity, mixing times, and spectral properties informs convergence and sampling quality.
ALTERNATIVE DEFINITIONS
Wikipedia
A Markov chain is a stochastic process that satisfies the Markov property — future states depend only on the current state, not on the events that occurred before it.
Classic definition highlighting the memoryless (Markov) property.
Encyclopaedia Britannica
A Markov process is a stochastic process with the property that, given the present state, the future is independent of the past.
Generalized Markov process definition applicable to chains.
RELATED TERMS
Markov Property Transition Matrix Stationary Distribution Ergodicity Hidden Markov Model Random Walk Markov Decision Process Stochastic Process Mixing Time Monte Carlo Simulation
COMPETING TERMS
Non-Markovian Process
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist AI Engineer Data Analyst
EXTERNAL SOURCES
- Wikipedia - Markov chain
- Wolfram MathWorld - Formal notes on Markov chains
- Stanford - Tutorial introduction to Markov chains
TOP QUESTIONS
What is the Markov property in a Markov Chain?
It states that the next state depends only on the current state, not the full history.
Category: Concepts
How are Markov Chains used in machine learning?
They underpin HMMs, MCMC sampling, and algorithms like PageRank for ranking and inference.
Category: Example
What is a stationary distribution in a Markov Chain?
A distribution that remains unchanged after transitions; it describes long-run behavior under ergodic conditions.
Category: Concepts
What are common mistakes when modeling with Markov Chains?
Assuming the Markov property holds without validation and ignoring mixing time can yield misleading results.
Category: Mistakes
How do I estimate transition probabilities?
Use frequency counts from observed transitions or maximum likelihood, possibly with smoothing for sparsity.
Category: Best practices
What’s the risk of using too few states?
Over-aggregation hides dynamics and biases estimates; consider model selection and diagnostics.
Category: Risk
How do continuous-time Markov Chains differ?
They evolve via rates in a generator matrix instead of discrete transition steps.
Category: Concepts
Where can I learn about Markov Chains?
Start with Wikipedia and lecture notes, then texts on stochastic processes and MCMC.
Category: Getting Started
What is an ergodic Markov Chain?
A chain that is irreducible and aperiodic with a unique stationary distribution, ensuring convergence.
Category: Concepts