Runs one step of Hamiltonian Monte Carlo.

Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) algorithm that takes a series of gradient-informed steps to produce a Metropolis proposal. This class implements one random HMC step from a given current_state. Mathematical details and derivations can be found in Neal (2011).

mcmc_hamiltonian_monte_carlo(
  target_log_prob_fn,
  step_size,
  num_leapfrog_steps,
  state_gradients_are_stopped = FALSE,
  step_size_update_fn = NULL,
  seed = NULL,
  store_parameters_in_results = FALSE,
  name = NULL
)

Arguments

target_log_prob_fn	Function which takes an argument like current_state (if it's a list current_state will be unpacked) and returns its (possibly unnormalized) log-density under the target distribution.
step_size	Tensor or list of Tensors representing the step size for the leapfrog integrator. Must broadcast with the shape of current_state. Larger step sizes lead to faster progress, but too-large step sizes make rejection exponentially more likely. When possible, it's often helpful to match per-variable step sizes to the standard deviations of the target distribution in each variable.
num_leapfrog_steps	Integer number of steps to run the leapfrog integrator for. Total progress per HMC step is roughly proportional to step_size * num_leapfrog_steps.
state_gradients_are_stopped	logical indicating that the proposed new state be run through tf$stop_gradient. This is particularly useful when combining optimization over samples from the HMC chain. Default value: FALSE (i.e., do not apply stop_gradient).
step_size_update_fn	Function taking current step_size (typically a tf$Variable) and kernel_results (typically collections$namedtuple) and returns updated step_size (Tensors). Default value: NULL (i.e., do not update step_size automatically).
seed	integer to seed the random number generator.
store_parameters_in_results	If TRUE, then step_size and num_leapfrog_steps are written to and read from eponymous fields in the kernel results objects returned from one_step and bootstrap_results. This allows wrapper kernels to adjust those parameters on the fly. This is incompatible with step_size_update_fn, which must be set to NULL.
name	string prefixed to Ops created by this function. Default value: NULL (i.e., 'hmc_kernel').

Value

a Monte Carlo sampling kernel

Details

The one_step function can update multiple chains in parallel. It assumes that all leftmost dimensions of current_state index independent chain states (and are therefore updated independently). The output of target_log_prob_fn(current_state) should sum log-probabilities across all event dimensions. Slices along the rightmost dimensions may have different target distributions; for example, current_state[0, :] could have a different target distribution from current_state[1, :]. These semantics are governed by target_log_prob_fn(current_state). (The number of independent chains is tf$size(target_log_prob_fn(current_state)).)

References

• Radford Neal. MCMC Using Hamiltonian Dynamics. Handbook of Markov Chain Monte Carlo, 2011.

• Bernard Delyon, Marc Lavielle, Eric, Moulines. Convergence of a stochastic approximation version of the EM algorithm, Ann. Statist. 27 (1999), no. 1, 94--128.

See also

Other mcmc_kernels: mcmc_dual_averaging_step_size_adaptation(), mcmc_metropolis_adjusted_langevin_algorithm(), mcmc_metropolis_hastings(), mcmc_no_u_turn_sampler(), mcmc_random_walk_metropolis(), mcmc_replica_exchange_mc(), mcmc_simple_step_size_adaptation(), mcmc_slice_sampler(), mcmc_transformed_transition_kernel(), mcmc_uncalibrated_hamiltonian_monte_carlo(), mcmc_uncalibrated_langevin(), mcmc_uncalibrated_random_walk()