Ab_sort {multinomineq}

Sort Inequalities by Acceptance Rate

Description

Uses samples from the prior/posterior to order the inequalities by the acceptance rate.

Usage

Ab_sort(A, b, k = 0, options, M = 1000, drop_irrelevant = TRUE)

Arguments

A

a matrix with one row for each linear inequality constraint and one column for each of the free parameters. The parameter space is defined as all probabilities x that fulfill the order constraints A*x <= b.

b

a vector of the same length as the number of rows of A.

k

optional: number of observed frequencies (only for posterior sampling).

options

optional: number of options per item type/category system. Uniform sampling on [0,1] for each parameter is used if omitted.

M

number of samples.

drop_irrelevant

whether to drop irrelevant constraints for probabilities such as theta[1] >= 0, theta[1] <= 1, or sum(theta) <= 1.

Details

Those constraints that are rejected most often are placed at the first positions. This can help when computing the encompassing Bayes factor and counting how many samples satisfy the constraints (e.g., count_binom or bf_multinom). Essentially, it becomes more likely that the while-loop for testing whether the inequalities hold can stop earlier, thus making the computation faster.

The function could also be helpful to improve the efficiency of the stepwise sampling implemented in count_binom and count_multinom. First, one can use accept-reject sampling to test the first few, rejected inequalities. Next, one can use a Gibbs sampler to draw samples conditional on the first constraints.

Examples

### Binomial probabilities
b <- c(0, 0, .30, .70, 1)
A <- matrix(
  c(
    -1, 1, 0, # p1 >= p2
    0, 1, -1, # p2 <= p3
    1, 0, 0, # p1 <=.30
    0, 1, 0, # p2 <= .70
    0, 0, 1
  ), # p3 <= 1 (redundant)
  ncol = 3, byrow = 2
)
Ab_sort(A, b)


### Multinomial probabilities
# prior sampling:
Ab_sort(A, b, options = 4)
# posterior sampling:
Ab_sort(A, b, k = c(10, 3, 2, 14), options = 4)