# Confidence intervals fallacy once again — second attempt

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.

• 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 $P(\theta \in CI)$ ? – user83346 Sep 15 '16 at 17:45
• @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 $\Pr(X>y)$.) – Scortchi - Reinstate Monica Sep 15 '16 at 19:45