bf_nonlinear {multinomineq}

Bayes Factor for Nonlinear Inequality Constraints

Description

Computes the encompassing Bayes factor for a user-specified, nonlinear inequality constraint. Restrictions are defined via an indicator function of the free parameters c(p11,p12,p13, p21,p22,...) (i.e., the multinomial probabilities).

Usage

bf_nonlinear(
  k,
  options,
  inside,
  prior = rep(1, sum(options)),
  log = FALSE,
  ...
)

count_nonlinear(
  k = 0,
  options,
  inside,
  prior = rep(1, sum(options)),
  M = 5000,
  progress = TRUE,
  cpu = 1
)

Arguments

k

vector of observed response frequencies.

options

number of observable categories/probabilities for each item type/multinomial distribution, e.g., c(3,2) for a ternary and binary item.

inside

an indicator function that takes a vector with probabilities p=c(p11,p12, p21,p22,...) (where the last probability for each multinomial is dropped) as input and returns 1 or TRUE if the order constraints are satisfied and 0 or FALSE otherwise (see details).

prior

a vector with two positive numbers defining the shape parameters of the beta prior distributions for each binomial rate parameter.

log

whether to return the log-Bayes factor instead of the Bayes factor

...

further arguments passed to count_binom or count_multinom (e.g., M, steps).

M

number of posterior samples drawn from the encompassing model

progress

whether a progress bar should be shown (if cpu=1).

cpu

either the number of CPUs used for parallel sampling, or a parallel cluster (e.g., cl <- parallel::makeCluster(3)). All arguments of the function call are passed directly to each core, and thus the total number of samples is M*number_cpu.

Details

Inequality constraints are defined via an indicator function inside which returns inside(x)=1 (or 0) if the vector of free parameters x is inside (or outside) the model space. Since the vector x must include only free (!) parameters, the last probability for each multinomial must not be used in the function inside(x)!

Efficiency can be improved greatly if the indicator function is defined as C++ code via the function cppXPtr in the package RcppXPtrUtils (see below for examples). In this case, please keep in mind that indexing in C++ starts with 0,1,2... (not with 1,2,3,... as in R)!

References

Klugkist, I., & Hoijtink, H. (2007). The Bayes factor for inequality and about equality constrained models. Computational Statistics & Data Analysis, 51(12), 6367-6379. doi:10.1016/j.csda.2007.01.024

Klugkist, I., Laudy, O., & Hoijtink, H. (2010). Bayesian evaluation of inequality and equality constrained hypotheses for contingency tables. Psychological Methods, 15(3), 281-299. doi:10.1037/a0020137

Examples

##### 2x2x2 continceny table (Klugkist & Hojtink, 2007)
#
# (defendant's race) x (victim's race) x (death penalty)
# indexing: 0 = white/white/yes  ; 1 = black/black/no
# probabilities: (p000,p001,  p010,p011,  p100,p101,  p110,p111)
# Model2:
# p000*p101 < p100*p001  &   p010*p111 < p110*p011

# observed frequencies:
k <- c(19, 132, 0, 9, 11, 52, 6, 97)

model <- function(x) {
  x[1] * x[6] < x[5] * x[2] & x[3] * (1 - sum(x)) < x[7] * x[4]
}
# NOTE: "1-sum(x)"  must be used instead of "x[8]"!

# compute Bayes factor (Klugkist 2007: bf_0u=1.62)
bf_nonlinear(k, 8, model, M = 50000)


##### Using a C++ indicator function (much faster)
cpp_code <- "SEXP model(NumericVector x){
  return wrap(x[0]*x[5] < x[4]*x[1] & x[2]*(1-sum(x)) < x[6]*x[3]);}"
# NOTE: C++ indexing starts at 0!

# define C++ pointer to indicator function:
model_cpp <- RcppXPtrUtils::cppXPtr(cpp_code)

bf_nonlinear(
  k = c(19, 132, 0, 9, 11, 52, 6, 97), M = 100000,
  options = 8, inside = model_cpp
)