**How to misunderstand confidence intervals**

It is very tempting to misunderstand this statement in the following way. We used capital letters *U* and *V* for random variables; it is conventional to use lower-case letters *u* and *v* for their observed values in a particular instance. The misunderstanding is the conclusion that

Consequently

so the interval from *X* - 1.645 to *X* + 1.645 is a 90% confidence interval for θ. But when *X* = 82 is observed, can we then say that

This conclusion does not follow from the laws of probabilty because θ is not a "random variable"; i.e., no probability distribution has been assigned to it. Confidence intervals are generally a frequentist method, i.e., employed by those who interpret "90% probability" as "occurring in 90% of all cases". Suppose, for example, that θ is the mass of the planet Neptune, and the randomness in our measurement error means that 90% of the time our statement that the mass is between this number and that number will be correct. The mass is not what is random. Therefore, given that we have measured it to be 82 units, we cannot say that in 90% of all cases, the mass is between 82 - 1.645 and 82 + 1.645. There are no such cases; there is, after all, only one planet Neptune.

But if probabilities are construed as degrees of belief rather than as relative frequencies of occurrence of random events, i.e., if we are Bayesians rather than frequentists, can we *then* say we are 90% sure that the mass is between 82 − 1.645 and 82 + 1.645? Many answers to this question have been proposed, and are philosophically controversial. The answer will not be a mathematical theorem, but a philosophical tenet.

[*I will add an example of a "recognizable subset" here; i.e., a case in which the data themselves make the epistemic conclusion dubious.*]

**Concrete practical examples**

Here is one of the most familiar realistic examples. Suppose *X*_{1}, ..., *X*_{n} are an independent sample from a normally distributed population with mean μ and variance σ^{2}. Let

Consequently

and we have a 90% confidence interval for μ.