sample_unbiasedestimator: Unbiased MCMC estimators

In pierrejacob/debiasedmcmc: Unbiased MCMC with couplings

Unbiased MCMC estimators

Description

Sample two Markov chains, each following 'single_kernel' marginally, and 'coupled_kernel' jointly, until min(max(tau, m), max_iterations), where tau is the first time the two chains meet (the "meeting time"). An unbiased estimator of the expectation of a test function h is computed on the fly, between step k and step m, and returned.

Allows for an arbitrary lag, i.e. X_t = Y_t-lag, and lag is one by default.

Compared to sample_coupled_chains this function requires specifying the test function, but does not record the trajectories, and thus is memory-light.

If you're only interested in sampling meeting times, see sample_meetingtime.

Usage

sample_unbiasedestimator(
  single_kernel,
  coupled_kernel,
  rinit,
  h = function(x) x,
  k = 0,
  m = 1,
  lag = 1,
  max_iterations = Inf
)

Arguments

single_kernel	A list taking a state and returning a state, performing one step of a Markov kernel
coupled_kernel	A list taking two states and returning two states, performing one step of a coupled Markov kernel; it also returns a boolean "identical" indicating whether the two states are identical.
rinit	A list representing the initial state of the chain, that can be given to 'single_kernel'
h	A test function of interest, which should take a chain state ("chain_state" entry of the output of "rinit", for instance) and return a numeric vector
k	An integer at which to start computing the unbiased estimator
m	A time horizon: the chains are sampled until the maximum between m and the meeting time
lag	A time lag, equal to one by default
max_iterations	A maximum number of iterations, at which to interrup the while loop; Inf by default

Value

A list with

• mcmcestimator: an MCMC estimator computed on the first chain, from step k to m

• correction: the bias correction term

• uestimator: unbiased estimator, equal to the sum of mcmcestimator and correction

• meetingtime: the meeting time; equal to Inf if while loop was interrupted

• iteration: final iteration; could be equal to m, to meetingtime, or to max_iterations

• elapsedtime: elapsed wall-clock time, in seconds

• cost: computing cost in terms of calls to Markov kernels (counting coupled kernel as twice the cost)

pierrejacob/debiasedmcmc documentation built on Aug. 22, 2022, 12:41 a.m.