Markov chain: Difference between revisions

A Markov chain (discrete-time Markov chain or DTMC^[1]) named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process usually characterized as memoryless: 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.

Formally, a Markov chain is a random process with the Markov property. Often, the term "Markov chain" is used to mean a Markov process which has a discrete (finite or countable) state-space. Usually a Markov chain is defined for a discrete set of times (i.e., a discrete-time Markov chain)^[2] although some authors use the same terminology where "time" can take continuous values.^[3][4] The use of the term in Markov chain Monte Carlo methodology covers cases where the process is in discrete time (discrete algorithm steps) with a continuous state space. The following concentrates on the discrete-time discrete-state-space case.

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.

The changes of state of the system are called transitions, and the probabilities associated with various state-changes are called transition probabilities. The set of all states and transition probabilities completely characterizes a Markov chain. By convention, we assume all possible states and transitions have been included in the definition of the processes, so there is always a next state and the process goes on forever.

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, it will not eat lettuce again tomorrow but will eat grapes with probability 4/10 or cheese with probability 6/10.

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 even farther 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.

A Markov chain is a sequence of random variables X₁, X₂, X₃, ... with the Markov property, namely that, given the present state, the future and past states are independent. Formally,

${\displaystyle \Pr(X_{n+1}=x|X_{1}=x_{1},X_{2}=x_{2},\ldots ,X_{n}=x_{n})=\Pr(X_{n+1}=x|X_{n}=x_{n}).\,}$

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 directed graph, where the edges are labeled by the probabilities of going from one state to the other states.

• Continuous-time Markov processes have a continuous index.
• Time-homogeneous Markov chains (or stationary Markov chains) are processes where

${\displaystyle \Pr(X_{n+1}=x|X_{n}=y)=\Pr(X_{n}=x|X_{n-1}=y)\,}$

for all n. The probability of the transition is independent of n.

• A Markov chain of order m (or a Markov chain with memory m), where m is finite, is a process satisfying

${\displaystyle {\begin{aligned}{}&\Pr(X_{n}=x_{n}|X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{1}=x_{1})\\=&\Pr(X_{n}=x_{n}|X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{n-m}=x_{n-m}){\text{ for }}n>m\end{aligned}}}$

In other words, the future state depends on the past m states. It is possible to construct a chain (Y_n) from (X_n) which has the 'classical' Markov property by taking as state space the ordered m-tuples of X values, ie. Y_n = (X_n, X_n−1, ..., X_n−m+1).

• An additive Markov chain of order m is determined by an additive conditional probability,

${\displaystyle \Pr(X_{n}=x_{n}|X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{n-m}=x_{n-m})=\sum _{r=1}^{m}f(x_{n},x_{n-r},r).}$

The value f(x_n,x_n-r,r) is the additive contribution of the variable x_n-r to the conditional probability.^{[5][clarification needed]}

A simple example is shown in the figure on the right, using a directed graph to picture the state transitions. The states represent whether the economy is in a bull market, a bear market, or a recession, during a given week. According to the figure, a bull week is followed by another bull week 90% of the time, a bear market 7.5% of the time, and a recession the other 2.5%. Labelling the state space {1=bull, 2=bear, 3=recession} the transition matrix for this example is

${\displaystyle P={\begin{bmatrix}0.9&0.075&0.025\\0.15&0.8&0.05\\0.25&0.25&0.5\end{bmatrix}}.}$

The distribution over states can be written as a stochastic row vector x with the relation x^(n + 1) = x⁽ⁿ⁾P. So if at time n the system is in state 2=bear then 3 time periods later at time n + 3 the distribution is

${\displaystyle {\begin{aligned}x^{(n+3)}&=x^{(n+2)}P=\left(x^{(n+1)}P\right)P\\\\&=x^{(n+1)}P^{2}=\left(x^{(n)}P^{2}\right)P\\&=x^{(n)}P^{3}\\&={\begin{bmatrix}0&1&0\end{bmatrix}}{\begin{bmatrix}0.9&0.075&0.025\\0.15&0.8&0.05\\0.25&0.25&0.5\end{bmatrix}}^{3}\\&={\begin{bmatrix}0&1&0\end{bmatrix}}{\begin{bmatrix}0.7745&0.17875&0.04675\\0.3575&0.56825&0.07425\\0.4675&0.37125&0.16125\\\end{bmatrix}}\\&={\begin{bmatrix}0.3575&0.56825&0.07425\end{bmatrix}}.\end{aligned}}}$

From this figure it is possible to calculate, for example, the long-term fraction of time during which the economy is in a recession, or on average how long it will take to go from a recession to a bull market. Using the transition probabilities, the steady-state probabilities indicate that 62.5% of weeks will be in a bull market, 31.25% of weeks will be in a bear market and 6.25% of weeks will be in a recession.

A thorough development and many examples can be found in the on-line monograph Meyn & Tweedie 2005.^[6] The appendix of Meyn 2007,^[7] also available on-line, contains an abridged Meyn & Tweedie.

A finite state machine can be used as a representation of a Markov chain. Assuming a sequence of independent and identically distributed input signals (for example, symbols from a binary alphabet chosen by coin tosses), if the machine is in state y at time n, then the probability that it moves to state x at time n + 1 depends only on the current state.

The probability of going from state i to state j in n time steps is

${\displaystyle p_{ij}^{(n)}=\Pr(X_{n}=j\mid X_{0}=i)\,}$

and the single-step transition is

${\displaystyle p_{ij}=\Pr(X_{1}=j\mid X_{0}=i).\,}$

For a time-homogeneous Markov chain:

${\displaystyle p_{ij}^{(n)}=\Pr(X_{k+n}=j\mid X_{k}=i)\,}$

and

${\displaystyle p_{ij}=\Pr(X_{k+1}=j\mid X_{k}=i).\,}$

The n-step transition probabilities satisfy the Chapman–Kolmogorov equation, that for any k such that 0 < k < n,

${\displaystyle p_{ij}^{(n)}=\sum _{r\in S}p_{ir}^{(k)}p_{rj}^{(n-k)}}$

where S is the state space of the Markov chain.

The marginal distribution Pr(X_n = x) is the distribution over states at time n. The initial distribution is Pr(X₀ = x). The evolution of the process through one time step is described by

${\displaystyle \Pr(X_{n}=j)=\sum _{r\in S}p_{rj}\Pr(X_{n-1}=r)=\sum _{r\in S}p_{rj}^{(n)}\Pr(X_{0}=r).}$

Note: The superscript (n) is an index and not an exponent.

A state j is said to be accessible from a state i (written i → j) 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 ≥ 0 such that

${\displaystyle \Pr(X_{n}=j|X_{0}=i)=p_{ij}^{(n)}>0.\,}$

Allowing n to be zero means that every state is defined to be accessible from itself.

A state i is said to communicate with state j (written i ↔ j) if both i → j and j → i. A set of states C is a communicating class if every pair of states in C communicates with each other, and no state in C communicates with any state not in C. It can be shown that communication in this sense is an equivalence relation and thus that communicating classes are the equivalence classes of this relation. A communicating class is closed if the probability of leaving the class is zero, namely that if i is in C but j is not, then j is not accessible from i.

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.^[8]

Finally, 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.

A state i has period k if any return to state i must occur in multiples of k time steps. Formally, the period of a state is defined as

${\displaystyle k=\operatorname {gcd} \{n:\Pr(X_{n}=i|X_{0}=i)>0\}}$

(where "gcd" is the greatest common divisor). Note that even though a state has period k, it may not be possible to reach the state in k steps. For example, suppose it is possible to return to the state in {6, 8, 10, 12, ...} time steps; k would be 2, even though 2 does not appear in this list.

If k = 1, then the state is said to be aperiodic: returns to state i can occur at irregular times. In other words, a state i is aperiodic if there exists n such that for all n' ≥ n,

${\displaystyle \Pr(X_{n'}=i|X_{0}=i)>0.}$

Otherwise (k > 1), the state is said to be periodic with period k. A Markov chain is aperiodic if every state is aperiodic. An irreducible markov chain only needs one aperiodic state to imply all states are aperiodic.

Every state of a bipartite graph has an even period.

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 "hitting time"):

${\displaystyle T_{i}=\inf\{n\geq 1:X_{n}=i|X_{0}=i\}.}$

The number

${\displaystyle f_{ii}^{(n)}=\Pr(T_{i}=n)}$

is the probability that we return to state i for the first time after n steps. Therefore, state i is transient if

${\displaystyle \Pr(T_{i}<{\infty })=\sum _{n=1}^{\infty }f_{ii}^{(n)}<1.}$

State i is recurrent (or persistent) if it is not transient. Recurrent states have finite hitting time with probability 1.

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:

${\displaystyle M_{i}=E[T_{i}]=\sum _{n=1}^{\infty }n\cdot f_{ii}^{(n)}.\,}$

State i is positive recurrent (or non-null persistent) if M_i is finite; otherwise, state i is null recurrent (or null persistent).

It can be shown that a state i is recurrent if and only if the expected number of visits to this state is infinite, i.e.,

${\displaystyle \sum _{n=0}^{\infty }p_{ii}^{(n)}=\infty .}$

A state i is called absorbing if it is impossible to leave this state. Therefore, the state i is absorbing if and only if

${\displaystyle p_{ii}=1{\text{ and }}p_{ij}=0{\text{ for }}i\not =j.}$

If every state can reach an absorbing state, then the Markov chain is an absorbing Markov chain.

A state i is said to be ergodic if it is aperiodic and positive recurrent. In other words, a state i is ergodic if it is recurrent, has a period of 1 and it has finite mean recurrence time. If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic.

It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state. A model has the ergodic property if there's a finite number N such that any state can be reached from any other state in exactly N steps. In case of a fully connected transition matrix where all transitions have a non-zero probability, this condition is fulfilled with N=1. A model with more than one state and just one out-going transition per state cannot be ergodic.

If the Markov chain is a time-homogeneous Markov chain, so that the process is described by a single, time-independent matrix ${\displaystyle p_{ij}}$, then the vector ${\displaystyle {\boldsymbol {\pi }}}$ is called a stationary distribution (or invariant measure) if its entries ${\displaystyle \pi _{j}}$ are non-negative and sum to 1 and if it satisfies

${\displaystyle \pi _{j}=\sum _{i\in S}\pi _{i}p_{ij}.}$

An irreducible chain has a stationary distribution if and only if all of its states are positive recurrent. In that case, π is unique and is related to the expected return time:

${\displaystyle \pi _{j}={\frac {C}{M_{j}}}\,,}$

where ${\displaystyle C}$ is the normalizing constant. Further, if the chain is both irreducible and aperiodic, then for any i and j,

${\displaystyle \lim _{n\rightarrow \infty }p_{ij}^{(n)}={\frac {C}{M_{j}}}.}$

Note that there is no assumption on the starting distribution; the chain converges to the stationary distribution regardless of where it begins. Such π is called the equilibrium distribution of the chain.

If a chain has more than one closed communicating class, its stationary distributions will not be unique (consider any closed communicating class ${\displaystyle C_{i}}$ in the chain; each one will have its own unique stationary distribution ${\displaystyle \pi _{i}}$. Extending these distributions to the overall chain, setting all values to zero outside the communication class, yields that the set of invariant measures of the original chain is the set of all convex combinations of the ${\displaystyle \pi _{i}}$'s). However, if a state j is aperiodic, then

${\displaystyle \lim _{n\rightarrow \infty }p_{jj}^{(n)}={\frac {C}{M_{j}}}}$

and for any other state i, let f_ij be the probability that the chain ever visits state j if it starts at i,

${\displaystyle \lim _{n\rightarrow \infty }p_{ij}^{(n)}=C{\frac {f_{ij}}{M_{j}}}.}$

If a state i is periodic with period k > 1 then the limit

${\displaystyle \lim _{n\rightarrow \infty }p_{ii}^{(n)}}$

does not exist, although the limit

${\displaystyle \lim _{n\rightarrow \infty }p_{ii}^{(kn+r)}}$

does exist for every integer r.

A Markov chain need not necessarily be time-homogeneous to have an equilibrium distribution. If there is a probability distribution over states ${\displaystyle {\boldsymbol {\pi }}}$ such that

${\displaystyle \pi _{j}=\sum _{i\in S}\pi _{i}\,\Pr(X_{n+1}=j\mid X_{n}=i)}$

for every state j and every time n then ${\displaystyle {\boldsymbol {\pi }}}$ is an equilibrium distribution of the Markov chain. Such can occur in Markov chain Monte Carlo (MCMC) methods in situations where a number of different transition matrices are used, because each is efficient for a particular kind of mixing, but each matrix respects a shared equilibrium distribution.

If the state space is finite, the transition probability distribution can be represented by a matrix, called the transition matrix, with the (i, j)th element of P equal to

${\displaystyle p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).\,}$

Since each row of P sums to one and all elements are non-negative, P is a right stochastic matrix.

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If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k-step transition probability can be computed as the k-th power of the transition matrix, P^k.

The stationary distribution π is a (row) vector, whose entries are non-negative and sum to 1, that satisfies the equation

${\displaystyle \pi =\pi \mathbf {P} .\,}$

In other words, the stationary distribution π is a normalized (meaning that the sum of its entries is 1) left eigenvector of the transition matrix associated with the eigenvalue 1.

Alternatively, π can be viewed as a fixed point of the linear (hence continuous) transformation on the unit simplex associated to the matrix P. As any continuous transformation in the unit simplex has a fixed point, a stationary distribution always exists, but is not guaranteed to be unique, in general. However, if the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π. Additionally, in this case P^k converges to a rank-one matrix in which each row is the stationary distribution π, that is,

${\displaystyle \lim _{k\rightarrow \infty }\mathbf {P} ^{k}=\mathbf {1} \pi }$

where 1 is the column vector with all entries equal to 1. This is stated by the Perron–Frobenius theorem. If, by whatever means, ${\displaystyle \scriptstyle \lim _{k\to \infty }\mathbf {P} ^{k}}$ is found, then the stationary distribution of the Markov chain in question can be easily determined for any starting distribution, as will be explained below.

For some stochastic matrices P, the limit ${\displaystyle \scriptstyle \lim \limits _{k\to \infty }\mathbf {P} ^{k}}$ does not exist, as shown by this example:

${\displaystyle {\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}$

Because there are a number of different special cases to consider, the process of finding this limit if it exists can be a lengthy task. However, there are many techniques that can assist in finding this limit. Let P be an n×n matrix, and define ${\displaystyle \scriptstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.}$

It is always true that

${\displaystyle \mathbf {QP} =\mathbf {Q} .}$

Subtracting Q from both sides and factoring then yields

${\displaystyle \mathbf {Q} (\mathbf {P} -\mathbf {I} _{n})=\mathbf {0} _{n,n},}$

where I_n is the identity matrix of size n, and 0_n,n is the zero matrix of size n×n. Multiplying together stochastic matrices always yields another stochastic matrix, so Q must be a stochastic matrix (see the definition above). It is sometimes sufficient to use the matrix equation above and the fact that Q is a stochastic matrix to solve for Q. Including the fact that the sum of each the rows in P is 1, there are n+1 equations for determining n unknowns, so it is computationally easier if on the one hand one selects one row in Q and substitute each of its elements by one, and on the other one substitute the corresponding element (the one in the same column) in the vector 0, and next left-multiply this latter vector by the inverse of transformed former matrix to find Q.

Here is one method for doing so: first, define the function f(A) to return the matrix A with its right-most column replaced with all 1's. If [f(P − I_n)]⁻¹ exists then^{[citation needed]}

${\displaystyle \mathbf {Q} =f(\mathbf {0} _{n,n})[f(\mathbf {P} -\mathbf {I} _{n})]^{-1}.}$

Explain: The original matrix equation is equivalent to a system of n×n linear equations in n×n variables. And there are n more linear equations from the fact that Q is a right stochastic matrix whose each row sums to 1. So it needs any n×n independent linear equations of the (n×n+n) equations to solve for the n×n variables. In this example, the n equations from “Q multiplied by the right-most column of (P-In)” have been replaced by the n stochastic ones.

One thing to notice is that if P has an element P_i,i on its main diagonal that is equal to 1 and the ith row or column is otherwise filled with 0's, then that row or column will remain unchanged in all of the subsequent powers P^k. Hence, the ith row or column of Q will have the 1 and the 0's in the same positions as in P.

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As stated earlier, from the equation ${\displaystyle \mathbf {\pi } =\mathbf {\pi P} }$, (if exists) the stationary (or steady state) distribution π is a left eigenvector of row stochastic matrix P. Then assuming that P is diagonalizable or equivalently that P has n linearly independent eigenvectors, speed of convergence is elaborated as follows. For non-diagonalizable matrices, one may start with "Jordan Canonical Form" (almost diagonal form) of P and proceed with a bit more involved set of arguments in a similar way.^[9]

Let U be the matrix of eigenvectors (each normalized to having an L2 norm equal to 1) where each column is a left eigenvector of P and let Σ be the diagonal matrix of left eigenvalues of P, i.e. Σ = diag(λ₁,λ₂,λ₃,...,λ_n). Then by eigendecomposition

${\displaystyle \mathbf {P} =\mathbf {U\Sigma U} ^{-1}.}$

Let the eigenvalues be enumerated such that 1=|λ₁|>|λ₂|≥|λ₃|≥...≥|λ_n|. Since P is a row stochastic matrix, its largest left eigenvalue is 1. If there is a unique stationary distribution, then the largest eigenvalue and the corresponding eigenvector is unique too (because there is no other π which solves the stationary distribution equation above). Let u_i be the ith column of U matrix, i.e. u_i is the left eigenvector of P corresponding to λ_i. Also let x be an arbitrary length n row vector in the span of the eigenvectors u_i, that is

${\displaystyle \mathbf {x} ^{T}=\sum _{i=1}^{n}a_{i}\mathbf {u} _{i}}$

for some set of a_i∈ℝ. If we start multiplying P with x from left and continue this operation with the results, in the end we get the stationary distribution π. In other words π = u_i ← xPPP...P = xP^k as k goes to infinity. That means

${\displaystyle \mathbf {\pi } ^{(k)}=\mathbf {x} (\mathbf {U\Sigma U} ^{-1})(\mathbf {U\Sigma U} ^{-1})...(\mathbf {U\Sigma U} ^{-1})}$

${\displaystyle =\mathbf {xU\Sigma } ^{k}\mathbf {U} ^{-1}}$

since UU^-1 = I the identity matrix and power of a diagonal matrix is also a diagonal matrix where each entry is taken to that power.

${\displaystyle =(a_{1}\mathbf {u} _{1}^{T}+a_{2}\mathbf {u} _{2}^{T}+...+a_{n}\mathbf {u} _{n}^{T})\mathbf {U\Sigma } ^{k}\mathbf {U} ^{-1},}$

${\displaystyle =a_{1}\lambda _{1}^{k}\mathbf {u} _{1}+a_{2}\lambda _{2}^{k}\mathbf {u} _{2}+...+a_{n}\lambda _{n}^{k}\mathbf {u} _{n},}$

since the eigenvectors are orthonormal. Then^[10]

${\displaystyle =\lambda _{1}^{k}\left\{a_{1}\mathbf {u} _{1}+a_{2}\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{2}+a_{3}\left({\frac {\lambda _{3}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{3}+...+a_{n}\left({\frac {\lambda _{n}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{n}\right\}.}$

Since π = u₁, π^(k) approaches to π as k goes to infinity with a speed in the order of λ₂/λ₁ exponentially. This follows because |λ₂|≥|λ₃|≥...≥|λ_n|, hence λ₂/λ₁ is the dominant term. Random noise in the state distribution π can also speed up this convergence to the stationary distribution.^[11]

A Markov chain is said to be reversible if there is a probability distribution over states, π, such that

${\displaystyle \pi _{i}\Pr(X_{n+1}=j\mid X_{n}=i)=\pi _{j}\Pr(X_{n+1}=i\mid X_{n}=j)}$

for all times n and all states i and j. This condition is also known as the detailed balance condition (some books refer the local balance equation). With a time-homogeneous Markov chain, Pr(X_n+1 = j | X_n = i) does not change with time n and it can be written more simply as ${\displaystyle p_{ij}}$. In this case, the detailed balance equation can be written more compactly as

${\displaystyle \pi _{i}p_{ij}=\pi _{j}p_{ji}\,.}$

Summing the original equation over i gives

${\displaystyle {\begin{aligned}\sum _{i}\pi _{i}\Pr(X_{n+1}=j\mid X_{n}=i)&=\sum _{i}\pi _{j}\Pr(X_{n+1}=i\mid X_{n}=j)\\&=\pi _{j}\sum _{i}\Pr(X_{n+1}=i\mid X_{n}=j)=\pi _{j}\,,\end{aligned}}}$

so, for reversible Markov chains, π is always a steady-state distribution of Pr(X_n+1 = j | X_n = i) for every n.

If the Markov chain begins in the steady-state distribution, i.e., if Pr(X₀ = i) = π_i, then Pr(X_n = i) = π_i for all n and the detailed balance equation can be written as

${\displaystyle \Pr(X_{n}=i,X_{n+1}=j)=\Pr(X_{n+1}=i,X_{n}=j)\,.}$

The left- and right-hand sides of this last equation are identical except for a reversing of the time indices n and n + 1.

Reversible Markov chains are common in Markov chain Monte Carlo (MCMC) approaches because the detailed balance equation for a desired distribution π necessarily implies that the Markov chain has been constructed so that π 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 π distribution, this necessarily implies that π is a steady-state distribution of the Markov chain.

A Bernoulli scheme is a special case of a Markov chain where the transition probability matrix has identical rows, which means that the next state is even independent of the current state (in addition to being independent of the past states). A Bernoulli scheme with only two possible states is known as a Bernoulli process.

Many results for Markov chains with finite state space can be generalized to chains with uncountable state space through Harris chains. The main idea is to see if there is a point in the state space that the chain hits with probability one. Generally, it is not true for continuous state space, however, we can define sets A and B along with a positive number ε and a probability measure ρ, such that

1. ${\displaystyle {\text{If }}\tau _{A}=\inf\{n\geq 0:X_{n}\in A\},{\text{ then }}P_{z}(\tau _{A}<\infty )>0{\text{ for all }}z.}$
2. ${\displaystyle {\text{If }}x\in A{\text{ and }}C\subset B,{\text{ then }}p(x,C)\geq \varepsilon \rho (C).}$

Then we could collapse the sets into an auxiliary point α, and a recurrent Harris chain can be modified to contain α. Lastly, the collection of Harris chains is a comfortable level of generality, which is broad enough to contain a large number of interesting examples, yet restrictive enough to allow for a rich theory.

Considering a collection of Markov chains whose evolution takes in account the state of other Markov chains, is related to the notion of locally interacting Markov chains. This corresponds to the situation when the state space has a (Cartesian-) product form. See interacting particle system and stochastic cellular automata. See for instance Interaction of Markov Processes^[12] or ^[13]

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Research has reported the application and usefulness of Markov chains in a wide range of topics such as physics, chemistry, medicine, music, game theory and sports.

Markovian systems appear extensively in thermodynamics and statistical mechanics, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description.^{[citation needed]}

Chemistry is often a place where Markov chains and continuous-time Markov processes are especially useful because these simple physical systems tend to satisfy the Markov property quite well. The classical model of enzyme activity, Michaelis-Menten kinetics, can be viewed as a Markov chain, where at each time step the reaction proceeds in some direction. While Michaelis-Menten is fairly straightforward, far more complicated reaction networks can also be modeled with Markov chains.

An algorithm based on a Markov chain was also used to focus the fragment-based growth of chemicals in silico towards a desired class of compounds such as drugs or natural products.^[14] As a molecule is grown, a fragment is selected from the nascent molecule as the "current" state. It is not aware of its past (i.e., it is not aware of what is already bonded to it). It then transitions to the next state when a fragment is attached to it. The transition probabilities are trained on databases of authentic classes of compounds.

Also, the growth (and composition) of copolymers may be modeled using Markov chains. Based on the reactivity ratios of the monomers that make up the growing polymer chain, the chain's composition may be calculated (e.g., whether monomers tend to add in alternating fashion or in long runs of the same monomer). Due to steric effects, second-order Markov effects may also play a role in the growth of some polymer chains.

Similarly, it has been suggested that the crystallization and growth of some epitaxial superlattice oxide materials can be accurately described by Markov chains.^[15]

Several theorists have proposed the idea of the Markov chain statistical test (MCST), a method of conjoining Markov chains to form a "Markov blanket", arranging these chains in several recursive layers ("wafering") and producing more efficient test sets—samples—as a replacement for exhaustive testing. MCSTs also have uses in temporal state-based networks; Chilukuri et al.'s paper entitled "Temporal Uncertainty Reasoning Networks for Evidence Fusion with Applications to Object Detection and Tracking" (ScienceDirect) gives a background and case study for applying MCSTs to a wider range of applications.

Markov chains are used throughout information processing. Claude Shannon's famous 1948 paper A mathematical theory of communication, which in a single step created the field of information theory, opens by introducing the concept of entropy through Markov modeling of the English language. Such idealized models can capture many of the statistical regularities of systems. Even without describing the full structure of the system perfectly, such signal models can make possible very effective data compression through entropy encoding techniques such as arithmetic coding. They also allow effective state estimation and pattern recognition. Markov chains also play an important role in reinforcement learning.

Markov chains are also the basis for Hidden Markov Models, which are an important tool in such diverse fields as telephone networks (which use the Viterbi algorithm for error correction), speech recognition and bioinformatics.

Markov chains are the basis for the analytical treatment of queues (queueing theory). Agner Krarup Erlang initiated the subject in 1917.^[16] This makes them critical for optimizing the performance of telecommunications networks, where messages must often compete for limited resources (such as bandwidth).^[7]

The PageRank of a webpage as used by Google is defined by a Markov chain.^[17] It is the probability to be at page ${\displaystyle i}$ in the stationary distribution on the following Markov chain on all (known) webpages. If ${\displaystyle N}$ is the number of known webpages, and a page ${\displaystyle i}$ has ${\displaystyle k_{i}}$ links to it then it has transition probability ${\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}$ for all pages that are linked to and ${\displaystyle {\frac {1-\alpha }{N}}}$ for all pages that are not linked to. The parameter ${\displaystyle \alpha }$ is taken to be about 0.85.^[18]

Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on a particular website can be modeled using first- or second-order Markov models and can be used to make predictions regarding future navigation and to personalize the web page for an individual user.

Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called Markov chain Monte Carlo (MCMC). In recent years this has revolutionized the practicability of Bayesian inference methods, allowing a wide range of posterior distributions to be simulated and their parameters found numerically.

Markov chains are used in Finance and Economics to model a variety of different phenomena, including asset prices and market crashes. The first financial model to use a Markov chain was from Prasad et al. in 1974.^[19] Another was the regime-switching model of James D. Hamilton (1989), in which a Markov chain is used to model switches between periods of high volatility and low volatility of asset returns.^[20] A more recent example is the Markov Switching Multifractal asset pricing model, which builds upon the convenience of earlier regime-switching models.^[21] It uses an arbitrarily large Markov chain to drive the level of volatility of asset returns.

Dynamic macroeconomics heavily uses Markov chains. An example is using Markov chains to exogenously model prices of equity (stock) in a general equilibrium setting.^[22]

Markov chains are generally used in describing path-dependent arguments, where current structural configurations condition future outcomes. An example is the reformulation of the idea, originally due to Karl Marx's Das Kapital, tying economic development to the rise of capitalism. In current research, it is common to use a Markov chain to model how once a country reaches a specific level of economic development, the configuration of structural factors, such as size of the commercial bourgeoisie, the ratio of urban to rural residence, the rate of political mobilization, etc., will generate a higher probability of transitioning from authoritarian to democratic regime.^[23]

Markov chains also have many applications in biological modelling, particularly population processes, which are useful in modelling processes that are (at least) analogous to biological populations. The Leslie matrix is one such example, though some of its entries are not probabilities (they may be greater than 1). Another example is the modeling of cell shape in dividing sheets of epithelial cells.^[24] Yet another example is the state of ion channels in cell membranes.

Markov chains are also used in simulations of brain function, such as the simulation of the mammalian neocortex.^[25]

Markov chains have been used in population genetics in order to describe the change in gene frequencies in small populations affected by genetic drift, for example in diffusion equation method described by Motoo Kimura.^{[citation needed]}

Markov chains can be used to model many games of chance. The children's games Snakes and Ladders and "Hi Ho! Cherry-O", for example, are represented exactly by Markov chains. At each turn, the player starts in a given state (on a given square) and from there has fixed odds of moving to certain other states (squares).

Markov chains are employed in algorithmic music composition, particularly in software programs such as CSound, Max or SuperCollider. In a first-order chain, the states of the system become note or pitch values, and a probability vector for each note is constructed, completing a transition probability matrix (see below). An algorithm is constructed to produce and output note values based on the transition matrix weightings, which could be MIDI note values, frequency (Hz), or any other desirable metric.^[26]

A second-order Markov chain can be introduced by considering the current state and also the previous state, as indicated in the second table. Higher, nth-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense of phrasal structure, rather than the 'aimless wandering' produced by a first-order system.^[27]

Markov chains can be used structurally, as in Xenakis's Analogique A and B.^[28] Markov chains are also used in systems which use a Markov model to react interactively to music input.^[29]

Usually musical systems need to enforce specific control constraints on the finite-length sequences they generate, but control constraints are not compatible with Markov models, since they induce long-range dependencies that violate the Markov hypothesis of limited memory. In order to overcome this limitation, a new approach has been proposed.^[30]

Markov chain models have been used in advanced baseball analysis since 1960, although their use is still rare. Each half-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered. During any at-bat, there are 24 possible combinations of number of outs and position of the runners. Mark Pankin shows that Markov chain models can be used to evaluate runs created for both individual players as well as a team.^[31] He also discusses various kinds of strategies and play conditions: how Markov chain models have been used to analyze statistics for game situations such as bunting and base stealing and differences when playing on grass vs. astroturf.^[32]

Markov processes can also be used to generate superficially "real-looking" text given a sample document: they are used in a variety of recreational "parody generator" software (see dissociated press, Jeff Harrison,^[33] Mark V Shaney^[34][35] ).

These processes are also used by spammers to inject real-looking hidden paragraphs into unsolicited email and post comments in an attempt to get these messages past spam filters.

When fitting a Markov chain to data, situations where parameters poorly describe the situation may highlight interesting trends.^[36]

Andrey Markov produced the first results (1906) for these processes, purely theoretically. A generalization to countably infinite state spaces was given by Kolmogorov (1936). Markov chains are related to Brownian motion and the ergodic hypothesis, two topics in physics which were important in the early years of the twentieth century, but Markov appears to have pursued this out of a mathematical motivation, namely the extension of the law of large numbers to dependent events. In 1913, he applied his findings for the first time to the first 20,000 letters of Pushkin's Eugene Onegin.^{[citation needed]}. By 1917, more practical application of his work was made by Erlang to obtain formulas for call loss and waiting time in telephone network.^[16]

Seneta provides an account of Markov's motivations and the theory's early development.^[37] The term "chain" was used by Markov (1906).^[38]