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You say (e.g here and here)

The reason why it is won't to say there is a 95% chance the confidence interval contains the true parameter (or the probability of the interval containing the true parameter is .95) is because the parameter is either contained in the interval or not. There are two cases here:

  1. The parameter is contained in the interval. What is the probability (or chance) the confidence interval contains the parameter? 1 since it is in the interval.

  2. The parameter is not contained in the interval. What is the probability (or chance) the confidence interval contains the parameter? 0 since it is not in the interval.

In short, you say that we know that the parameter is within CI or not for sure and, therefore, it is technically illegal to speak about probability of certain event. At the same time, you say that this does not apply to Bayesian credible intervals. Is it double standard in the purest form? Why not? Heads either occur or tails. You see, that the parameter of interest is either found in the credible interval or not. So, credible intervals are meaningless. There is no probability. Why do you arrest me instead of answering the question?

Kodiologist suggests that There is a randomness on the frequentist side; it's just that the interval is random, not the parameter. The Bayesian approach has it the other way round, by treating the data as fixed and the parameter as random.

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What I said in chat was:—

Saying that either the true value of the parameter falls in a given interval or it doesn't isn't an argument - as you point out it applies regardless of whether that interval's labeled with "credible" or "confidence". Rather it's all you can say unless you define what you mean by a probability distribution over parameter values; until you've done that you can't make sense of statements like "there's a 90% probability that the true value of the parameter falls in this interval".

It's the therefore in your first sentence that's incorrect: taking a Bayesian approach doesn't require you to deny the Law of Excluded Middle. As @Kodiologist says, the parameter is now considered a random variable; in the subjective Bayesian formulation probability represents credence, degree of belief. And there's no contradiction in saying "The true parameter value's either inside this interval or outside, but I'd lay odds of 19 to 1 against its being outside".

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  • $\begingroup$ Maybe just an idea, but wouldn't it be less confusing if we say that ''there is a 90% chance that the (random) confidence interval contains the true value'' so $P(CI \ni \theta)$ in stead of saying "there's a 90% probability that the true value of the parameter falls in this interval" or $P(\theta \in CI)$ ? $\endgroup$ – user83346 Sep 15 '16 at 17:45
  • $\begingroup$ I ask why I cannot say but you can say that? I refer you the answers that do that. I have copy-pasted the answer that I refer. Can you tell me where I misinterpret it? $\endgroup$ – Little Alien Sep 15 '16 at 18:49
  • $\begingroup$ @LittleAlien: I don't think you do misinterpret the passage you've quoted. It's misleading in my opinion. The subsequent explanation seems clearer. $\endgroup$ – Scortchi - Reinstate Monica Sep 15 '16 at 19:37
  • $\begingroup$ @fcop: Quite so - the subject of falls is strongly implied to be the random variable. (I'm not so sure the order of variables reliably signals which is random in mathematical notation. The convention that random variables only appear in upper case is more common & allows expressions like $\Pr(X>Y)$ to be distinguished from $\Pr(x>Y)$ or $\Pr(X>y)$.) $\endgroup$ – Scortchi - Reinstate Monica Sep 15 '16 at 19:45

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