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Suppose that I have access to the observations from the entire population $x_1,...,x_N$. If I fit a distribution to this population $x_i\sim F(\cdot ; \theta)$, and I calculate confidence intervals/regions for $\theta$,
what is the interpretation of these intervals/regions?
Even if you can compute the confidence interval in usual manner, those interval are senseless. The usual story behind the confidence interval rules suppose that you collect a sample by $iid$ sampling on infinite large population. For finite population some ad hoc adjustments exist (read here: http://www.utstat.toronto.edu/~brunner/oldclass/utm218s07/FinitePop.pdf). From those adjustments you can understand/remember that "confidence interval" regard the inference from sample to population. If you have data at population level the estimators are no more random variables; then no standard deviation exist and, therefore, no confidence intervals.
Haven't I already obtained the true value of the parameter as there are no more individuals?
If your distribution, or model more in general, are correctly specified the reply is yes. Otherwise your estimates are biased. Note that biasedness can remain at population level also. This discussion is strongly related: Do we need hypothesis testing when we have all the population? I added there some consideration about prediction al causal inference.
