marginalMPT: Marginal Likelihood for Simple MPT

Description

Computes the marginal likelihood for simple (fixed-effects, nonhierarchical) MPT models.

Usage

marginalMPT(
  eqnfile,
  data,
  restrictions,
  alpha = 1,
  beta = 1,
  dataset = 1,
  method = "importance",
  posterior = 500,
  mix = 0.05,
  scale = 0.9,
  samples = 10000,
  batches = 10,
  show = TRUE,
  cores = 1
)

Arguments

eqnfile: The (relative or full) path to the file that specifies the MPT model (standard .eqn syntax). Note that category labels must start with a letter (different to multiTree) and match the column names of data. Alternatively, the EQN-equations can be provided within R as a character value (cf. readEQN). Note that the first line of an .eqn-file is reserved for comments and always ignored.
data: The (relative or full) path to the .csv file with the data (comma separated; category labels in first row). Alternatively: a data frame or matrix (rows=individuals, columns = individual category frequencies, category labels as column names)
restrictions: Specifies which parameters should be (a) constant (e.g., "a=b=.5") or (b) constrained to be identical (e.g., "Do=Dn") or (c) treated as fixed effects (i.e., identical for all participants; "a=b=FE"). Either given as the path to a text file with restrictions per row or as a list of restrictions, e.g., list("D1=D2","g=0.5"). Note that numbers in .eqn-equations (e.g., d*(1-g)*.50) are directly interpreted as equality constraints.
alpha: first shape parameter(s) for the beta prior-distribution of the MPT parameters \(\theta_s\) (can be a named vector to use a different prior for each MPT parameter)
beta: second shape parameter(s)
dataset: for which data set should Bayes factors be computed?
method: either "importance" (importance sampling using a mixture of uniform and beta-aproximation of the posterior) or "prior" (brute force Monte Carlo sampling from prior)
posterior: number of posterior samples used to approximate importance-sampling densities (i.e., beta distributions)
mix: mixture proportion of the uniform distribution for the importance-sampling density
scale: how much should posterior-beta approximations be downscaled to get fatter importance-sampling density
samples: total number of samples from parameter space
batches: number of batches. Used to compute a standard error of the estimate.
show: whether to show progress
cores: number of CPUs used

Details

Currently, this is only implemented for a single data set!

If method = "prior", a brute-force Monte Carlo method is used and parameters are directly sampled from the prior.Then, the likelihood is evaluated for these samples and averaged (fast, but inefficient).

Alternatively, an importance sampler is used if method = "importance", and the posterior distributions of the MPT parameters are approximated by independent beta distributions. Then each parameter \(s\) is sampled from the importance density:

\(mix*U(0,1) + (1-mix)*Beta(scale*a_s, scale*b_s)\)

References

Vandekerckhove, J. S., Matzke, D., & Wagenmakers, E. (2015). Model comparison and the principle of parsimony. In Oxford Handbook of Computational and Mathematical Psychology (pp. 300-319). New York, NY: Oxford University Press.

Examples

Run this code

# 2-High-Threshold Model
eqn <- "## 2HTM ##
   Target  Hit  d
   Target  Hit  (1-d)*g
   Target  Miss (1-d)*(1-g)
   Lure    FA   (1-d)*g
   Lure    CR   (1-d)*(1-g)
   Lure    CR   d"
data <- c(
  Hit = 46, Miss = 14,
  FA = 14, CR = 46
)

# weakly informative prior for guessing
aa <- c(d = 1, g = 2)
bb <- c(d = 1, g = 2)
curve(dbeta(x, aa["g"], bb["g"]))

# compute marginal likelihood
htm <- marginalMPT(eqn, data,
  alpha = aa, beta = bb,
  posterior = 200, samples = 1000
)
# second model: g=.50
htm.g50 <- marginalMPT(eqn, data, list("g=.5"),
  alpha = aa, beta = bb,
  posterior = 200, samples = 1000
)

# Bayes factor
# (per batch to get estimation error)
bf <- htm.g50$p.per.batch / htm$p.per.batch
mean(bf) # BF
sd(bf) / sqrt(length(bf)) # standard error of BF estimate