compute posterior probabilities and Bayesian credible intervals such as equal-tailed intervals and highest posterior density (HPD) intervals. Bayesian HPD intervals are the shortest intervals containing the parameter of interest with the desired posterior coverage probability. They are more desirable than the commonly used equal-tailed intervals when the posterior distribution is highly skewed, but are more difficult to compute. @ChenShao1999 develop the Monte Carlo method to compute the HPD intervals. In their paper, they also discussed how to compute Monte-Carlo-based Bayesian credible intervals under importance sampling.

Marginal Predictive Distribution and Bayes Factor {#sub5sec4}
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In this section, we derive the marginal predictive distribution and the Bayes factor under Jeffreys’ prior and Bernardo’s prior. For simplicity and clarity of the presentation, we work with the parameterization $(\gamma, U, V)$. In addition, we derive a single Bayes factor formula for these two priors. To accomplish this single formulation, we use a family of prior distributions that encapsulates both Jeffreys’ prior and the reference prior as special cases.

### $H_0$: $\lambda_0=\lambda_1$ versus $H_1$: $\lambda_1\neq
\lambda_0$ {#sub1sub4sec4}

Consider the hypotheses $H_0$: $\lambda_0=\lambda_1=\lambda$ and $H_1$: $\lambda_1\neq\lambda_0$ or equivalently $H_0$: $U=V=\theta$ against $H_1$: $ U \neq V$. Under $H_1$, we consider the family of prior distributions $$\begin{aligned}
