Markov Chain

Essential properties of Markov Chain

In the setup of MCMC algorithms, we construct Markov chains from a transition kernel KK, a conditional probability density such that Xn+1K(Xn,Xn+1)X_{n+1}\sim K(X_n,X_{n+1}).

The chain encountered in MCMC settings enjoy a very strong stability property, namely a stationary probability distribution; that is, a distribution π\pi such that if XnπX_n\sim\pi, then Xn+1πX_{n+1}\sim \pi, if the kernel KK allows for free moves all over the state space. This freedom is called irreducibility in the theory of Markov chains and is formalized as the existence of nNn\in\mathbb{N} such that P(XnAX0)>0P(X_n\in A\mid X_0)>0 for every AA such that π(A)>0\pi(A)>0. This property also ensures that most of the chains involved in MCMC algorithms are recurrent (that is, that the average number of visits to an arbitrary set AA is infinite), or even Harris recurrent (that is, such that the probability of an infinite number of returns to AA is 1).

Harris recurrence ensures that the chain has the same limiting behavior for every starting value instead of almost every starting value.

The stationary distribution is also a limiting distribution in the sense that the limiting distribution of Xn+1X_{n+1} is π\pi under the total variation norm, notwithstanding the initial value of X0X_0.

Strong forms of convergence are also encountered in MCMC settings, like geometric and uniform convergences.

If the marginals are proper, for convergence we only need our chain to be aperiodic. A sufficient condition is that K(xn,)>0K(x_n,\cdot)>0 (or, equivalently, f(xn)>0f(\cdot\mid x_n)>0) in a neighborhood of xnx_n.

If the marginal are not proper, or if they do not exist, then the chain is not positive recurrent. It is either null recurrent, and both cases are bad.

The detailed balance condition is not necessary for ff to be a stationary measure associated with the transition kernel KK, but it provides a sufficient condition that is often easy to check and that can be used for most MCMC algorithms.

Ergodicity: independence of initial conditions

  • geometrically h-ergodic: decreasing at least at a geometric speed.

  • uniform ergodicity: stronger than geometric ergodicity in the sense that the rate of geometric convergence must be uniform over the whole space.

Irreducibility + Aperiodic = Ergodicity ?

Basic Notions

Bernoulli-Laplace Model

The finite chain is indeed irreducible since it is possible to connect the status xx and yy in xy\vert x-y\vert steps with probability

i=xyxy1(MiM)2.\prod_{i=x\land y}^{x\vee y-1}\Big(\frac{M-i}{M}\Big)^2\,.

The Bernoulli-Laplace chain is aperiodic and even strongly aperiodic since the diagonal terms satisfy Pxx>0P_{xx}>0 for every x{0,,K}x\in \{0,\ldots,K\}.

Given the quasi-diagonal shape of the transition matrix, it is possible to directly determine the invariant distribution, π=(π0,,πK)\pi=(\pi_0,\ldots,\pi_K). From the equation πP=π\pi P = \pi,

π0=P00π0+P10π1π1=P01π1+P11π1+P21π2=πK=P(K1)KπK1+PKKπK.\begin{aligned} \pi_0 &= P_{00}\pi_0 + P_{10}\pi_1\\ \pi_1 &= P_{01}\pi_1 + P_{11}\pi_1 + P_{21}\pi_2\\ \cdots &=\cdots\\ \pi_K &= P_{(K-1)K}\pi_{K-1} + P_{KK}\pi_K\,. \end{aligned}

Thus,

πk=(Kk)2π0,k=0,,K,\pi_k=\binom{K}{k}^2\pi_0\,,\qquad k=0,\ldots,K\,,

and through normalization,

πk=(Kk)2(2KK),\pi_k=\frac{\binom{K}{k}^2}{\binom{2K}{K}}\,,

by using Chu-Vandermonde identity

(m+nr)=k=0r(mk)(nrk)\binom{m+n}{r}=\sum_{k=0}^r\binom{m}{k}\binom{n}{r-k}

with m=n=r=Km=n=r=K. It turns out that the hypergeometric distribution H(2K,K,1/2)H(2K,K,1/2) is the invariant distribution for the Bernoulli-Laplace model.

AR(1) Models

A simple illustration of Markov chains on continuous state-space.

Xn=θXn1+εn,θIR,X_n = \theta X_{n-1}+\varepsilon_n\;,\theta\in \mathrm{I\!R}\,,

with εnN(0,σ2)\varepsilon_n\in N(0,\sigma^2), and if the εn\varepsilon_n's are independent, XnX_n is indeed independent from Xn2,Xn3,X_{n-2},X_{n-3},\ldots conditionally on Xn1X_{n-1}.

  • The Markovian properties of an AR(q) can be derived from (Xn,,Xnq+1)(X_n,\ldots,X_{n-q+1}).

  • ARMA(p, q) doesn't fit in the Markovian framework.

Since Xnxn1N(θxn1,σ2)X_n\mid x_{n-1}\sim N(\theta x_{n-1},\sigma^2), consider the lower bound of the transition kernel (θ>0\theta > 0):

K(xn1,xn)=12πexp{12σ2(xnθxn1)2}12πσexp{12σ2max{(xnθw)2,(xnθwˉ)2}}12πσexp{12σ2[max{2θxnw,2θxnwˉ}+xn2+θ2w2wˉ2]}={12πσexp{12σ2[2θxnw+xn2+θ2w2wˉ2]}if xn>012πσexp{12σ2[2θxnwˉ+xn2+θ2w2wˉ2]}if xn<0,\begin{aligned} K(x_{n-1},x_n) &= \frac{1}{\sqrt{2\pi}}\exp\Big\{-\frac{1}{2\sigma^2}(x_n-\theta x_{n-1})^2\Big\}\\ &\ge \frac{1}{\sqrt{2\pi}\sigma}\exp\Big\{-\frac{1}{2\sigma^2}\max\{(x_n-\theta \underline w)^2, (x_n-\theta \bar w)^2\}\Big\}\\ &\ge \frac{1}{\sqrt{2\pi}\sigma}\exp\Big\{-\frac{1}{2\sigma^2}\Big[ \max\{-2\theta x_n\underline w,-2\theta x_n\bar w\}+x_n^2 + \theta^2\underline w^2\land \bar w^2 \Big]\Big\}\\ &=\begin{cases} \frac{1}{\sqrt{2\pi}\sigma}\exp\Big\{-\frac{1}{2\sigma^2}\Big[ -2\theta x_n\underline w+x_n^2 + \theta^2\underline w^2\land \bar w^2 \Big]\Big\}& \text{if }x_n>0\\ \frac{1}{\sqrt{2\pi}\sigma}\exp\Big\{-\frac{1}{2\sigma^2}\Big[ -2\theta x_n\bar w+x_n^2 + \theta^2\underline w^2\land \bar w^2 \Big]\Big\}& \text{if }x_n<0 \end{cases}\,, \end{aligned}

when xn1[w,wˉ]x_{n-1}\in[\underline w, \bar w]. The set C=[w,wˉ]C = [\underline w, \bar w] is a small set, as the measure ν1\nu_1 with density

exp{(x2+2θxw)/2σ2}Ix>0+exp{(x2+2θxwˉ)/2σ2}Ix<02πσ{[1Φ(θw/σ2)]exp(θ2w2/2σ2)+Φ(θwˉ/σ)exp(θ2wˉ2/2σ2)},\frac{\exp\{(-x^2+2\theta x\underline w)/2\sigma^2\}I_{x>0} + \exp\{(-x^2+2\theta x\bar w)/2\sigma^2\}I_{x<0} }{\sqrt{2\pi}\sigma\{[1-\Phi(-\theta\underline w/\sigma^2)]\exp(\theta^2\underline w^2/2\sigma^2)+\Phi(-\theta\bar w/\sigma)\exp(\theta^2\bar w^2/2\sigma^2)\}}\,,

satisfy

Given that the transition kernel corresponds to the N(θxn1,σ2)N(\theta x_{n-1},\sigma^2) distribution, a normal distribution N(μ,τ2)N(\mu,\tau^2) is stationary for the AR(1) chain only if

μ=θμandτ2=τ2θ2+σ2.\mu=\theta\mu\qquad\text{and}\qquad \tau^2=\tau^2\theta^2+\sigma^2\,.

These conditions imply that μ=0\mu=0 and that τ2=σ2/(1θ2)\tau^2=\sigma^2/(1-\theta^2), which can only occur for θ<1\vert \theta\vert < 1. In this case, N(0,σ2/(1θ2))N(0,\sigma^2/(1-\theta^2)) is indeed the unique stationary distribution of the AR(1) chain.

Branching process

  • If ϕ\phi doesn't have a constant term, i.e., P(X1=0)=0P(X_1=0)=0, then chain StS_t is necessarily transient since it is increasing.

  • If P(X1=0)>0P(X_1=0)>0, the probability of a return to 0 at time tt is ρ(t)=P(St=0)=gt(0)\rho(t)=P(S_t=0)=g_t(0), which thus satisfies the recurrence equation ρt=ϕ(ρt1)\rho_t=\phi(\rho_{t-1}). There exists a limit ρ\rho different from 1, such that ρ=ϕ(ρ)\rho=\phi(\rho), iff ϕ(1)>1\phi'(1)>1; namely if E[X]>1E[X]>1. The chain is thus transient when the average number of siblings per individual is larger than 1. If there exists a restarting mechanism in 0, St+1St=0ϕS_{t+1}\mid S_t=0\sim\phi, it is easily shown that when ϕ(1)>1\phi'(1)>1, the number of returns to 0 follows a geometric distribution with parameter ρ\rho.

  • If ϕ(1)1\phi'(1)\le 1, the chain is recurrent.