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I've read this before and I'm still confused at the difference... can someone make it intuitive? In my textbook, this is written:

The correct way to speak of confidence is intervals is to say: 'We are XX% confident that the population parameter is between Z and Y...'

Incorrect language might try to describe the confidence interval as capturing the population parameter with a certain probability. This is a common error: while it might be useful to think of it as a probability, the confidence level only quantifies how plausible it is that the parameter is in the interval.

How are those two different?

Let's refer to the statements using A) and B)

A) 'We are XX% confident that the population parameter is between Z and Y...'

and

B) 'The confidence interval from Y to Z captures the population mean 95% of the time`.

Jwan622
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  • Confidence interval calculations are based on independence assumptions and Gaussian approximations, which are not necessarily true for the given problem of interest. So they use language of "confidence" to mean that, while they do not have an accurate probability model to say the results are "probabilities," they are confident in the sense that "these numbers are often meaningful in related problems of interest." – Michael Mar 04 '17 at 19:05
  • In a model where independence and Gaussian assumptions hold exactly, but where a parameter $\mu$ is unknown, one can often compute probabilities of the form $P[A \leq \mu \leq B]$ where $A$ and $B$ are random observations. But for given observations $A=a,B=b$, one cannot compute conditional probabilities $P[ a \leq \mu \leq b | A=a,B=b]$ since we cannot use Baye’s rule without an a-priori probability on $\mu$. So you can say “given our experiment produced $A=a, B=b$, we are $P[A \leq \mu \leq B]$ confident that $\mu \in [a,b]$,” and people known what you mean when you say this. – Michael Mar 04 '17 at 19:47
  • I'm a beginner student and I don't understand. – Jwan622 Mar 05 '17 at 03:43

1 Answers1

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I do not think my comments above used advanced language. In any case, you did not specify what was confusing. A summary of my above comments would be: The "confidence" language lets people know you are talking about “confidence interval type problems” and is useful for approximations and/or for cases when no probability rule is known for some parameters of interest.

Specifically on the statements “A” and “B” of your question:

Let’s suppose you are applying confidence interval formulas on a batch of data in order to estimate a parameter $\mu$ given “random” observations $Y$ and $Z$ (where $Y$ and $Z$ are obtained from the data). There are confidence interval formulas that allow you to approximate the probability $P[Y \leq \mu \leq Z]$. Those formulas are exact when your data satisfies certain Gaussian assumptions. So statement A just means $P[Y\leq \mu \leq Z] \approx 0.95$, while statement B means $P[Y \leq \mu \leq Z] =0.95$. Statements A and B are the same if the Gaussian assumptions hold.

Why should we be “confident” of the approximation if the Gaussian assumptions do not hold? It is because those approximations have been observed to be useful in many cases even when Gaussian assumptions fail. There is also a “central limit theorem” which mathematically shows that Gaussian approximations are good in certain cases (particularly when “independence” assumptions hold).

Note that both statements A and B are different from the conditional probability statement $P[9.4 \leq \mu \leq 11.2 | Y=9.4, Z=11.2]=0.95$. It is impossible to compute this conditional probability without knowing an a-priori probability distribution for the unknown parameter $\mu$. Some people get statement A mixed up with this conditional probability statement, and that is what your book is likely reminding you not to do. If you do not know what conditional probability is, then don’t worry: You are not in danger of mixing things up!

Michael
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