Author: Bilal Ahmad
Date: 2026-06-04

# Bayesian Estimation with BetaDanish

#### Bilal Ahmad

#### 2026-06-04

## 1. Overview

This vignette walks through Bayesian estimation of the three-parameter Exponentiated Danish (ED) submodel and the full four-parameter Beta-Danish distribution using `bayes_betadanish()`.

## 2. ED submodel on bladder cancer remission times

    library(BetaDanish)
    data("remission")
    fit_bayes <- bayes_betadanish(
      time     = remission$time,
      status   = remission$status,
      submodel = TRUE,
      burnin   = 2000, mcmc = 5000,
      tune     = 0.5,
      seed     = 1
    )
    print(fit_bayes)

## 3. Posterior diagnostics

    draws <- fit_bayes$draws
    op <- par(mfrow = c(3, 1), mar = c(3, 4, 2, 1))
    coda::traceplot(draws)
    par(op)

## 4. Posterior survival vs Kaplan-Meier

    post_mean <- summary(draws)$statistics[, "Mean"]
    b <- post_mean["b"]; c <- post_mean["c"]; k <- post_mean["k"]
    km <- survival::survfit(survival::Surv(time, status) ~ 1, data = remission)
    plot(km, conf.int = FALSE, xlab = "Time (months)",
         ylab = "Survival probability",
         main = "Posterior mean ED fit on remission data")
    t_grid <- seq(0.1, max(remission$time), length.out = 200)
    S_post <- pbetadanish(t_grid, a = 1, b = b, c = c, k = k,
                          lower.tail = FALSE)
    lines(t_grid, S_post, col = "red", lwd = 2)
    legend("topright",
           legend = c("Kaplan-Meier", "Posterior-mean ED"),
           col = c("black", "red"), lty = 1, lwd = c(1, 2), bty = "n")

## 5. Bayesian vs MLE comparison

    fit_mle <- fit_betadanish(survival::Surv(time, status) ~ 1,
                              data = remission, submodel = TRUE,
                              n_starts = 3)
    cbind(
      MLE  = round(coef(fit_mle), 4),
      Bayes_post_mean = round(post_mean, 4)
    )