030507 - Lecture 12 Recall Metropolis Algorithm S big...

Lecture 12: 03/05/2007
Recall:
Metropolis Algorithm
S: big discrete set of states
A probability distribution on S: ( )∝ ∈( ) p x e x kT
Algorithm constructs a Markov Chain using “proposal followed by accept/reject.”
Start with symmetric Q
ij
; Q
ij
=prob{given I, propose j}
Accept j as next state with probability ( , )= < ( ) -( - ) > ( ) α i j 1 if ϵj ϵ i e ϵj ϵi kT if ϵj ϵ i
Overall transition probabilities for Markov chain are: P(I,j)=Q(I,j)α(I,j)
Recall detailed balance ideas given a probabilitiy distribution π on S, their πis stationary distribution for
MC with transition probs P(I,j) if
, = , , πiPi j Pj iπjfor all i j
(Detailed balance conditions)
Claim : with P(I,j)= Q(I,j)α(I,j), defined as above, stationary distribution is precisely π
*
, where
*= - ∈ - πi e ϵikTj Se ϵjkT
Ie, “π * =P”
Proof : Check detailed balance conditions:
Case 1: > ( ) ϵi ϵ j . Then P(I,j)=Q(I,j) and P(j,i)=Q(j,i) -( - ) e ϵi ϵj kT Since Q(j,i)=Q(I,j), this says
P(I,j)=Q(I,j)=Q(j,i)=P(j,i) ( - ) e ϵi ϵj kT
Note: ( - ) = ( ) ( )= * * e ϵi ϵj kT p j p i πj πi
Hence: * , = , * πi Pi j Pj iπj
Bottom Line : π * , the Boltzmann distribution, is the (unique*) stationary distribution for Markov chain
(easy to argue similarly when > . ϵj ϵi
*unique because, by construction, the Markov chain is generic -every state positively recurrent, only one
recurrence class, hence only one stationary distribution.