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A Markov chain (discrete-time Markov chain or DTMC), named after Andrey Markov, is a random process that undergoes transitions from one state to another on a state space. It must possess a property that is usually characterized as "memorylessness": the probability distribution of the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes. ==Introduction== A Markov chain is a stochastic process with the Markov property. The term "Markov chain" refers to the sequence of random variables such a process moves through, with the Markov property defining serial dependence only between adjacent periods (as in a "chain"). It can thus be used for describing systems that follow a chain of linked events, where what happens next depends only on the current state of the system. In the literature, different kinds of Markov process are designated as "Markov chains". Usually the term is reserved for a process with a discrete set of times, i.e. a discrete-time Markov chain (DTMC).〔Everitt,B.S. (2002) ''The Cambridge Dictionary of Statistics''. CUP. ISBN 0-521-81099-X〕 On the other hand, a few authors use the term "Markov process" to refer to a continuous-time Markov chain without explicit mention.〔Parzen, E. (1962) ''Stochastic Processes'', Holden-Day. ISBN 0-8162-6664-6 (Table 6.1))〕〔Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9 (entry for "Markov chain")〕 While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space.〔Meyn, S. Sean P., and Richard L. Tweedie. (2009) ''Markov chains and stochastic stability''. Cambridge University Press. (Preface, p. iii)〕 However, many applications of Markov chains employ finite or countably infinite (that is, discrete) state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalisations (see Variations). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise. The changes of state of the system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement. Formally, the steps are the integers or natural numbers, and the random process is a mapping of these to states. The Markov property states that the conditional probability distribution for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps. Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future. However, the statistical properties of the system's future can be predicted. In many applications, it is these statistical properties that are important. A famous Markov chain is the so-called "drunkard's walk", a random walk on the number line where, at each step, the position may change by +1 or −1 with equal probability. From any position there are two possible transitions, to the next or previous integer. The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0.5, and all other transition probabilities from 5 are 0. These probabilities are independent of whether the system was previously in 4 or 6. Another example is the dietary habits of a creature who eats only grapes, cheese, or lettuce, and whose dietary habits conform to the following rules: * It eats exactly once a day. * If it ate cheese today, tomorrow it will eat lettuce or grapes with equal probability. * If it ate grapes today, tomorrow it will eat grapes with probability 1/10, cheese with probability 4/10 and lettuce with probability 5/10. * If it ate lettuce today, tomorrow it will eat grapes with probability 4/10 or cheese with probability 6/10. It will not eat lettuce again tomorrow. This creature's eating habits can be modeled with a Markov chain since its choice tomorrow depends solely on what it ate today, not what it ate yesterday or any other time in the past. One statistical property that could be calculated is the expected percentage, over a long period, of the days on which the creature will eat grapes. A series of independent events (for example, a series of coin flips) satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next step depends non-trivially on the current state. Many other examples of Markov chains exist. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Markov chain」の詳細全文を読む スポンサード リンク
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