multibridge (version 1.1.0)

binom_bf_inequality: 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
)

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

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: $$\theta_k ~ Beta(\alpha_k, \beta_k)$$

References

gronau2017tutorialmultibridge

sarafoglou2020evaluatingPreprintmultibridge

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

Run this code
# 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))

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