markov Chain Definitions

Practical application for estimation: The general concept for the application of this lies within having a set of generated [math]\displaystyle{ x_i }[/math] which approach a distribution [math]\displaystyle{ f(x) }[/math] so that a variation of importance estimation can be used to estimate an integral in the form
[math]\displaystyle{ I = \displaystyle\int^\ h(x)f(x)\,dx }[/math] by [math]\displaystyle{ \hat{I} = \frac{1}{N}\displaystyle\sum_{i=1}^Nh(x_i) }[/math]
All that is required is a Markov chain which eventually converges to [math]\displaystyle{ f(x) }[/math].

In the previous example, the entries [math]\displaystyle{ p_{ij} }[/math] in the transition matrix [math]\displaystyle{ P }[/math] represent the probability of reaching state [math]\displaystyle{ j }[/math] from state [math]\displaystyle{ i }[/math] after one step. For this reason, the sum over all entries j in a particular column sum to 1, as this itself must be a pmf if a transition from [math]\displaystyle{ i }[/math] is to lead to a state still within the state set for [math]\displaystyle{ X_t }[/math].

Homogeneous Markov Chain
The probability matrix [math]\displaystyle{ P }[/math] is the same for all indicies [math]\displaystyle{ n\in T }[/math]. [math]\displaystyle{ Pr(X_n=j|X_{n-1}=i)= Pr(X_1=j|X_0=i) }[/math]

If we denote the pmf of X_n by a probability vector [math]\displaystyle{ \frac{}{}\mu_n = [P(X_n=x_1),P(X_n=x_2),..,P(X_n=x_i)] }[/math]
where [math]\displaystyle{ i }[/math] denotes an ordered index of all possible states of [math]\displaystyle{ X }[/math].
Then we have a definition for the
marginal probabilty [math]\displaystyle{ \frac{}{}\mu_n(i) = P(X_n=i) }[/math]
where we simplify [math]\displaystyle{ X_n }[/math] to represent the ordered index of a state rather than the state itself.

From this definition it can be shown that, [math]\displaystyle{ \frac{}{}\mu_{n-1}P=\mu_0P^n }[/math]

Proof:

[math]\displaystyle{ \mu_{n-1}P=[\sum_{\forall i}(\mu_{n-1}(i))P_{i1},\sum_{\forall i}(\mu_{n-1}(i))P_{i2},..,\sum_{\forall i}(\mu_{n-1}(i))P_{ij}] }[/math] and since

[math]\displaystyle{ \sum_{\forall i}(\mu_{n-1}(i))P_{ij}=\sum_{\forall i}P(X_n=x_i)Pr(X_n=j|X_{n-1}=i)=\sum_{\forall i}P(X_n=x_i)\frac{Pr(X_n=j,X_{n-1}=i)}{P(X_n=x_i)} }[/math] [math]\displaystyle{ =\sum_{\forall i}Pr(X_n=j,X_{n-1}=i)=Pr(X_n=j)=\mu_{n}(j) }[/math]

Therefore,
[math]\displaystyle{ \frac{}{}\mu_{n-1}P=[\mu_{n}(1),\mu_{n}(2),...,\mu_{n}(i)]=\mu_{n} }[/math]

With this, it is possible to define [math]\displaystyle{ P(n) }[/math] as an n-step transition matrix where [math]\displaystyle{ \frac{}{}P_{ij}(n) = Pr(X_n=j|X_0=i) }[/math]
Theorem: [math]\displaystyle{ \frac{}{}\mu_n=\mu_0P^n }[/math]
Proof: [math]\displaystyle{ \frac{}{}\mu_n=\mu_{n-1}P }[/math] From the previous conclusion
[math]\displaystyle{ \frac{}{}=\mu_{n-2}PP=...=\mu_0\prod_{i = 1}^nP }[/math] And since this is a homogeneous Markov chain, [math]\displaystyle{ P }[/math] does not depend on [math]\displaystyle{ i }[/math] so
[math]\displaystyle{ \frac{}{}=\mu_0P^n }[/math]

From this it becomes easy to define the n-step transition matrix as [math]\displaystyle{ \frac{}{}P(n)=P^n }[/math]

Summary

• transition matrix [math]\displaystyle{ ( P ) }[/math] is an NxN when [math]\displaystyle{ N=|X| }[/math] matrix with [math]\displaystyle{ P_{ij}=Pr(X_1=j|X_0=i) }[/math] where [math]\displaystyle{ i,j \in X }[/math]
  
• n-step transition matrix [math]\displaystyle{ ( P_n ) }[/math] also NxN with [math]\displaystyle{ P_{ij}(n)=Pr(X_n=j|X_0=i) }[/math]
  
• marginal (probability of X) [math]\displaystyle{ \mu_n(i) = Pr(X_n=i) }[/math]
  
• Theorem: [math]\displaystyle{ P_n=P^n }[/math]
  
• Theorem: [math]\displaystyle{ \mu_n=\mu_0P^n }[/math]
  

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Definitions of different types of state sets

Define [math]\displaystyle{ i,j \in }[/math] State Space
If [math]\displaystyle{ P_{ij}(n) \gt 0 }[/math] for some [math]\displaystyle{ n }[/math] , then we say [math]\displaystyle{ i }[/math] reaches [math]\displaystyle{ j }[/math] denoted by [math]\displaystyle{ i\longrightarrow j }[/math]
This also mean j is accessible by i [math]\displaystyle{ j\longleftarrow i }[/math]
If [math]\displaystyle{ i\longrightarrow j }[/math] and [math]\displaystyle{ j\longrightarrow i }[/math] then we say [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] communicate, denoted by [math]\displaystyle{ i\longleftrightarrow j }[/math]

Theorems
1) [math]\displaystyle{ i\longleftrightarrow i }[/math]
2) [math]\displaystyle{ i\longleftrightarrow j,j\longleftrightarrow k\Rightarrow i\longleftrightarrow k }[/math]
3) The set of states of [math]\displaystyle{ X }[/math] can be written as a unique disjoint union of subsets [math]\displaystyle{ X=X_1 \bigcup X_2\bigcup... \bigcup X_k,k\gt 0 }[/math] where two states [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] communicate [math]\displaystyle{ IFF }[/math] they belong to the same subset

More Definitions
A set is Irreducible if all states communicate with each other ([math]\displaystyle{ \Leftrightarrow k=1 }[/math])
A subset of states (Class) is Closed if once you enter it, you can never leave. Alternatively, a subset of states (Class) is Open if it is possible to leave the Class.
An Absorbing set is a closed set with only 1 element