Overview Applied to MCMC: given a target distribution p(x HMC provides proposals not for x but the augmented random variable z=(x, v) with stationary distribution p(z)a exp (H(z)), where H(z==logp(x)+v M1/2. note that p(x=p(r, so samples of z will provide correct samples of x For practice: simulate the hamiltonian dynamics by some discrete integrator(e. g. leap frog that keeps some certain properties of the hamilton dynamics(e.g. symplectic symmetric consistent) correct the discretization error by mh testOverview • Applied to MCMC: given a target distribution 𝑝 𝑥 , HMC provides proposals not for 𝑥 but the augmented random variable 𝑧 = 𝑥, 𝑣 with stationary distribution 𝑝෤ 𝑧 ∝ exp −𝐻 𝑧 , where 𝐻 𝑧 = −log 𝑝 𝑥 + 𝑣 ⊤𝑀𝑣/2. Note that 𝑝෤ 𝑥 = 𝑝 𝑥 , so samples of 𝑧 will provide correct samples of 𝑥. • For practice: simulate the Hamiltonian dynamics by some discrete integrator (e.g. leap frog) that keeps some certain properties of the Hamilton dynamics (e.g. symplectic, symmetric, consistent); correct the discretization error by MH test