Markov Process

/ˈmɑːr.kɒv ˈprəʊ.ses/

noun … “the future depends only on the present, not the past.”

Markov Process is a stochastic process in which the probability of transitioning to a future state depends solely on the current state, independent of the sequence of past states. This “memoryless” property, known as the Markov property, makes Markov Processes a fundamental tool for modeling sequential phenomena in probability, statistics, and machine learning, including Hidden Markov Models, reinforcement learning, and time-series analysis.

Formally, for a sequence of random variables {Xₜ}, the Markov property states:

P(Xₜ₊₁ | Xₜ, Xₜ₋₁, ..., X₀) = P(Xₜ₊₁ | Xₜ)

Markov Processes can be discrete or continuous in time and space. Discrete-time Markov Chains model transitions between a finite or countable set of states, often represented by a transition matrix P with elements Pᵢⱼ = P(Xₜ₊₁ = j | Xₜ = i). Continuous-state Markov Processes, such as the Wiener process, extend this framework to real-valued variables evolving continuously over time.

Markov Processes are intertwined with multiple statistical and machine learning concepts. They rely on Probability Distributions for state transitions, Expectation Values for long-term behavior, Variance to measure uncertainty, and sometimes Stochastic Processes as a general framework. They underpin Hidden Markov Models for sequence modeling, reinforcement learning policies, and time-dependent probabilistic forecasting.

Example conceptual workflow for a discrete-time Markov Process:

define the set of possible states
construct transition matrix P with probabilities for moving between states
choose initial state distribution
simulate state evolution over time using P
analyze stationary distribution, expected values, or long-term behavior

Intuitively, a Markov Process is like walking through a maze where your next step depends only on where you are now, not how you got there. Each move is probabilistic, yet the structure of the maze and the transition rules guide the overall journey, allowing analysts to predict patterns, equilibrium behavior, and future states efficiently.