Runs one step of the Replica Exchange Monte Carlo

Replica Exchange Monte Carlo is a Markov chain Monte Carlo (MCMC) algorithm that is also known as Parallel Tempering. This algorithm performs multiple sampling with different temperatures in parallel, and exchanges those samplings according to the Metropolis-Hastings criterion. The K replicas are parameterized in terms of inverse_temperature's, (beta[0], beta[1], ..., beta[K-1]). If the target distribution has probability density p(x), the kth replica has density p(x)**beta_k.

mcmc_replica_exchange_mc(
  target_log_prob_fn,
  inverse_temperatures,
  make_kernel_fn,
  swap_proposal_fn = tfp$mcmc$replica_exchange_mc$default_swap_proposal_fn(1),
  state_includes_replicas = FALSE,
  seed = NULL,
  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.
inverse_temperatures	1D Tensor of inverse temperatures to perform samplings with each replica. Must have statically known shape. inverse_temperatures[0] produces the states returned by samplers, and is typically == 1.
make_kernel_fn	Function which takes target_log_prob_fn and seed args and returns a TransitionKernel instance.
swap_proposal_fn	function which take a number of replicas, and return combinations of replicas for exchange.
state_includes_replicas	Boolean indicating whether the leftmost dimension of each state sample should index replicas. If TRUE, the leftmost dimension of the current_state kwarg to tfp.mcmc.sample_chain will be interpreted as indexing replicas.
seed	integer to seed the random number generator.
name	string prefixed to Ops created by this function. Default value: NULL (i.e., "remc_kernel").

Value

list of next_state (Tensor or Python list of Tensors representing the state(s) of the Markov chain(s) at each result step. Has same shape as and current_state.) and kernel_results (collections$namedtuple of internal calculations used to 'advance the chain).

Details

Typically beta[0] = 1.0, and 1.0 > beta[1] > beta[2] > ... > 0.0.

• beta[0] == 1 ==> First replicas samples from the target density, p.

• beta[k] < 1, for k = 1, ..., K-1 ==> Other replicas sample from "flattened" versions of p (peak is less high, valley less low). These distributions are somewhat closer to a uniform on the support of p. Samples from adjacent replicas i, i + 1 are used as proposals for each other in a Metropolis step. This allows the lower beta samples, which explore less dense areas of p, to occasionally be used to help the beta == 1 chain explore new regions of the support. Samples from replica 0 are returned, and the others are discarded.

See also

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