Supplementary Material for Conformal Prediction using Conditional Histograms Matteo Sesia Department of Data Sciences and Operations University of Southern California, USA

S1.1 Estimating conditional distributions and histograms For any fixed K > 1, define the sequence a Note that we allow multiple estimated quantiles to be identical to each other, to accommodate the possibility of point masses. We will discuss in the next section practical options for estimating ˆq(x). Although there are multiple way of doing this, a principled solution is to convert the information contained in ˆq into a piece-wise constant density estimate, and then integrate that density within each bin. As the tails of the above estimated conditional density may be particularly inaccurate because relatively little information is available to estimate extremely low or high quantiles, we smooth them. This ensures any estimation errors will not make ˆf decay too fast, forcing one to look much farther than necessary in the tails before finding sufficient mass for the desired prediction intervals.