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The tail probability can be estimated by two methods:

  1. In Bayesian method: $$P_B(X>a)=\int^{\infty}_{-\infty}\pi(\theta|x)[1-F(a|\theta)]d\theta$$
  2. In Plug-in frequency method: $$P_F(X>a)=1-F(a|\hat{\theta})$$ where $\hat{\theta}$ is the MLE of $\theta$.

The numerical results show that it's always $$P_B \geq P_F$$ no matter what the distribution is.

Any ideas or any resources related to this topic to explain why is that?

Many thanks~

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