# Confidence interval semantics

Suppose the 95th percentile confidence interval is $(a, b)$. Are the following two statements equivalent? If not, what is the difference?

Statement 1: There is a 95% chance that the interval $(a, b)$ contains the true (unknown) parameter.

Statement 2: There is a 95% chance that the true (unknown) parameter is between $a$ and $b$.

I saw a biostatistics lecture note that claims that Statement 1 is correct, while Statement 2 is incorrect.

EDIT: Additionally, the lecture note says that the uncertainty is associated with the confidence interval, not the true parameter. (This is a frequentist lecture taught by a Bayesian.)

EDIT2: Another example given in the context:

Incorrect statement: There is a 95% chance that Mozart was born between 1709 and 1799. Why? Mozart was born in 1756, and this fact does not change based on the estimation procedure.

The difference in these statements is that the "95% chance" refers to different things. That is in statement 1 we treat data $D$ as random, and thus also the interval as it is a function of $D$, and $\theta$ stays fixed

$$Conf_{\alpha}(\theta) = (l, u):\ P(l(\bar{D}) \le \theta \le u(\bar{D})|\bar{D} \sim P_{\theta}) = 1 - \alpha$$

So if we sample some new, future data $\bar{D}$ from true distribution $P_{\theta}$, then $(l(\bar{D}), u(\bar{D}))$ is a confidence interval if it contains $\theta$ roughly $1 - \alpha$ percent of time.

Whereas a bayesian credible interval is true for the second statement as

$$Cred_{\alpha}(D) = (l, u):\ P(l \le \theta \le u|D) = 1 - \alpha$$

We treat data $D$ as fixed and $\theta$ as a random variable. Thus we can say that there is some percent chance for $\theta$ to be between $(l, u)$.

• I see. So the emphasis is that $\theta$ is not a random variable in frequentist statistics, and what is random is the CI. Thanks for the clear distinction. IMHO, English is not clear for distinguishing the two, but the equations definitely are clear!! – Memming Mar 9 '17 at 14:34
• @Memming Yes I agree, the difference between those two statements is not clear for someone who doesn't know the formulas in the first place :) – Łukasz Grad Mar 9 '17 at 14:34
• The only thing to add: The statements as given in that lecture seem really poorly phrased to me (unless some important details are ommitted in the question), I wonder how the lecturer hoped people would get the correct understanding (as given in the answer above) from these statements... – Björn Mar 9 '17 at 14:42
• @Björn There were several slides on this issue, and I only reproduced what seems to be the core distinction. The actual lecture was probably more clear than just those words. :) – Memming Mar 9 '17 at 14:46

The definitions of 95% CIs I've seen tend to use the verb 'contain'. Surely any parameter that is between the start and end of the interval could also be said to be contained by it...

• That was my reasoning too, but I'm trying to understand what the intention of the professor who wrote the notes. – Memming Mar 9 '17 at 13:57