How do you find the invariant distribution of a Markov chain?
A probability distribution π = (πx ⩾ 0 : x ∈ X) such that ∑x∈X πx = 1 is said to be stationary distribution or invariant distribution for the Markov chain X if π = πP, that is πy = ∑x∈X πx pxy for all y ∈ X.
Does Markov chain converge?
Do all Markov chains converge in the long run to a single stationary distribution like in our example? No. It turns out only a special type of Markov chains called ergodic Markov chains will converge like this to a single distribution.
What is the fundamental matrix of a Markov chain?
The matrix F=(In−B)−1 is called the fundamental matrix for the absorbing Markov chain, where In is an identity matrix of the same size as B. The i, j-th entry of this matrix tells us the average number of times the process is in the non-absorbing state j before absorption if it started in the non-absorbing state i.
What is a Markov chain linear algebra?
In the Linear Algebra book by Lay, Markov chains are introduced in Sections 1.10 (Difference Equations) and 4.9. Markov processes concern fixed probabilities of making transitions between a finite number of states. We start by defining a probability transition matrix or stochastic matrix.
Is invariant distribution unique?
The Perron-Frobenius Theorem ensures that for a matrix with strictly positive entries, the invariant probability distribution is always unique.
What is the stationary distribution of a Markov chain?
The stationary distribution of a Markov chain describes the distribution of Xt after a sufficiently long time that the distribution of Xt does not change any longer. To put this notion in equation form, let π be a column vector of probabilities on the states that a Markov chain can visit.
Why does a Markov chain converge?
Convergence to stationary distribution means that if you run the chain many times starting at any X0=x0 to obtain many samples of Xn, the empirical distribution of Xn will be close to stationary (for large n) and will get closer to it (and converge) as n increases.
How do you know if a Markov chain converges to equilibrium?
Convergence to equilibrium means that, as the time progresses, the Markov chain ‘forgets’ about its initial distribution λ. In particular, if λ = δ(i), the Dirac delta concentrated at i, the chain ‘forgets’ about initial state i. Clearly, this is related to properties of the n-step matrix Pn as n → ∞.
What makes a Markov chain absorbing?
An absorbing Markov chain is a Markov chain in which it is impossible to leave some states, and any state could (after some number of steps, with positive probability) reach such a state. It follows that all non-absorbing states in an absorbing Markov chain are transient.
How do you tell if a matrix is a transition matrix?
Regular Markov Chain: A transition matrix is regular when there is power of T that contains all positive no zeros entries. c) If all entries on the main diagonal are zero, but T n (after multiplying by itself n times) contain all postive entries, then it is regular.
What is transition matrix in Markov chain?
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix.