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The quantile function is simply the inverse of the cumulative distribution function (if it exists). Its graph can be produced by flipping the cumulative distribution function’s graph over.

The quantile function is a mathematical function that takes a quantile (a percentage of a distribution, from 0 to 1) as input and outputs the value of a variable. It can answer questions like, “If I want to guarantee that 95% of my customers receive their orders within 24 hours, how much inventory do I need to keep on hand?” As such, the quantile function is commonly used in the context of forecasting questions.

In practical cases, however, we rarely have a tidy formula for computing the quantile function. Instead, statisticians usually use regression analysis to approximate it for a single quantile level at a time. That means that if you decide you want to compute it for a different quantile, you have to build a new regression model — which, today, often means retraining a neural network.

In a pair of papers presented at this year’s International Conference on Artificial Intelligence and Statistics (AISTATS), Amazon researchers describe an approach to learning an approximation of the entire quantile function at once, rather than simply approximating it for each quantile level.

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