Why do we need to add confidence intervals to model predictions in cases we don't know the true data distribution?

Question

There are cases (arguably the vast majority) where the data distribution is unknown. Confidence intervals make sense for the class of "nicely" defined theoretical pdfs ie Gaussian. In this case, we know that additional samples help to derive the characteristics (statistical moments) of the pdf. However, there are other distributions in which additional sampling is not helping to converge to estimates but on the contrary diverge, for example, Cauchy distribution

sources:

• https://en.wikipedia.org/wiki/Cauchy_distribution#/media/File:Mean_estimator_consistency.gif
• https://en.wikipedia.org/wiki/Cauchy_distribution

So with this context, my question is: Why do we need to add confidence intervals (or the equivalent reporting mean and variance) to model predictions in cases we don't know the true data distribution?

I would like to focus the question in the context of Machine learning inference evaluation scores a common example is to add +/- in the score interval usually by rerunning the model, with some different random seed conditions.

(note that this is distinct question from from k-fold method discussed here: https://datascience.stackexchange.com/questions/108792/why-is-the-k-fold-cross-validation-needed but it could be also a relevant method there if the data is violating some of Central limit theorem conditions such as being independently and identically distributed. )

Answer 1

We only assume that the distribution is Gaussian for convenience, and mathematical results like the Central Limit Theorem tell us that assuming a Gaussian is a good approximation to the true distribution of uncertain quantities.

If we did not make simplifying assumptions then we would not be able to communicate any uncertainty in our predictions. Yes, our mean and variance estimates are wrong (or perhaps, imperfect would be a better word), but reporting only a point estimate ignores all uncertainty, which is clearly much worse than making some simplifying assumptions.

Answer 2

Apart from simulated examples, we never know the true distribution of the quantities we are looking at, whether these are point predictions or summary statistics like the AUROC you give as an example. We don't even know whether they satisfy the conditions for the Central Limit Theorem so that we could assume asymptotic normality - and even if we could, we rarely know how good this asymptotic approximation is.

However, this does not matter. Rarely does reporting a mean plus/minus a standard deviation imply a confidence interval. Rather, it is simply a description of the variability of the quantity being reported. Thus, it is an example of descriptive statistics, rather than inferential statistics.

Of course, if someone claims a confidence interval with specific properties, e.g., based on a normal distribution with estimated means and variances, then your point certainly holds: insofar as we don't know the true distribution, we have to trust in the CLT that we are at least approximately right.

This makes more sense in certain applications than in others, and knowing when it does and when it does not is part of the domain knowledge of specific areas of application in statistics. For instance, since the AUROC is bounded, it is certainly not Cauchy, and I would have no qualms about accepting an asymptotically normal distribution of AUROCs, with corresponding confidence intervals, as long as the sample size is large enough.