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A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. This is in contrast to card games such as blackjack, where the cards represent a 'memory' of the past moves. To see the difference, consider the probability for a certain event in the game.
For example, a series of simple observations, such as a person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the Hierarchical hidden Markov model [2] and the Abstract Hidden Markov Model. [3]
A more recent example is the Markov switching multifractal model of Laurent E. Calvet and Adlai J. Fisher, which builds upon the convenience of earlier regime-switching models. [ 91 ] [ 92 ] It uses an arbitrarily large Markov chain to drive the level of volatility of asset returns.
Figure 1. Probabilistic parameters of a hidden Markov model (example) X — states y — possible observations a — state transition probabilities b — output probabilities. In its discrete form, a hidden Markov process can be visualized as a generalization of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step). [7]
We say is Markov with initial distribution and rate matrix to mean: the trajectories of are almost surely right continuous, let be a modification of to have (everywhere) right-continuous trajectories, (()) = + almost surely (note to experts: this condition says is non-explosive), the state sequence (()) is a discrete-time Markov chain with ...
Markov chain; Markov chain central limit theorem; Markov chain geostatistics; Markov chain Monte Carlo; Markov partition; Markov property; Markov switching multifractal; Markovian discrimination; Maximum-entropy Markov model; MegaHAL; Models of DNA evolution; MRF optimization via dual decomposition; Multiple sequence alignment
Markov decision process (MDP), also called a stochastic dynamic program or stochastic control problem, is a model for sequential decision making when outcomes are uncertain. [ 1 ] Originating from operations research in the 1950s, [ 2 ] [ 3 ] MDPs have since gained recognition in a variety of fields, including ecology , economics , healthcare ...
[1] [2]: 10 It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory , statistics, mathematical finance and ...