Markov Chain

Essential properties of Markov Chain
In the setup of MCMC algorithms, we construct Markov chains from a transition kernel KKK
, a conditional probability density such that Xn+1∼K(Xn,Xn+1)X_{n+1}\sim K(X_n,X_{n+1})Xn+1​∼K(Xn​,Xn+1​)
.
The chain encountered in MCMC settings enjoy a very strong stability property, namely a stationary probability distribution; that is, a distribution π\piπ
 such that if Xn∼πX_n\sim\piXn​∼π
, then Xn+1∼πX_{n+1}\sim \piXn+1​∼π
, if the kernel KKK
 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 n∈Nn\in\mathbb{N}n∈N
 such that P(Xn∈A∣X0)>0P(X_n\in A\mid X_0)>0P(Xn​∈A∣X0​)>0
 for every AAA
 such that π(A)>0\pi(A)>0π(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 AAA
 is infinite), or even Harris recurrent (that is, such that the probability of an infinite number of returns to AAA
 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}Xn+1​
 is π\piπ
 under the total variation norm, notwithstanding the initial value of X0X_0X0​
.
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)>0K(xn​,⋅)>0
 (or, equivalently, f(⋅∣xn)>0f(\cdot\mid x_n)>0f(⋅∣xn​)>0
) in a neighborhood of xnx_nxn​
.
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 fff
 to be a stationary measure associated with the transition kernel KKK
, 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 xxx
 and yyy
 in ∣x−y∣\vert x-y\vert∣x−y∣
 steps with probability
∏i=x∧yx∨y−1(M−iM)2 .\prod_{i=x\land y}^{x\vee y-1}\Big(\frac{M-i}{M}\Big)^2\,.i=x∧y∏x∨y−1​(MM−i​)2.
The Bernoulli-Laplace chain is aperiodic and even strongly aperiodic since the diagonal terms satisfy Pxx>0P_{xx}>0Pxx​>0
 for every x∈{0,…,K}x\in \{0,\ldots,K\}x∈{0,…,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)π=(π0​,…,πK​)
. From the equation πP=π\pi P = \piπP=π
,
π0=P00π0+P10π1π1=P01π1+P11π1+P21π2⋯=⋯πK=P(K−1)KπK−1+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}π0​π1​⋯πK​​=P00​π0​+P10​π1​=P01​π1​+P11​π1​+P21​π2​=⋯=P(K−1)K​πK−1​+PKK​πK​.​
Thus,
πk=(Kk)2π0 ,k=0,…,K ,\pi_k=\binom{K}{k}^2\pi_0\,,\qquad k=0,\ldots,K\,,πk​=(kK​)2π0​,k=0,…,K,
and through normalization,
πk=(Kk)2(2KK) ,\pi_k=\frac{\binom{K}{k}^2}{\binom{2K}{K}}\,,πk​=(K2K​)(kK​)2​,
by using Chu-Vandermonde identity​
(m+nr)=∑k=0r(mk)(nr−k)\binom{m+n}{r}=\sum_{k=0}^r\binom{m}{k}\binom{n}{r-k}(rm+n​)=k=0∑r​(km​)(r−kn​)
with m=n=r=Km=n=r=Km=n=r=K
. It turns out that the hypergeometric distribution H(2K,K,1/2)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=θXn−1+εn  ,θ∈I ⁣R ,X_n = \theta X_{n-1}+\varepsilon_n\;,\theta\in \mathrm{I\!R}\,,Xn​=θXn−1​+εn​,θ∈IR,
with εn∈N(0,σ2)\varepsilon_n\in N(0,\sigma^2)εn​∈N(0,σ2)
, and if the εn\varepsilon_nεn​
's are independent, XnX_nXn​
 is indeed independent from Xn−2,Xn−3,…X_{n-2},X_{n-3},\ldotsXn−2​,Xn−3​,…
 conditionally on Xn−1X_{n-1}Xn−1​
.
The Markovian properties of an AR(q) can be derived from (Xn,…,Xn−q+1)(X_n,\ldots,X_{n-q+1})(Xn​,…,Xn−q+1​)
.
ARMA(p, q) doesn't fit in the Markovian framework.
Since Xn∣xn−1∼N(θxn−1,σ2)X_n\mid x_{n-1}\sim N(\theta x_{n-1},\sigma^2)Xn​∣xn−1​∼N(θxn−1​,σ2)
, consider the lower bound of the transition kernel (θ>0\theta > 0θ>0
):
K(xn−1,xn)=12πexp⁡{−12σ2(xn−θxn−1)2}≥12πσexp⁡{−12σ2max⁡{(xn−θw‾)2,(xn−θwˉ)2}}≥12πσexp⁡{−12σ2[max⁡{−2θxnw‾,−2θxnwˉ}+xn2+θ2w‾2∧wˉ2]}={12πσexp⁡{−12σ2[−2θxnw‾+xn2+θ2w‾2∧wˉ2]}if xn>012πσexp⁡{−12σ2[−2θxnwˉ+xn2+θ2w‾2∧wˉ2]}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}K(xn−1​,xn​)​=2π​1​exp{−2σ21​(xn​−θxn−1​)2}≥2π​σ1​exp{−2σ21​max{(xn​−θw​)2,(xn​−θwˉ)2}}≥2π​σ1​exp{−2σ21​[max{−2θxn​w​,−2θxn​wˉ}+xn2​+θ2w​2∧wˉ2]}=⎩⎨⎧​2π​σ1​exp{−2σ21​[−2θxn​w​+xn2​+θ2w​2∧wˉ2]}2π​σ1​exp{−2σ21​[−2θxn​wˉ+xn2​+θ2w​2∧wˉ2]}​if xn​>0if xn​<0​,​
when xn−1∈[w‾,wˉ]x_{n-1}\in[\underline w, \bar w]xn−1​∈[w​,wˉ]
. The set C=[w‾,wˉ]C = [\underline w, \bar w]C=[w​,wˉ]
 is a small set, as the measure ν1\nu_1ν1​
 with density
exp⁡{(−x2+2θxw‾)/2σ2}Ix>0+exp⁡{(−x2+2θxwˉ)/2σ2}Ix<02πσ{[1−Φ(−θw‾/σ2)]exp⁡(θ2w‾2/2σ2)+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)\}}\,,2π​σ{[1−Φ(−θw​/σ2)]exp(θ2w​2/2σ2)+Φ(−θwˉ/σ)exp(θ2wˉ2/2σ2)}exp{(−x2+2θxw​)/2σ2}Ix>0​+exp{(−x2+2θxwˉ)/2σ2}Ix<0​​,
satisfy
K1(x,A)≥ν1(A),∀x∈C,∀A∈B(X) .K^1(x,A)\ge \nu_1(A),\qquad \forall x\in C, \forall A\in {\cal B(X)}\,.K1(x,A)≥ν1​(A),∀x∈C,∀A∈B(X).
Given that the transition kernel corresponds to the N(θxn−1,σ2)N(\theta x_{n-1},\sigma^2)N(θxn−1​,σ2)
 distribution, a normal distribution N(μ,τ2)N(\mu,\tau^2)N(μ,τ2)
 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\,.μ=θμandτ2=τ2θ2+σ2.
These conditions imply that μ=0\mu=0μ=0
 and that τ2=σ2/(1−θ2)\tau^2=\sigma^2/(1-\theta^2)τ2=σ2/(1−θ2)
, which can only occur for ∣θ∣<1\vert \theta\vert < 1∣θ∣<1
. In this case, N(0,σ2/(1−θ2))N(0,\sigma^2/(1-\theta^2))N(0,σ2/(1−θ2))
 is indeed the unique stationary distribution of the AR(1) chain.
Branching process
If ϕ\phiϕ
 doesn't have a constant term, i.e., P(X1=0)=0P(X_1=0)=0P(X1​=0)=0
, then chain StS_tSt​
 is necessarily transient since it is increasing.
If P(X1=0)>0P(X_1=0)>0P(X1​=0)>0
, the probability of a return to 0 at time ttt
 is ρ(t)=P(St=0)=gt(0)\rho(t)=P(S_t=0)=g_t(0)ρ(t)=P(St​=0)=gt​(0)
, which thus satisfies the recurrence equation ρt=ϕ(ρt−1)\rho_t=\phi(\rho_{t-1})ρt​=ϕ(ρt−1​)
. There exists a limit ρ\rhoρ
 different from 1, such that ρ=ϕ(ρ)\rho=\phi(\rho)ρ=ϕ(ρ)
, iff ϕ′(1)>1\phi'(1)>1ϕ′(1)>1
; namely if E[X]>1E[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+1∣St=0∼ϕS_{t+1}\mid S_t=0\sim\phiSt+1​∣St​=0∼ϕ
, it is easily shown that when ϕ′(1)>1\phi'(1)>1ϕ′(1)>1
, the number of returns to 0 follows a geometric distribution with parameter ρ\rhoρ
.
If ϕ′(1)≤1\phi'(1)\le 1ϕ′(1)≤1
, the chain is recurrent.
Simulation of Exp-Abs-xy
MC Approximation
Last modified 4yr ago