Why aren't these sentences about confidence intervals equivalent? Yes, I'm aware that there's similar/duplicated questions already open:

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*Aren't these statements about confidence intervals equivalent?


*Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?


*Are there any examples where Bayesian credible intervals are obviously inferior to frequentist confidence intervals


*When do (and don't) confidence intervals and credible intervals coincide?
But funny enough, NONE of these questions has any answer marked as the right one, so this question is, technically, still unsolved in this site. I have read the answers and other related questions but I don't exactly see the difference yet so I will try to explain why I think these are equivalent expressions.
Given a confidence interval $CI_D$ for some unknown fixed parameter $\theta$ calculated from a dataset $D$, why aren't these three sentences equivalent?

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*If I repeat the experiment an infinite amount of times and I get an infinite amount of calculated confidance intervals, 95% of them will contain $\theta$.

*There's a 95% probability that $CI_D$ contains $\theta$.

*There's a 95% probability that $\theta$ is in $CI_D$.

Why I think these sentences are equivalent?

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*Sentences 1 and 2 are equivalent for me because I don't know if $CI_D$ is one of the 95% of intervals that contains $\theta$, which is the same as saying that $CI_D$ has a probability of 95% of being one of intervals that contains $\theta$, which is the same as saying that there's a probability of 95% that $CI_D$ contains $\theta$.


*Senteces 2 and 3 are equivalent because "$CI_D$ contains $\theta$" is equivalent to say "$\theta$ is in $CI_D$", because both sentences are translated as $\theta\in CI_D$. So $P[CI_D\ contains\ \theta] = 0.95$ is the same as saying $P[\theta\ is\ in\ CI_D] = 0.95$, because they are both the same as saying $P[\theta\in CI_D]=0.95$.
Am I right?
I know that frequentists doesn't allow to say "probability of fact around $X$" when $X$ is not a random variable (and $\theta$ is not because it's a constant), but $CI_D$ is and 2 and 3 speak about the probability of the relationship between $\theta$ and $CI_D$. So I'm not fully convinced that "probability of $\theta\in CI_D$" goes against the fact that $\theta$ is not a random variable, because $CI_D$ is a random variable and is also present in the same sentence (it tells something about $\theta$, but it also tells something about the $CI_D$).

The key thing to know is that the frequentist confidence interval say nothing directly about the unknown fixed value of $\theta$, the local parameter of interest for the experiment that you ran. Instead, the frequentist confidence interval is the result of a method that yields intervals with on average the nominal coverage (with a few small caveats) when applied in the long run to analyses of experiments for all possible values of $\theta$.
Some types of frequentist confidence interval that yield exactly the nominal coverage for all possible values of the parameter of interest (e.g. the ordinary Student's t confidence interval for the random sample-based estimate of the mean of a normally distributed population). If your interval is of that type then you can indirectly infer something about the probability of your fixed unknown instance of the parameter falling within the interval. But that probability is not a 'proper' frequentist probability because that 'proper' probability is either one or zero. Of course, even that indirect inference is only well-formed if you know for certain that your data are obtained in circumstances that allow the distributional and sampling assumptions of the method to be correct.
Other types of frequentist interval do not have uniformly nominal coverage. Cases where the data are discrete provide useful examples. See here, for example: Discrete functions: Confidence interval coverage? If your fixed unknown value of $\theta$ lies in an 'unlucky' region of parameter space then the coverage of the method can be remarkably far from nominal.
Where the coverage is not exactly nominal over all possible values of the parameter you cannot assume that the coverage is nominal for your fixed unknown value of the parameter and so you cannot make any exact statement about the probability that you value lies inside the interval.
If you need to know the probability that an interval contains the unknown value of $\theta$ then you need to use a method that allows direct specification of probabilities of non-random parameters. In other words you need Bayesian probabilities.
It is true that very often the Baysian credible intervals are very similar to (or identical to) some frequentist intervals (as long as the priors allow) and so Bayesian statements about the presence of fixed parameter values within frequentist intervals are not too misleading even if they are incorrect.
