Synchrotron-based high-pressure research in materials science Page: 3 of 5
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In Proc. Third Int. Symp. on Sensitivity Analysis of Model Output (2001)
P (t +-' = pi(t ) -a x(t )] (2)
pt(t+-)
xi(t+')=x (t+r)+2 2 (3)
The parameter t represents an increment in time. After m leapfrog steps corresponding to a
total trajectory time of T = mt, a Metropolis acceptance/rejection decision is made to guaran-
tee that the sequence is in statistical equilibrium with the target pdf. Clearly large steps in the
parameter space are possible with only a few evaluations of T and the gradient of T. Note that
the gradient of T can often be done in a time comparable to the (forward) calculation of p by
applying adjoint differentiation to the computer code used to calculate p [5]. In practice, the
length of the Hamiltonian trajectories must be randomised to realise adequate sampling of
2(x). Once an MCMC sequence has been generated, the properties of n(x) may be character-
ised by considering just the x samples. The momentum aspects of the extended pdf, exp(-H),
are marginalised out because they are independent of the x dependence.
3. RESULTS
Figure 1 shows typical paths followed by the Hamiltonian MCMC algorithm for a one-
dimensional target pdf, which is a Gaussian with unit standard deviation. The vertical jumps
correspond to the Gibbs sampling of momentum from the Gaussian pdf, exp(-p2), for unity
mass. The circular arcs correspond to the trajectories of constant H followed in five steps of
the leapfrog method using t = 0.4, yielding a total trajectory length of T = 5t = 2.
4
Figure 1. Example of several trajectories
in the momentum-parameter space for the
Hamiltonian method for a 1D Gaussian 2
distribution. For each trajectory, the E
momentum is drawn from the assumed
Gaussian momentum distribution (verti- E
cal jumps), which is followed by several
steps along a trajectory of constant
Hamiltonian value (circular paths). -2
-4 . .
-4 -2 0 2 4
Parameter
Figure 2 shows the behaviour of the Hamiltonian method for an asymmetric two-
dimensional Gaussian distribution with a standard deviation of four in one direction and unity
in the other. For this example, the maximum value for t is 0.4. The total length of each
Hamiltonian trajectory is randomly chosen from a distribution that is uniform from 0 to Tmax =2
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Synchrotron-based high-pressure research in materials science, article, Date Unknown; [Los Alamos, New Mexico]. (https://digital.library.unt.edu/ark:/67531/metadc933531/m1/3/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.