hamiltonian monto carlo principles details

• The HMC sampling procedure alternates between sampling the Gaussian momenta and letting the position of the particle evolve by integrating its Hamiltonian equations of motion. In most models, the latter cannot be integrated exactly, so the resulting position is used as a Metropo- lis proposal, with an acceptance probability that depends exponentially on the energy gained due to the numerical error.
因为进行离散化积分会导致一些误差，所以才产生了metropolis 进行误差纠正。要不然直接进行积分就可以解决。
• Several properties of Hamiltonian dynamics are crucial to its use in constructing Markov chain Monte Carlo updates.
构建马尔可夫链需要几个重要性质

• the acceptance probability is one if H kept invariant
当总能量始终不变的时候接受率始终为1

• For example, in a regression model with many predictor variables, the regression coefficients might be given Gaussian prior distributions, with mean of zero and a variance that is a hyperparameter. This hyperparameter could be given a broad prior distribution, so that its posterior distribution is determined mostly by the data.
这种sgima2 是只能取正值的然后全部采用HMC，但是可以进行分层，只对下层的分布抽样采用hmc.

• HMC variation

• leap frog method 分为三步，第一步和第三步是对称的，所以是reversible .

• Too short trajectories will cause a failure to suppress random walks and too long trajectories will be wasteful of computation

• This can usually be done with some experimentation, e.g., by monitoring the auto covariance function for parameters and increasing L until roughly independent samples are obtained
调节参数进行的办法 