Runs one step of the Metropolis-Hastings algorithm.

Source: R/mcmc-kernels.R

The Metropolis-Hastings algorithm is a Markov chain Monte Carlo (MCMC) technique which uses a proposal distribution to eventually sample from a target distribution.

mcmc_metropolis_hastings(inner_kernel, seed = NULL, name = NULL)

Arguments

inner_kernel

TransitionKernel-like object which has collections$namedtuple kernel_results and which contains a target_log_prob member and optionally a log_acceptance_correction member.
seed

integer to seed the random number generator.
name

string prefixed to Ops created by this function. Default value: NULL (i.e., "mh_kernel").

Value

a Monte Carlo sampling kernel

Details

Note: inner_kernel$one_step must return kernel_results as a collections$namedtuple which must:

  • have a target_log_prob field,

  • optionally have a log_acceptance_correction field, and,

  • have only fields which are Tensor-valued.

The Metropolis-Hastings log acceptance-probability is computed as:

log_accept_ratio = (current_kernel_results.target_log_prob
                   - previous_kernel_results.target_log_prob
                   + current_kernel_results.log_acceptance_correction)

If current_kernel_results$log_acceptance_correction does not exist, it is presumed 0 (i.e., that the proposal distribution is symmetric). The most common use-case for log_acceptance_correction is in the Metropolis-Hastings algorithm, i.e.,

accept_prob(x' | x) = p(x') / p(x) (g(x|x') / g(x'|x))
where,
p  represents the target distribution,
g  represents the proposal (conditional) distribution,
x' is the proposed state, and,
x  is current state

The log of the parenthetical term is the log_acceptance_correction. The log_acceptance_correction may not necessarily correspond to the ratio of proposal distributions, e.g, log_acceptance_correction has a different interpretation in Hamiltonian Monte Carlo.

See also

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