MCMC基本概念以及简单范例
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ISyE8843A, Brani Vidakovic Handout 10
1 MCMC Methodology.
Independence of X
1
, . . . , X
n
is not critical for an approximation of the form E
θ|x
h(X) =
1
n
P
n
i=1
h(X
i
), X
i
∼
π(θ|x). In fact, when X’s are dependent, the ergodic theorems describe the approximation.
An easy and convenient form of dependence is Markov chain dependence. The Markov dependence is
perfect for computer simulations since for producing a future realization of the chain, only the current state
is needed.
1.1 Theoretical Background and Notation
Random variables X
1
, X
2
, . . . , X
n
, . . . constitute a Markov Chain on continuous state space if they possess
a Markov property,
P (X
n+1
∈ A|X
1
, . . . , X
n
) = P (X
n+1
∈ A|X
1
, . . . , X
n
) = Q(X
n
, A) = Q(A|X
n
),
for some probability distribution Q. Typically, Q is assumed a time-homogeneous, i.e., independent on n
(“time”). The transition (from the state n to the state n + 1) kernel defines a probability measure on the state
space and we will assume that the density q exists, i.e.,
Q(A|X
n
= x) =
Z
A
q(x, y)dy =
Z
A
q(y|x)dy.
Distribution Π is invariant, if for all measurable sets A
Π(A) =
Z
Q(A|x)Π(dx).
If the transition density π exists, it is stationary if q(x|y)π(y) = q(y|x)π(x). Here and in the sequel we
assume that the density for Π exists, Π(A) =
R
A
π(x)dx.
A distribution Π is an equilibrium distribution if for Q
n
(A|x) = P (X
n
∈ A|X
0
= x),
lim
n→∞
Q
n
(A|x) = Π(A).
In plain terms, the Markov chain will forget the initial distribution and will converge to the stationary distri-
bution.
The Markov Chain is irreducible if for each A for which Π(A) > 0, and for each x, one can find n, so
that Q
n
(A|x) > 0.
The Markov Chain X
1
, . . . , X
n
, . . . is recurrent if for each B such that Π(B) > 0,
P (X
n
∈ B i.o.|X
0
= x) = 1, a.s.(in distribution of X
0
)
It is Harris recurrent if P (X
n
∈ B i.o.|X
0
= x) = 1, (∀x). The acronym i.o. stands for infinitely often.
1
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