Interpreting a confidence interval

Question

Everything I read about confidence intervals warns against the following interpretation: If I construct a 95% confidence interval (a,b) for some parameter p, it is incorrect to say that the probability that p is in (a,b) is .95.

Mathematically, I totally get it. a, b, and p are constants, so the probability is either 1 or 0.

However, philosophically, I don't see the problem with constructing an interval based on a sample and saying that the probability that your interval contains the unknown parameter is .95 since probability is, after all, a consequence of uncertainty in the face of imperfect information.

For example, before I flip a coin, there is a 50% chance it lands on heads. Suppose I flip the coin but I don't look at the outcome. Is it wrong to say that the probability that the coin is currently heads up is 50%, even though it's and 'unknown constant?'

Answer 1

This issue for interpreting a confidence interval in that manner is more that the frequentist statistical theory says the true parameter value is constant so a randomly generated interval will either contain or exclude that fixed parameter value (slightly different look than yours).

If you want to make that kind of probability statement, a Bayesian credible interval will allow for it because it allows parameters to be random variables and via the incorporation of prior information and current information you can make a probability statement about an interval containing a parameter which is a random variable (contrary to a fixed value of a parameter).

The coin question depends on the definition of probability for whoever you ask.

Answer 2

For a sample as opposed to a population, the mean and the confidence interval are estimates. Consequently, to obtain a prediction interval as opposed to a confidence interval, you would need to take account of the potential errors in the calculation of the mean and the variance. This reference provides a more complete explanation. In the specific case of tossing a coin, it would be reasonable to presume that the probability of getting an head or a tail is equal, apart from the incredibly rare case where it lands and stays on its edge.