When explaining the meaning of a confidence interval don’t say “the probability that the parameter is in the interval is 0.95” because probability is a precious concept and this statement does not match the meaning of this term. Instead, say “We are 95% confident that the parameter is in the interval”. Admittedly, I don’t know what people will make of the word “confident”. But I also don’t know what they will make of the word “probability”

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## 3 comments

April 7, 2016 at 12:28 pm

AaronBut “the probability that the parameter is in the interval is 0.95” is exactly correct. Its just that the randomness over which the probability is taken is not the randomness of the parameter, but the randomness of the interval.

April 7, 2016 at 12:37 pm

EranGood catch Aaron !

The situation I had in mind is after we already came up with the interval, say [3.7, 5.4]. So we should not say there is 95% chance that the parameter is between [3.7, 5.4].

April 7, 2016 at 2:08 pm

JI’m not entirely certain I agree with you. Your entire argument rests on the premise that probability is not valid way to represent uncertainty.

That is to say, your argument is dependent on saying that the true parameter is either in or not in the interval. Thus, there cannot be a chance that it is in the interval (other than 1 and 0) because either p=1 or p=0 is the true probability.

However, if we allow the probability to represent our uncertainty, a probabilistic statement is totally valid. I don’t know if the parameter is in the interval. Based on the math I’ve done, 95% of the time it will be. Thus, I can say there is a 95% chance, because I’m uncertain.

Now, the question is “should we let probability represent uncertainty?” To which I would answer yes unequivocally. You may disagree, but then you need a different system of representing uncertainty. Probability, being the system used in forecasting as well as gambling is natural in this context. If you disagree, I would ask what it means when Nate Silver says the chance of someone winning an election is 60%. There is a response regarding repeating the election over and over, but it isn’t as sensible as saying that the probability is representing his uncertainty.