bf_equality {multinomineq}

Bayes Factor with Inequality and (Approximate) Equality Constraints

Description

To obtain the Bayes factor for the equality constraints C*x = d, a sequence of approximations abs(C*x - d) < delta is used.

Usage

bf_equality(
  k,
  options,
  A,
  b,
  C,
  d,
  prior = rep(1, sum(options)),
  M1 = 1e+05,
  M2 = 20000,
  delta = 0.5^(1:8),
  return_Ab = FALSE,
  ...
)

Arguments

k

the number of choices for each alternative ordered by item type (e.g. c(a1,a2,a3, b1,b2) for a ternary and a binary item type). The length of k must be equal to the sum of options. The default k=0 is equivalent to sampling from the prior.

options

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

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.

C

a matrix specifying the equality constraints C*x = d with columns refering to the free parameters (similar to A)

d

a vector with the same number of elements as the rows of C.

prior

the prior parameters of the Dirichlet-shape parameters. Must have the same length as k.

M1

number of independent samples from the encompassing model to test whether A*x < b.

M2

number of Gibbs-sampling iterations for each step of the approximation of C*x = d.

delta

a vector of stepsizes that are used for the approximation.

return_Ab

if TRUE, the function returns a list with the additional inequality constraints (specified via A, b, and steps) that are used in the stepwise approximation abs(C*x - d) < delta[i].

...

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

Details

First, the encompassing Bayes factor for the equality constraint A*x<b is computed using M1 independent Dirichlet samples. Next, the equality constraint C*x=d is approximated by drawing samples from the model A*x<b and testing whether abs(C*x - d) < delta[1]. Similarly, the stepsize delta is reduced step by step until abs(C*x - d) < min(delta). Klugkist et al. (2010) show that this procedure provides a valid approximation of the exact equality constraints if the step size becomes sufficiently small.

References

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

# Equality constraints:  C * x = d
d <- c(.5, .5, 0)
C <- matrix(
  c(
    1, 0, 0, 0, # p1 = .50
    0, 1, 0, 0, # p2 = .50
    0, 0, 1, -1
  ), # p3 = p4
  ncol = 4, byrow = TRUE
)
k <- c(3, 7, 6, 4, 2, 8, 5, 5)
options <- c(2, 2, 2, 2)
bf_equality(k, options,
  C = C, d = d, delta = .5^(1:5),
  M1 = 50000, M2 = 5000
) # only for CRAN checks

# check against exact equality constraints (see ?bf_binom)
k_binom <- k[seq(1, 7, 2)]
bf_binom(k_binom,
  n = 10, A = matrix(0), b = 0,
  map = c(0, 0, 1, 1)
)