vignettes/intervals.Rmd
intervals.RmdTL; DR: You almost certainly want a predictive interval.
uncertainty is key. the philosophy of safepredict is that you nearly certainly fit a wrong model. i.e. we would like to make predictionsand we have fit the wrong model
regression problems: you typically estimate the conditional mean \(E[Y|X]\) (or perhaps the conditional median or mode) – it doesn’t really matter, the key take away is that you are estimating the center of a distribution.
example: linear regression
\[y = XB + \varepsilon\]
and if you then assume that \(\varepsilon \sim \mathrm{Normal}(0, \sigma^2)\), we get that
suppose you have a distribution, you want to know where most of that
if you are a bayesian, pretty much the same thinking applies, except instead of looking at a sampling distribution of your prediction, you look at the posterior predictive distribution of your prediction and take quantiles of that. different interpretation, and different name (credible interval), but the same idea.
pointwise confidence interval for a new \(x_i'\) in linear regression
you’ve probably been taught to think about confidence intervals. i suggest instead that you always default to predictive intervals, that you sanity check coverage visually, and that you think about multiple testing and whether it will impact you.