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#' Full Bayes Bias Prior for Binomial Data
#'
#' @param x number of historical successes
#' @param n number historical patients
#' @param verbose Print messages
#' @param mc.cores number of cores for parallel
#'
#' @return A function of the probability parmater p
#' @export
#'
#'
binom.Bias.FB <- function(x, n, verbose=FALSE, mc.cores=1){
n.hist <- length(x)
dat <- data.frame(x, n, z=1:n.hist)
# if(missing(tau.prior)){
prior <- list(prior= "logtnormal", param=c(0,1))
# } else {
# prior <- tau.prior
# }
formula <- x ~ 1 + f(z, model="iid",
hyper = list(prec = prior))
result <- INLA::inla(formula,
data = dat,
family = "binomial",
control.fixed = list(mean.intercept = 0, prec.intercept = 1/1000),
Ntrials=n,
control.predictor = list(compute=TRUE, link=1))
k <- integrate(Vectorize(function(p) prod(sapply(result$marginals.fitted.values, inla.dmarginal, x=p))),
lower=0,
upper=1)$value
f <- function(p) sapply(p, function(p) prod(sapply(result$marginals.fitted.values, inla.dmarginal, x=p))/k)
}
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