1 L`algorithme de Metropolis-Hastings

E V :E→R
x∈E V (x)E
E V
E x, y
x∼y
E V
V
X0∈E
β∈[0,+∞)
T∈N
t= 0 T−1
y Xt
V(y)< V (Xt)
Xt+1 =y
V(y)≥V(Xt)
Xt+1 =y e−β(V(y)−V(Xt))
XT
E
e−β(V(y)−V(Xt))
1Xt
y
x7→ V(x)
β= 0 β= +∞

E β > 0
P(Xt=x)t→+∞ →1
Zβ
e−βV (x), Zβ=X
z∈E
e−βV (z).
V
•E={1,2, . . . , k}k= 40
•i i −1i+ 1
•V(x) = cos(4πx/k)−p4πx/k
•β T = 2000 β= 0.4
t7→ V(Xt)V
(X1, Y1),...,(Xn, Yn)n
σ?Snn
σ?= argmaxσV(σ) := argmaxσ
n−1
X
i=1 k(Xσ(i+1), Yσ(i+1))−(Xσ(i), Yσ(i))k
Snσ∼σ0σ σ0
n= 4 1234 2134 1324 1243 1432 3214 4231 σ
n
2
n×2
i j
•
•T= 50000 β= 2
V(σ?) (Xi, Yi)
[0,1]2c V (σ?)∼c√n
c≈0.712