# The Markov Property

The Markov Property makes a very strong assumption that the output of a random variable in a sequence only depends on the current state. Consider a sequence of state variables . The Markov assumption states that

A discrete-time process satisfying this property is called a Markov Chain and a continuous process is called a Markov process.

## Markov Chain

A Markov Chain is defined by a state space and the probabilities of transitioning within this state space. Given states there are possible transitions. This can be represented as a finite state machine with the symbols above the transition indicating the probability of that transition.

Consider the discrete process of predicting the weather for a given day. The set of values it can take include . These correspond to the states. Each discrete quanta is one day.

The Markov assumption states that today’s weather depends only on yesterday’s weather. Assume that there is a high chance that today will be if yesterday was and a moderate chance that today will be if yesterday was . These values are added as probabilities. The outgoing probabilities of a state must add up to .

Using this Markov Chain, the weather can be predicted by following the transitions. A test run will result in the following output. stands for and stands for

## Memoryless

The Markov property makes the model memoryless. The sequence of states generated are not context-sensitive and do not rely on the information captured in the full chain of prior states. For example, a Markov chain would do a poor job in mimicking the musical structure of a song. Higher order Markov chains use the assumption of the current state relying on multiple prior states.

## Mathematical Representation

A Markov Chain can be mathematically represented as transition matrix. Given possible states, the matrix will have dimensions , where each entry is the probability of transitioning from to . Also, by the law of total probability

The Markov chain also has an initial state vector of dimensions , where each entry is the probability of starting at the -th state.