Recycled-Momentum HMC Sampler (REC) — sampler

Recycled-Momentum HMC is a sampling algorithm that uses Hamiltonian Dynamics to approximate a posterior distribution. Unlike in standard HMC, proposals are autocorrelated, as the momentum of the current trajectory is not independent of the last trajectory, but is instead updated by a parameter alpha (see Details).

sampler_rec(
  start,
  distr_name = NULL,
  distr_params = NULL,
  epsilon = 0.5,
  L = 10,
  alpha = 0.1,
  iterations = 1024L,
  weights = NULL,
  custom_density = NULL
)

Arguments

start: Vector. Starting position of the sampler.
distr_name: Name of the distribution from which to sample from.
distr_params: Distribution parameters.
epsilon: Size of the leapfrog step
L: Number of leapfrog steps per iteration
alpha: Recycling factor, from -1 to 1 (see Details).
iterations: Number of iterations of the sampler.
weights: If using a mixture distribution, the weights given to each constituent distribution. If none given, it defaults to equal weights for all distributions.
custom_density: Instead of providing names, params and weights, the user may prefer to provide a custom density function.

Value

A named list containing

Samples: the history of visited places (an n x d x c array, n = iterations; d = dimensions; c = chain index, with c==1 being the 'cold chain')
Momentums: the history of momentum values (an n x d matrix, n = iterations; d = dimensions). Nothing is proposed in the first iteration (the first iteration is the start value) and so the first row is NA
Acceptance Ratio: The proportion of proposals that were accepted (for each chain).

Details

While in HMC the momentum in each iteration is an independent draw,, here the momentum of the last utterance $p^{n-1}$ is also involved. In each iteration, the momentum $p$ is obtained as follows $$p \gets \alpha \times p^{n-1} + (1 - \alpha^2)^{\frac{1}{2}} \times v$$; where $v \sim N(0, I)$.

Recycled-Momentum HMC does not support discrete distributions.

This algorithm has been used to model human data in Castillo et al. (2024)

References

Castillo L, León-Villagrá P, Chater N, Sanborn A (2024). “Explaining the Flaws in Human Random Generation as Local Sampling with Momentum.” PLOS Computational Biology, 20(1), 1–24. doi:10.1371/journal.pcbi.1011739 .

Examples


result <- sampler_rec(
    distr_name = "norm", distr_params = c(0,1), 
    start = 1, epsilon = .01, L = 100
)
cold_chain <- result$Samples