Markov property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined  for an interconnected network of items. An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the Markov property is known as a Markov chain. Introduction. A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past. A process with this property is said to be Markov or Markovian and known as a Markov process. The most famous  Markov process is a Markov chain. Brownian motion is another well-known Markov process. Definition. Let formula_1 be a probability space with a filtration formula_2, for some (totally ordered) index set formula_3; and let formula_4 be a measurable space. A formula_4-valued stochastic process formula_6 adapted to the filtration is said to possess the "Markov property" if, for each formula_7 and each formula_8 with formula_9, In the case where formula_11 is a discrete set with the discrete sigma algebra and formula_12, this can be reformulated as follows: Alternative formulations. Alternatively, the Markov property can be formulated as follows. for all  formula_15 and formula_16 bounded and measurable. Strong Markov property. Suppose that formula_17 is a stochastic process on a probability space formula_1 with natural filtration formula_19. Then for any stopping time formula_20 on formula_21, we can define Then formula_23 is said to have the strong Markov property if, for each stopping time formula_24, conditioned on the event formula_25, we have that for each formula_26, formula_27 is independent of formula_28 given formula_29. The strong Markov property implies the ordinary Markov property since by taking the stopping time formula_30, the ordinary Markov property can be deduced. In forecasting. In the fields of predictive  modelling and probabilistic forecasting, the Markov property is considered desirable since it may enable the reasoning and resolution of the problem that otherwise would not be possible to be resolved because of its intractability. Such a model is known as a Markov model. Examples. Assume that an urn contains two red balls and one green ball. One ball was drawn yesterday, one ball was drawn today, and the final ball will be drawn tomorrow. All of the draws are "without replacement". Suppose you know that today's ball was red, but you have no information about yesterday's ball. The  chance that tomorrow's ball will be red is 1/2. That's because the only two remaining outcomes for this random experiment are: On the other hand, if you know that both today and yesterday's balls were red, then you are guaranteed to get a green ball tomorrow. This discrepancy shows that the probability distribution for tomorrow's color depends not only on the present value, but is also affected by information about the past. This stochastic process of observed colors doesn't have the Markov property. Using the same experiment above, if sampling "without replacement" is changed to  sampling "with replacement," the process of observed colors will have the Markov property. An application of the Markov property in a generalized form is in Markov chain Monte Carlo computations in the context of Bayesian statistics.