# Computes Bayes Factors For Inequality Constrained Independent Binomial Parameters 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. ```r 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. ## Returns 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 ## 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: $$ \theta_k ~ Beta(\alpha_k, \beta_k) $$ ## 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` ## Examples ```r # 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, niter = 100, 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 = 500, prior=TRUE, seed = 2020) results_prior <- binom_bf_inequality(prior_samples, inequalities, niter = 100, seed=2020) cbind(exp(results_prior$logml), 1/factorial(4)) ``` ## References \insertRef gronau2017tutorialmultibridge \insertRef 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()`

binom_bf_inequality function