# How to uniquely define the credible interval?

Consider we are in the Bayesian paradigm and consider that we know the posterior $P(\tau|d)$ of the single parameter $\tau$, whose max of the posterior is given by $\tau_{\text{best}}$.

The credible interval $[\tau_{\text{min}}, \tau_{\text{max}}]$ is defined by defining the indicador function

$$I(\tau) = \begin{cases} 1 & \text{if } \tau_{\text{min}} < \tau < \tau_{\text{max}} \\ 0 & \text{else} \end{cases}$$

and solve $P(I|d) = 0.95$, right?

My issue is that this does not uniquely define the interval since there is only one condition to compute two independent variables (i.e. the system is underdetermined).

Given this, I tried defining it using two conditions

$$P(\tau_{\text{min}} < \tau < \tau_{\text{best}}) = 0.95/2$$ $$P(\tau_{\text{best}} < \tau < \tau_{\text{max}}) = 0.95/2$$

which is to say that I want the mass "0.95" equally distributed on both sides of $\tau_{\text{best}}$.

However, I'm working on an example on which 0.95/2 is not on the left side of the distribution (i.e. the distribution is not symmetrical).

So, my question is: how do I define a unique credible interval that is generic? Is it even possible to define it generically and uniquely?

• As you say, you need to add another condition. Some people seek a shortest interval (often, but not always, unique), others seek a symmetric interval (equal proportion in each tail); it's not for us to tell you what to prefer/optimize. Jan 6 '16 at 0:29

There are some issues with your wording. I assume when you say $\tau_\text{best}$ you are referring to the Bayes estimator (posterior mean) or perhaps the posterior mode. Also, the min max subscripts would best be defined by lower and upper limits of the credible set.