I found some writings (prominently on the web) that mention a sort of confidence set (c.f. confidence interval) with the ratio of coverage at which the concerned parameter is believed to belong to the set.

Assume we have observed a sample with a sample statistic . Further we are given a random set based on the statistic to estimate a parameter .

Then, what does it mean by saying is a p% confidence set for ? In other words, over which distribution the probability p, at which the is confidently covered by , is evaluated?

Note that the parameter is fixed and unknown (to be estimated). Then for each observation , the derived set determines whether or not, which is just an indicator taking a value of 1 or 0, as a probability.

Assuming ranges over the sample space , then for some and not for the others. In this case, we can naively define (replace the integral with sum for discrete random variable )

as a measurement of “well-coveredness” of by , where is the distribution function of . This is equivalent to define by the conditional expectation over the distribution .

The confusion comes when is derived from an observation and therefore fixed. As noted above, holds indeed.

This is just a memo, not to claim a sort of insight or discovery. Caveat!