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Spherical Hamiltonian Monte Carlo for Constrained Target Distributions Shiwei Lan∗, Bo Zhou∗ , Babak Shahbaba† January 8, 2014 Supplementary: geodesic on a sphere To find the geodesic on a sphere, we need to solve the following equations: θ̇ = v v̇ = −v T Γv (1) (2) for which we need to calculate the Christoffel symbols, Γ, first. Note that the (i, j)-th element of GS is 4 2 2 . + 2θi θj θk /θD+1 , and the (i, j, k)-th element of dGS is gij,k = (δik θj + θi δjk )/θD+1 gij = δij + θi θj /θD+1 Therefore 1 Γkij = g kl [glj,i + gil,j − gij,l ] 2 2 = θk [δij + θi θj /θD+1 ] = [GS ⊗ θ]ijk Using these results, we can write equation (2) as v̇ = −v T GS vθ = −kṽk22 θ. Further, we have p θ̇D+1 = dtd 1 − kθk22 = vD+1 v̇D+1 = T = −kṽk22 θD+1 v − dtd θθD+1 Therefore, we can rewrite the geodesic equations (1)(2) as θ̃˙ = ṽ ṽ˙ = −kṽk22 θ̃ Multiplying both sides of Equation (4) by ṽ T to obtain differential equations as follows: θ̃(t) = θ̃(0) cos(kṽ(0)k2 t) + dkṽk22 dt (3) (4) = 0, we can solve the above system of ṽ(0) sin(kṽ(0)k2 t) kṽ(0)k2 ṽ(t) = −θ̃(0)kṽ(0)k2 sin(kṽ(0)k2 t) + ṽ(0) cos(kṽ(0)k2 t) ∗ † Department of Statistics, University of California, Irvine, USA. Department of Statistics and Department of Computer Science, University of California, Irvine, USA. 1