The HMC Algorithm with Overrelaxation and Adaptive--Step ...

P
Background
Aim of Talk
The Hamiltonian Monte Carlo (HMC) Algorithm
Improving Performance of HMC Algorithm
Numerical Experiments & Results
Practical Implementation
The Algorithm–Example Implementation
Algorithm
Initialize C (0)
for i = 1 to N − 1
Sample u ∼ U [0,1] and P ∗ ∼ N(0, I)
C 0 = C (i) and P 0 = P ∗ + ε 2 ∇V(C 0)
For l = 1 to L
P l = P l−1 − ε 2 ∇V(C l)
C l = C l−1 +εP l−1
P l = P l−1 − ε 2 ∇V(C l)
end For
dH = H(C L , P L ) − H(C (i) , P ∗ )
if u < min{1, exp(−dH)}
(C (i+1) , P (i+1) ) = (C L , P L )
else
(C (i+1) , P (i+1) ) = (C (i) , P (i) )
end for
return C = [C (1) , C (2) , ..., C (N−1) ]
Example
4
3
2
1
0
−1
−2
−3
−4
−4 −3 −2 −1 0 1 2 3 4
C
π(C) , Frequency
600
500
400
300
200
100
Truth
0
−4 −3 −2 −1 0 1 2 3 4
C , Bin Centers
Phase-space and distribution plots for 2D correlated
Gaussian distribution.
M. Alfaki, S. Subbey, and D. Haugland The Hamiltonian Monte Carlo (HMC) Algorithm