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We know confidence interval can't be used for probability statement, this is something reserved for credible interval.

However, the most commonly used frequentist techniques (e.g. confidence intervals for means and proportions) are equivalent to Bayesian credible intervals for some specific prior. A common example is the flat prior. (Reference: William Bolstad on Bayesian Statistics)

If this is true, if I follow a frequentist text-book and calculate confidence interval. Can I say:

"This is my confidence interval. I'm actually a Bayesian because this is also credible interval with flat prior. I'm going to interpret this interval as Bayesian probability statement about my parameters."

So all statistical students learning statistics are Bayesian? We're all Bayesian?

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    $\begingroup$ Confidence intervals are probabilistic if you conceive of the frequentist type of probability just as credible intervals deal with probability as a degree of belief about a certain parameter. $\endgroup$ – AdamO Mar 10 '17 at 0:34
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    $\begingroup$ @AdamO In this question, I stress the Bayesian probability, for example, the parameters are not fixed as in classical frequentist. $\endgroup$ – SmallChess Mar 10 '17 at 0:35
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I am going to be snotty and say "no." Of course, an element of this is your wording of the question "can I." No. I forbid it. You cannot say that or anything at all like it. I also forbid you to say "turnip" for the entire month of May. Not just this May, but every May.

In a more serious vein, the answer is still "no," but only for a couple of very picky reasons that should be thought of as more of a personal/professional opinion rather than a canonical answer. Bayesian, Likelihoodist and Frequentist statistics do not answer the same type of question they answer three, often similar, questions.

The Frequentist answers the question of the probability of seeing the observed data, given a null hypothesis is the true state of nature. The Bayesian is answering a question regarding the probability a hypothesis is true, given an observed sample and any prior knowledge. Incidentally, the calculation formula and/or value is conceptually similar and may map to the same value.

So, 2+2=4 and 6-2=4, but they are not the same question. It can be a bit more complex because some tests for particular priors look notationally the same but are not the same. Consider the simple case of a normal density function/likelihood with a known variance of one where the open question is whether the center of location is less than four and a sample size of n. Because of how the problem marginalizes out both appear to use the same formula $$z=\sqrt{n}(\bar{x}-\mu),$$ but they are not the same formula at all. What is a constant for one is a variable for another and vice versa.

The interval is a bit more complex, though. There are an infinite number of possible confidence intervals and infinite possible numbers of credible intervals for the same problem, but for different reasons. You are deciding that the formulas, as above, are the same. For the same reason, they are not the same at all.

There is another, more subtle, problem here. You are forcing the prior to be flat, but it is rare to have no information at all. So the Bayesian answer is invalid in the presence of actual, but unused, information. Of course, if it is a true flat prior due to true, total ignorance then the answer is still "no," but not for the reason of this objection.

Finally, the answer is still "no" because you should be creating an interval that solves a problem as statistics is a branch of rhetoric, not mathematics. One of the two schools will be better at creating the argument you are trying solve and the other will not be as good. There is a utility or cost function question here as to how you should decide between the schools.

You risk being "irrational," and as "scholars," we should not do that. Again, as above, I forbid it. Indeed, if it will help, I will throw in a "verily." So, verily, I say unto you that it is forbidden for you to say the contents of your long quote above. And, forsooth, it is forbidden to you to say "turnip," for the entire month of May.

This inability to say "turnip," is your penance for having tried to mix and match two schools of thought. You are fain to obey this command, for otherwise, the end of all is nigh should you ever say the long quote above, or "turnip" during any month of May.

I do hope you have confidence that the answer is a credible solution to your problem (couldn't pass it up, no matter how hard I tried).

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