I read the answers provided to the question "are all values in a 95% confidence interval equally likely?". These answers raise questions for me regarding interpretation. I have read that under certain (large sample) circumstances, a confidence interval can approximate a likelihood interval. I often find myself in the circumstance with a lot of data related to the same trial (i.e. a large sample), but that trial is not replicable, or rather no one will pay to replicate it if I do not show promising results (to completely non-technical people). Given that I ask many questions that relate to the interpretation of reality based on the observed data from a single study (not n repeated trials), that Bayesian methods are still computationally intensive and not routinely used by those around me, and that when we have called in Bayesian experts, they constructed models constructed with non-informative priors (because that was all the information available), are there circumstances in which it would be okay to interpret a confidence interval as giving a range of likely values for the true (but unknown parameter)? Exactly what can one say about a confidence interval that is both reasonably accurate, and allows for a more intuitive understanding than is possible within a frequentist framework? Can one say that the values closest to the center of the interval are most consistent with the data? Would there be circumstances under which it is reasonable to use a 65% of 75% confidence interval to show the range of values that are most consistent with the data, acknowledging that many reasonable answers lie outside this interval, but that the answers that are most consistent with the data lie within the interval?
I acknowledge that a Bayesian approach might be better, but I am just acknowledging reality at this specific point in time regarding the statistical methods are being used around me.