Discrete-time Markov chain

In probability, a discrete-time Markov chain is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past. For instance, a machine may have two states, A and E. When it is in state A, there is a 40% chance of it moving to state E and a 60% chance of it remaining in state A. When it is in state E, there is a 70% chance of it moving to A and a 30% chance of it staying in E. The sequence of states of the machine is a Markov chain. If we denote the chain by then is the state which the machine starts in and is the random variable describing its state after 10 transitions. The process continues forever, indexed by the natural numbers.
An example of a stochastic process which is not a Markov chain is the model of a machine which has states A and E and moves to A from either state with 50% chance if it has ever visited A before, and 20% chance if it has never visited A before. This is because the behavior of the machine depends on the whole history—if the machine is in E, it may have a 50% or 20% chance of moving to A, depending on its past values. Hence, it does not have the Markov property.
A Markov chain can be described by a stochastic matrix, which lists the probabilities of moving to each state from any individual state. From this matrix, the probability of being in a particular state n steps in the future can be calculated. A Markov chain's state space can be partitioned into communicating classes that describe which states are reachable from each other. Each state can be described as transient or recurrent, depending on the probability of the chain ever returning to that state. Markov chains can have properties including periodicity, reversibility and stationarity. A continuous-time Markov chain is like a discrete-time Markov chain, but it moves states continuously through time rather than as discrete time steps. Other stochastic processes can satisfy the Markov property, the property that past behavior does not affect the process, only the present state.

Definition

A discrete-time Markov chain is a sequence of random variables with the Markov property, namely that the probability of moving to the next state depends only on the present state and not on the previous states:
The possible values of X_i form a countable set S called the state space of the chain.
Markov chains are often described by a sequence of directed graphs, where the edges of graph n are labeled by the probabilities of going from one state at time n to the other states at time n + 1, The same information is represented by the transition matrix from time n to time n + 1. However, Markov chains are frequently assumed to be time-homogeneous, in which case the graph and matrix are independent of n and are thus not presented as sequences.
These descriptions highlight the structure of the Markov chain that is independent of the initial distribution When time-homogeneous, the chain can be interpreted as a state machine assigning a probability of hopping from each vertex or state to an adjacent one. The probability of the machine's state can be analyzed as the statistical behavior of the machine with an element of the state space as input, or as the behavior of the machine with the initial distribution of states as input, where is the Iverson bracket.

Variations

Time-homogeneous Markov chains are processes where
A Markov chain with memory
''n''-step transitions

The probability of going from state i to state j in n time steps is
and the single-step transition is
For a time-homogeneous Markov chain:
and
The n-step transition probabilities satisfy the Chapman–Kolmogorov equation, that for any k such that 0 < k < n,
where S is the state space of the Markov chain.
The marginal distribution Pr is the distribution over states at time n. The initial distribution is Pr. The evolution of the process through one time step is described by
.

Communicating classes and properties

A state j is said to be accessible from a state i if a system started in state i has a non-zero probability of transitioning into state j at some point. Formally, state j is accessible from state i if there exists an integer n_ij ≥ 0 such that
A state i is said to communicate with state j if both i → j and j → i. A communicating class is a maximal set of states C such that every pair of states in C communicates with each other. Communication is an equivalence relation, and communicating classes are the equivalence classes of this relation.
A communicating class is closed if the probability of leaving the class is zero, namely if i is in C but j is not, then j is not accessible from i. The set of communicating classes forms a directed, acyclic graph by inheriting the arrows from the original state space. A communicating class is closed if and only if it has no outgoing arrows in this graph.
A state i is said to be essential or final if for all j such that i → j it is also true that j → i. A state i is inessential if it is not essential. A state is final if and only if its communicating class is closed.
A Markov chain is said to be irreducible if its state space is a single communicating class; in other words, if it is possible to get to any state from any state.

Periodicity

A state has period if any return to state must occur in multiples of time steps. Formally, the period of state is defined as
provided that this set is not empty. Otherwise the period is not defined. Note that even though a state has period, it may not be possible to reach the state in steps. For example, suppose it is possible to return to the state in time steps; would be, even though does not appear in this list.
If, then the state is said to be aperiodic. Otherwise, the state is said to be periodic with period . Periodicity is a class property—that is, if a state has period then every state in its communicating class has period.

Transience and recurrence

A state i is said to be transient if, given that we start in state i, there is a non-zero probability that we will never return to i. Formally, let the random variable T_i be the first return time to state i :
The number
is the probability that we return to state i for the first time after n steps.
Therefore, state i is transient if
State i is recurrent if it is not transient. Recurrence and transience are class properties, that is, they either hold or do not hold equally for all members of a communicating class.
A state i is recurrent if and only if the expected number of visits to i is infinite:

Positive recurrence

Even if the hitting time is finite with probability 1, it need not have a finite expectation. The mean recurrence time at state i is the expected return time M_i:
State i is positive recurrent if M_i is finite; otherwise, state i is null recurrent. Positive and null recurrence are classes properties.

Absorbing states

A state i is called absorbing if it is impossible to leave this state. Therefore, the state i is absorbing if and only if
If every state can reach an absorbing state, then the Markov chain is an absorbing Markov chain.

Reversible Markov chain

A Markov chain is said to be reversible if there is a probability distribution ' over its states such that
for all times n and all states i and j. This condition is known as the detailed balance condition.
Considering a fixed arbitrary time n and using the shorthand
the detailed balance equation can be written more compactly as
The single time-step from n to n + 1 can be thought of as each person i having _i dollars initially and paying each person j a fraction p_ij of it. The detailed balance condition states that upon each payment, the other person pays exactly the same amount of money back. Clearly the total amount of money ' each person has remains the same after the time-step, since every dollar spent is balanced by a corresponding dollar received. This can be shown more formally by the equality
which essentially states that the total amount of money person j receives during the time-step equals the amount of money he pays others, which equals all the money he initially had because it was assumed that all money is spent. The assumption is a technical one, because the money not really used is simply thought of as being paid from person j to himself.
As n was arbitrary, this reasoning holds for any n, and therefore for reversible Markov chains ' is always a steady-state distribution of Pr for every n.
If the Markov chain begins in the steady-state distribution, that is, if, then for all and the detailed balance equation can be written as
The left- and right-hand sides of this last equation are identical except for a reversing of the time indices n and n + 1.
Kolmogorov's criterion gives a necessary and sufficient condition for a Markov chain to be reversible directly from the transition matrix probabilities. The criterion requires that the products of probabilities around every closed loop are the same in both directions around the loop.
Reversible Markov chains are common in Markov chain Monte Carlo approaches because the detailed balance equation for a desired distribution ' necessarily implies that the Markov chain has been constructed so that ' is a steady-state distribution. Even with time-inhomogeneous Markov chains, where multiple transition matrices are used, if each such transition matrix exhibits detailed balance with the desired ' distribution, this necessarily implies that is a steady-state distribution of the Markov chain.