“A Markov model is a statistical tool for stochastic (random) processes where the future state depends only on the current state, not the entire past history-this is the Markov Property or “memoryless” property, making them useful for modeling systems like weather, finance, etc.” – Markov model
A Markov model is a statistical tool for stochastic (random) processes where the future state depends only on the current state, not the entire past history. This defining characteristic is known as the Markov property or “memoryless” property, rendering it highly effective for modelling systems such as weather patterns, financial markets, speech recognition, and chronic diseases in healthcare.1,2,4,5
Core Principles and Components
The simplest form is the Markov chain, which represents systems with fully observable states. It models transitions between states using a transition matrix, where rows denote current states and columns indicate next states, with each row’s probabilities summing to one. Graphically, states are circles connected by arrows labelled with transition probabilities.1,2,4
Formally, for a discrete-time Markov chain, the probability of transitioning from state i to j is given by the transition matrix P, where P_{ij} = Pr(X_{t+1}=j \mid X_t = i). The state at time t follows Pr(X_t = j) = \sum_i Pr(X_{t-1} = i) P_{ij}.4
Advanced variants include Markov decision processes (MDPs) for decision-making in stochastic environments, incorporating actions and rewards, and partially observable MDPs (POMDPs) where states are not fully visible. These extend to fields like AI, economics, and robotics.1,7
Applications Across Domains
- Finance: Predicting market crashes or stock price movements via transition probabilities from historical data.1,5
- Healthcare: Modelling disease progression for economic evaluations of interventions.6
- Machine Learning: Markov chain Monte Carlo (MCMC) for Bayesian inference and sampling complex distributions.3,4
- Other: Weather forecasting, search algorithms, fault-tolerant systems, and speech processing.1,4,8
Key Theorist: Andrey Andreyevich Markov
The preeminent theorist behind the Markov model is Russian mathematician Andrey Andreyevich Markov (1856-1922), who formalised these concepts in probability theory. Born in Ryazan, Russia, Markov studied at St. Petersburg University under Pafnuty Chebyshev, a pioneer in probability. He earned his doctorate in 1884 and became a professor there, though academic rivalries with colleagues like Dmitri Mendeleev led to his resignation in 1905.5
Markov’s seminal work began in 1906 with his analysis of Pushkin’s novel Eugene Onegin, applying chains to model letter sequences and refute Chebyshev’s independence assumptions in language-a direct precursor to modern Markov chains. He generalised this to stochastic processes satisfying the memoryless property, publishing key papers from 1906-1913. His contributions underpin applications in statistics, physics, and computing, earning the adjective “Markovian.” Markov’s rigorous mathematical framework proved invaluable for modelling real-world random systems, influencing fields from Monte Carlo simulations to AI.2,4,5
Despite personal hardships, including World War I and the Russian Revolution, Markov’s legacy endures through the foundational Markov chains that enable tractable predictions in otherwise intractable systems.2,4
References
1. https://www.techtarget.com/whatis/definition/Markov-model
2. https://en.wikipedia.org/wiki/Markov_model
3. https://www.publichealth.columbia.edu/research/population-health-methods/markov-chain-monte-carlo
4. https://en.wikipedia.org/wiki/Markov_chain
5. https://blog.quantinsti.com/markov-model/
6. https://pubmed.ncbi.nlm.nih.gov/10178664/
7. https://labelstud.io/blog/markov-models-chains-to-choices/
8. https://ntrs.nasa.gov/api/citations/20020050518/downloads/20020050518.pdf
10. https://www.youtube.com/watch?v=d0xgyDs4EBc
