binom_bf_inequality {multibridge}

Computes Bayes Factors For Inequality Constrained Independent Binomial Parameters

Description

Computes Bayes factor for inequality constrained binomial parameters using a bridge sampling routine. Restricted hypothesis H_r states that binomial proportions follow a particular trend. Alternative hypothesis H_e states that binomial proportions are free to vary.

Usage

binom_bf_inequality(
  samples = NULL,
  restrictions = NULL,
  x = NULL,
  n = NULL,
  Hr = NULL,
  a = rep(1, ncol(samples)),
  b = rep(1, ncol(samples)),
  factor_levels = NULL,
  prior = FALSE,
  index = 1,
  maxiter = 1000,
  seed = NULL,
  niter = 5000,
  nburnin = niter * 0.05
)

Arguments

samples

matrix of dimension (nsamples x nparams) with samples from truncated Dirichlet density

restrictions

list of class bmult_rl or of class bmult_rl_ineq as returned from generate_restriction_list that encodes inequality constraints for each independent restriction

x

a vector of counts of successes, or a two-dimensional table (or matrix) with 2 columns, giving the counts of successes and failures, respectively

n

numeric. Vector of counts of trials. Must be the same length as x. Ignored if x is a matrix or a table

Hr

string or character. Encodes the user specified informed hypothesis. Use either specified factor_levels or indices to refer to parameters. See “Note” section for details on how to formulate informed hypotheses

a

numeric. Vector with alpha parameters. Must be the same length as x. Default sets all alpha parameters to 1

b

numeric. Vector with beta parameters. Must be the same length as x. Default sets all beta parameters to 1

factor_levels

character. Vector with category names. Must be the same length as x

prior

logical. If TRUE the function will ignore the data and evaluate only the prior distribution

index

numeric. Index of current restriction. Default is 1

maxiter

numeric. Maximum number of iterations for the iterative updating scheme used in the bridge sampling routine. Default is 1,000 to avoid infinite loops

seed

numeric. Sets the seed for reproducible pseudo-random number generation

niter

numeric. Vector with number of samples to be drawn from truncated distribution

nburnin

numeric. A single value specifying the number of burn-in samples when drawing from the truncated distribution. Minimum number of burn-in samples is 10. Default is 5% of the number of samples. Burn-in samples are removed automatically after the sampling.

Details

The model assumes that the data in x (i.e., x_1, ..., x_K) are the observations of K independent binomial experiments, based on n_1, ..., n_K observations. Hence, the underlying likelihood is the product of the k = 1, ..., K individual binomial functions:

(x_1, ... x_K) ~ \prod Binomial(N_k, \theta_k)

Furthermore, the model assigns a beta distribution as prior to each model parameter (i.e., underlying binomial proportions). That is:

\theta_k ~ Beta(\alpha_k, \beta_k)

Value

List consisting of the following elements:

$eval

• q11: log prior or posterior evaluations for prior or posterior samples

• q12: log proposal evaluations for prior or posterior samples

• q21: log prior or posterior evaluations for samples from proposal

• q22: log proposal evaluations for samples from proposal

$niter

number of iterations of the iterative updating scheme

$logml

estimate of log marginal likelihood

$hyp

evaluated inequality constrained hypothesis

$error_measures

• re2: the approximate relative mean-squared error for the marginal likelihood estimate

• cv: the approximate coefficient of variation for the marginal likelihood estimate (assumes that bridge estimate is unbiased)

• percentage: the approximate percentage error of the marginal likelihood estimate

Note

The following signs can be used to encode restricted hypotheses: "<" and ">" for inequality constraints, "=" for equality constraints, "," for free parameters, and "&" for independent hypotheses. The restricted hypothesis can either be a string or a character vector. For instance, the hypothesis c("theta1 < theta2, theta3") means

• theta1 is smaller than both theta2 and theta3

• The parameters theta2 and theta3 both have theta1 as lower bound, but are not influenced by each other.

The hypothesis c("theta1 < theta2 = theta3 & theta4 > theta5") means that

• Two independent hypotheses are stipulated: "theta1 < theta2 = theta3" and "theta4 > theta5"

• The restrictions on the parameters theta1, theta2, and theta3 do not influence the restrictions on the parameters theta4 and theta5.

• theta1 is smaller than theta2 and theta3

• theta2 and theta3 are assumed to be equal

• theta4 is larger than theta5

References

Gronau QF, Sarafoglou A, Matzke D, Ly A, Boehm U, Marsman M, Leslie DS, Forster JJ, Wagenmakers E, Steingroever H (2017). “A tutorial on bridge sampling.” Journal of Mathematical Psychology, 81, 80–97.

Sarafoglou A, Haaf JM, Ly A, Gronau QF, Wagenmakers EJ, Marsman M (2021). “Evaluating Multinomial Order Restrictions with Bridge Sampling.” Psychological Methods.

See Also

generate_restriction_list

Other functions to evaluate informed hypotheses: binom_bf_equality(), binom_bf_informed(), mult_bf_equality(), mult_bf_inequality(), mult_bf_informed()

Examples

# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)

# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

results_prior  <- binom_bf_inequality(Hr=Hr, a=a, b=b, 
factor_levels=factor_levels, prior=TRUE, seed = 2020)
# corresponds to
cbind(exp(results_prior$logml), 1/factorial(4))

# alternative - if you have samples and a restriction list
inequalities  <- generate_restriction_list(Hr=Hr, a=a,b=b,
factor_levels=factor_levels)$inequality_constraints
prior_samples <- binom_tsampling(inequalities, niter = 2e3, 
prior=TRUE, seed = 2020)
results_prior <- binom_bf_inequality(prior_samples, inequalities, seed=2020)
cbind(exp(results_prior$logml), 1/factorial(4))