THE GEOMETRIC FOUNDATIONS OF HAMILTONIAN MONTE CARLO 7 where f is the density of with respect to the lebesgue measure on R".When targeting complex distributions either cor the support of the indicator will be small and the resulting translations barely perturb the initial state. EXAMPLE 2.The random scan Gibbs sampler is induced by axis-aligned translations, tn:9→P() i~U1,,n} n~00,1刂. where is the cumulative distribution function of the ith conditional measure.When the target distribution is strongly correlated,the conditional measures concentrate near the initial q and,as above,the translations are stunted. ) are not li e sions (Okser mple,yield measure-preserving maps tha cross the entire target distribut ion.Unfortunately that diffusion tends to expand across the target measures only slowly(Figure 2):for any finite diffu on time the resulting Langevin kernels are localized around the initial point(Figure 3).What we need are more coherent maps that avoid such diffusive behavior One potential candidate for coherent maps are flows.A flow,is a family of isomor- phisms parameterized by a time,t, o:Q→Q,teR, that form a one-dimensional Lie group on composition, =s+t 1=6t where Ido is the natural identity map on Q.Because the inverse of a map is given only by negating t,as the time is increased the resulting pushes points away from their initial positions and avoids localized exploration (Figure 4).Our final obstacle is in engineering a flow comprised of measure-preserving maps. Flows are particularly natural on the smooth manifolds of differential geometry,and flows that preserve a given target measure can be engineered on one exceptional class of smooth manifolds known as symplectic manifolds.If we can understand these manifolds THE GEOMETRIC FOUNDATIONS OF HAMILTONIAN MONTE CARLO 7 where f is the density of ̟ with respect to the Lebesgue measure on R n . When targeting complex distributions either ǫ or the support of the indicator will be small and the resulting translations barely perturb the initial state. Example 2. The random scan Gibbs sampler is induced by axis-aligned translations, ti,η : qi → P −1 i (η) i ∼ U{1, . . . , n} η ∼ U[0, 1] , where Pi(qi) = Z qi ∞ ̟(d˜qi |q) is the cumulative distribution function of the ith conditional measure. When the target distribution is strongly correlated, the conditional measures concentrate near the initial q and, as above, the translations are stunted. In order to define a Markov kernel that remains efficient in difficult problems we need measure-preserving maps whose domains are not limited to local exploration. Realizations of Langevin diffusions (Øksendal, 2003), for example, yield measure-preserving maps that diffuse across the entire target distribution. Unfortunately that diffusion tends to expand across the target measures only slowly (Figure 2): for any finite diffusion time the resulting Langevin kernels are localized around the initial point (Figure 3). What we need are more coherent maps that avoid such diffusive behavior. One potential candidate for coherent maps are flows. A flow, {φt}, is a family of isomor￾phisms parameterized by a time, t, φt : Q → Q, ∀t ∈ R, that form a one-dimensional Lie group on composition, φt ◦ φs = φs+t φ −1 t = φ−t φ0 = IdQ, where IdQ is the natural identity map on Q. Because the inverse of a map is given only by negating t, as the time is increased the resulting φt pushes points away from their initial positions and avoids localized exploration (Figure 4). Our final obstacle is in engineering a flow comprised of measure-preserving maps. Flows are particularly natural on the smooth manifolds of differential geometry, and flows that preserve a given target measure can be engineered on one exceptional class of smooth manifolds known as symplectic manifolds. If we can understand these manifolds