[ivansavz]

> Ideally one should use the whole posterior distribution of your model parameters which is not the case for point estimates.

This is a historical issue because of some hard-headed frequentist founders, but in modern days the frequentist concept of confidence distribution is gaining acceptance, which is the proper frequentist equivalent of the posterior, so this distinction between Bayesian and Frequentist is disappearing.

Rather than giving specific point estimates or interval estimates, calculating a frequentist confidence distribution allows you to compute confidence intervals for all possible confidence levels, just like the posterior does. See this excellent review paper for more info on this: https://statweb.rutgers.edu/mxie/RCPapers/insr.12000.pdf

The key insights is that a confidence distribution is an estimator for the parameter of interest, instead of an inherent distribution of the parameter.

[zozbot234]

The confidence distribution is generally derived from normalizing a likelihood function, and the likelihood function is arguably the proper underlying concept that provides a link to both Bayesian and frequentist inference, per https://en.wikipedia.org/wiki/Likelihood_principle

[kgwgk]

> the frequentist concept of confidence distribution is gaining acceptance, which is the proper frequentist equivalent of the posterior, so this distinction between Bayesian and Frequentist is disappearing

The major distinction remains: Frequentist confidence intervals are something quite different from Bayesian credible intervals. I don't think that having a distribution that can be used to calculate any desired confidence interval - like the posterior distribution can be used to calculate different credible intervals - changes much.