bf_binom() R function from [multinomineq]

Bayes Factor for Linear Inequality Constraints

Computes the Bayes factor for product-binomial/-multinomial models with linear order-constraints (specified via: A*x <= b or the convex hull V).


bf_binom(k, n, A, b, V, map, prior = c(1, 1), log = FALSE, ...)

bf_multinom(
  k,
  options,
  A,
  b,
  V,
  prior = rep(1, sum(options)),
  log = FALSE,
  ...
)

Arguments

k: vector of observed response frequencies.
n: the number of choices per item type. If k=n=0, Bayesian inference is relies on the prior distribution only.
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.
V: a matrix of vertices (one per row) that define the polytope of admissible parameters as the convex hull over these points (if provided, A and b are ignored). Similar as for A, columns of V omit the last value for each multinomial condition (e.g., a1,a2,a3,b1,b2 becomes a1,a2,b1). Note that this method is comparatively slow since it solves linear-programming problems to test whether a point is inside a polytope (Fukuda, 2004) or to run the Gibbs sampler.
map: optional: numeric vector of the same length as k with integers mapping the frequencies k to the free parameters/columns of A/V, thereby allowing for equality constraints (e.g., map=c(1,1,2,2)). Reversed probabilities 1-p are coded by negative integers. Guessing probabilities of .50 are encoded by zeros. The default assumes different parameters for each item type: map=1:ncol(A)
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).
options: number of observable categories/probabilities for each item type/multinomial distribution, e.g., c(3,2) for a ternary and binary item.

Returns

a matrix with two columns (Bayes factor and SE of approximation) and three rows:

bf_0u: constrained vs. unconstrained (saturated) model
bf_u0: unconstrained vs. constrained model
bf_00': constrained vs. complement of inequality-constrained model (e.g., pi>.2 becomes pi<=.2; this assumes identical equality constraints for both models)

Details

For more control, use count_binom to specifiy how many samples should be drawn from the prior and posterior, respectively. This is especially recommended if the same prior distribution (and thus the same prior probability/integral) is used for computing BFs for multiple data sets that differ only in the observed frequencies k and the sample size n. In this case, the prior probability/proportion of the parameter space in line with the inequality constraints can be computed once with high precision (or even analytically), and only the posterior probability/proportion needs to be estimated separately for each unique vector k.

Examples


k <- c(0, 3, 2, 5, 3, 7)
n <- rep(10, 6)

# linear order constraints:
#             p1 <p2 <p3 <p4 <p5 <p6 <.50
A <- matrix(
  c(
    1, -1, 0, 0, 0, 0,
    0, 1, -1, 0, 0, 0,
    0, 0, 1, -1, 0, 0,
    0, 0, 0, 1, -1, 0,
    0, 0, 0, 0, 1, -1,
    0, 0, 0, 0, 0, 1
  ),
  ncol = 6, byrow = TRUE
)
b <- c(0, 0, 0, 0, 0, .50)

# Bayes factor: unconstrained vs. constrained
bf_binom(k, n, A, b, prior = c(1, 1), M = 10000)
bf_binom(k, n, A, b, prior = c(1, 1), M = 2000, steps = c(2, 4, 5))
bf_binom(k, n, A, b, prior = c(1, 1), M = 1000, cmin = 200)

References

Karabatsos, G. (2005). The exchangeable multinomial model as an approach to testing deterministic axioms of choice and measurement. Journal of Mathematical Psychology, 49(1), 51-69. tools:::Rd_expr_doi("10.1016/j.jmp.2004.11.001")

Regenwetter, M., Davis-Stober, C. P., Lim, S. H., Guo, Y., Popova, A., Zwilling, C., … Messner, W. (2014). QTest: Quantitative testing of theories of binary choice. Decision, 1(1), 2-34. tools:::Rd_expr_doi("10.1037/dec0000007")

bf_binom function