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As far as I know, credible intervals usually involve a single contiguous interval and allow one to make statements such as: "Based on the model, data, and priors, $X$ is within the bounds $(x_1, x_2)$ with probability 89%."

But what if I'm interested in making a statement like the following: "Based on the model, data, and priors, $X$ is outside of the bounds $(x_3, x_4)$ with probability 89%"? Would it be valid to use the bounds from an 11% CI as $x_3$ and $x_4$? Or is there a better way to construct such intervals?

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    $\begingroup$ That is a perfectly valid probability statement. I wouldn't call it a Credible Interval, because these are typically defined in terms of their probability of containing the parameter, not excluding it, but the advantage of the Bayesian approach is that we get more than a CI, we get a whole posterior to do anything we want with, and what you are suggesting is a perfectly legitimate use of the posterior. $\endgroup$
    – John Madden
    Commented Jan 5 at 17:01

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A credible interval $I$ has credibility $X\%$ because the posterior probability of $I$ has $P(I) = X/100$. Consequently, the probability of not falling within $I$ is $1-X/100$.

Therefore, if you construct an $11\%$ credible interval $I = (x_1,x_2)$, then $P(\neg I) = 0.89$, yes.

I could see something like this making sense if you have to be within $\varepsilon$ of some point $\theta_0$ and wind up with posterior $P\left(\neg(\theta_0 - \varepsilon, \theta_0 + \varepsilon)\right) = 0.89$.

Boss, there's an 89% chance that the true value is outside of our acceptable range. Rats!

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