Runs one step of Metropolis-adjusted Langevin algorithm.

Metropolis-adjusted Langevin algorithm (MALA) is a Markov chain Monte Carlo (MCMC) algorithm that takes a step of a discretised Langevin diffusion as a proposal. This class implements one step of MALA using Euler-Maruyama method for a given current_state and diagonal preconditioning volatility matrix.

mcmc_metropolis_adjusted_langevin_algorithm(
  target_log_prob_fn,
  step_size,
  volatility_fn = NULL,
  seed = NULL,
  parallel_iterations = 10,
  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.
volatility_fn	function which takes an argument like current_state (or *current_state if it's a list) and returns volatility value at current_state. Should return a Tensor or list of Tensors that must broadcast with the shape of current_state. Defaults to the identity function.
seed	integer to seed the random number generator.
parallel_iterations	the number of coordinates for which the gradients of the volatility matrix volatility_fn can be computed in parallel.
name	String prefixed to Ops created by this function. Default value: NULL (i.e., 'mala_kernel').

Details

Mathematical details and derivations can be found in Roberts and Rosenthal (1998) and Xifara et al. (2013).

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 reduce 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

• Gareth Roberts and Jeffrey Rosenthal. Optimal Scaling of Discrete Approximations to Langevin Diffusions. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60: 255-268, 1998.

• T. Xifara et al. Langevin diffusions and the Metropolis-adjusted Langevin algorithm. arXiv preprint arXiv:1309.2983, 2013.

See also

Other mcmc_kernels: mcmc_dual_averaging_step_size_adaptation(), mcmc_hamiltonian_monte_carlo(), 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()