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 $CI_D$).

Sentence 1 is the de jure interpretation of confidence intervals. I like to say it as follows:

The 95% in "95% confidence interval" refers to the long term relative frequency of these estimators containing the true estimand upon repeated construction under ideal and identical circumstances.

Before embarking, I'd like to point out that $ \theta \in CI_D$ can be interpreted as either $\theta$ being in $CI_D$ as in sentence 3, or $CI_D$ covers $\theta$ as in sentence 2.  Hence, sentences 2 and 3 are (from where I stand) equivalent.  Let's discuss why sentence 1 and 2 (or 1 and 3) are thus not equivalent.
The statement made in sentence 2 seems to imply that $\theta$ is the random quantity. Frequentists treat estimands as fixed.  Hence sentences 1 and 2 are not equivalent merely based on this fact.  Now, people will often defend interpreting confidence intervals in the style of sentence 2 (or 3) by referencing the estimators long term relative frequency, however I think this is an error.  Any given confidence interval either contains the estimand (assuming it is fixed) or it does not.  There is no probabalistic element to this, unless we were to repeat the construction of the interval with different data (which would appeal to the definition). Its like asking "what is the probability that the card on the top of this deck is an ace?".  The card is either an ace or it isn't, there is no randomness here.  When people answer "the probability is 4/52" what they really mean is "In an infinite sequence, were I to come to well shuffled decks then 4/52 decks would have an ace on top".  These are two very different scenarios.
That's my argument, let's take a look at some of yours:

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

Here, I think you are making the mistake I spoke of earlier using the decks.  The interval is fixed, the parameter is fixed, the interval either covers it or it does not.  This means the probability is either 0 or 1, but we don't know.  All we know is the long term relative frequency of intervals capturing the estimated under ideal conditions, and we just hope that this happens to be one of those times where our interval covers the estimand.

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 $P[\theta\in CI_D]=0.95$.

I disagree with the statement $P[CI_D\ contains\ \theta] = 0.95$, because as noted before, any given interval either covers the estimand or not. The probabalistic statement is about infinite sequences of intervals.
If you're interested, I wrote a little about this here on my blog.
