# Markov chains

This means the number of cells grows quadratically as we add states to our Markov chain. See Also Basic Concept A Markov chain is a stochastic processbut it differs from a general stochastic process in that a Markov chain must be "memory-less".

That is, the probability of future actions are not dependent upon the steps that led up to the present state. This is called the Markov property. While the theory of Markov chains is important precisely because so many "everyday" processes satisfy the Markov property, there are many common examples of stochastic properties that do not satisfy the Markov property.

A common probability question asks what is the probability of getting a certain color ball, when selecting uniformly and at random from a bag of multicolored balls. It could also ask what the probability of the next ball is, and so on. In such a way, a stochastic process begins to exist with color for the random variable, and it does not satisfy the Markov property.

Depending upon which balls are removed, the probability of getting a certain color ball later may be drastically different. A variant of the same question asks once again for ball color, but it allows replacement each time a ball is drawn.

Once again, this creates a stochastic process with color for the random variable. This process, however, does satisfy the Markov property. Can you figure out why? In probability theory, the most immediate example is that of a time-homogeneous Markov chain, in which the probability of any state transition is independent of time.

A time-homogeneous Markov chain built on states A and B is depicted in the diagram below. What is the probability that a process beginning on A will be on B after 2 moves?

In order to move from A to B, the process must either stay on A the first move, then move to B the second move; or move to B the first move, then stay on B the second move. According to the diagram, the probability of that is Alternatively, the probability that the process will be on A after 2 moves is.

Since there are only two states in the chain, the process must be on B if it is not on A, and therefore, the probability that the process will be on B after 2 moves is In the language of conditional probability and random variablesa Markov chain is a sequence of random variables satisfying the rule of conditional independence:Markov Chains Introduction Most of our study of probability has dealt with independent trials processes.

These processes are the basis of classical probability theory and much of statistics. We have discussed two of the principal theorems for these processes: the . Probabilities in the game Monopoly including land on frequencies, expected earnings, and payback times. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition (Chapman & Hall/CRC Texts in Statistical Science) May 10, and the sequence is called a Markov chain (Papoulis , p. ). A simple random walk is an example of a Markov chain.

The Season 1 episode "Man Hunt" () of the television crime drama NUMB3RS features Markov chains. From this table, we can determine that while the n-gram co is followed by n % of the time, and while the n-gram on is followed by d % of the time, the n-gram de is followed by s 50% of the time, and n the rest of the time.

Likewise, the n-gram es is followed by c 50% of the time, and followed by the end of the text the other 50% of the time.. Generative text with Markov chains. Jan 13,  · In this video, I discuss Markov Chains, although I never quite give a definition as the video cuts off!

Markov Chains, Part 3 - Regular Markov Chains - Duration: What is a Markov chain.

Origin of Markov chains (video) | Khan Academy