Metropolis-coupled MCMC sampler (MC3) — sampler

This sampler is a variant of MH in which multiple parallel chains are run at different temperatures. The chains stochastically swap positions which allows the coldest chain to visit regions far from its starting point (unlike in MH). Because of this, an MC3 sampler can explore far-off regions, whereas an MH sampler may become stuck in a particular point of high density.

sampler_mc3(
  start,
  distr_name = NULL,
  distr_params = NULL,
  sigma_prop = NULL,
  nChains = 6,
  delta_T = 4,
  swap_all = TRUE,
  iterations = 1024L,
  weights = NULL,
  custom_density = NULL,
  alpha = 0
)

Arguments

start: Either a vector or a matrix. If it is a vector, it will be the starting point of all the chains (with length = number of dimensions). If it's a matrix, every row will be the starting point of one chain (and so it must have as many rows as nChains, and as many columns as number of dimensions in the space).
distr_name: Name of the distribution from which to sample from.
distr_params: Distribution parameters.
sigma_prop: Covariance matrix of the proposal distribution. If sampling in 1D space, it can be instead a number.
nChains: Number of chains to run.
delta_T: numeric, >1. Temperature increment parameter. The bigger this number, the steeper the increase in temperature between the cold chain and the next chain
swap_all: Boolean. If true, every iteration attempts floor(nChains / 2) swaps. If false, only one swap per iteration.
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.
alpha: autocorrelation of proposals parameter, from -1 to 1, with 0 being independent proposals

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')
Proposals: the history of proposed places (an n x d x c array, n = iterations; d = dimensions; c = chain index, with c==1 being the 'cold chain'). 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).
Beta Values: The set of temperatures used in each chain
Swap History: the history of chain swaps
Swap Acceptance Ratio: The ratio of swap acceptances

Details

This algorithm has been used to model human data in Castillo et al. (2024) , Zhu et al. (2022) and Zhu et al. (2018) among others.

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 .

Zhu J, León-Villagrá P, Chater N, Sanborn AN (2022). “Understanding the Structure of Cognitive Noise.” PLoS Computational Biology, 18(8), e1010312. doi:10.1371/journal.pcbi.1010312 .

Zhu J, Sanborn AN, Chater N (2018). “Mental Sampling in Multimodal Representations.” Advances in Neural Information Processing Systems, 31, 5748–5759.

Examples


# Sample from a normal distribution
result <- sampler_mc3(
    distr_name = "norm", distr_params = c(0,1), 
    start = 1, sigma_prop = diag(1)
)
cold_chain <- result$Samples[,,1]