Tim van Erven

Rényi Divergence and Kullback-Leibler Divergence

Rényi divergence is related to Rényi entropy much like Kullback-Leibler divergence is related to Shannons entropy, and comes up in many settings. It was introduced by Rényi as a measure of information that satisfies almost the same axioms as Kullback-Leibler divergence, and depends on a parameter that is called its order. In particular, the Rényi divergence of order 1 equals the Kullback-Leibler divergence. We review and extend the most important properties of Rényi divergence and Kullback- Leibler divergence, including convexity, continuity, limits of σ-algebras, and the relation of the special order 0 to the Gaussian dichotomy and contiguity. We also show how to generalize the Pythagorean inequality to orders different from 1, and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs (for all orders) and present several other minimax results.

Rényi divergence and majorization

Rényi divergence is related to Rényi entropy much like information divergence (also called Kullback-Leibler divergence or relative entropy) is related to Shannons entropy, and comes up in many settings. It was introduced by Rényi as a measure of information that satisfies almost the same axioms as information divergence. We review the most important properties of Rényi divergence, including its relation to some other distances. We show how Rényi divergence appears when the theory of majorization is generalized from the finite to the continuous setting. Finally, Rényi divergence plays a role in analyzing the number of binary questions required to guess the values of a sequence of random variables.

Game-Theoretically Optimal Reconciliation of Contemporaneous Hierarchical Time Series Forecasts

In hierarchical time series (HTS) forecasting, the hierarchical relation between multiple time series is exploited to make better forecasts. This hierarchical relation implies one or more aggregate consistency constraints that the series are known to satisfy. Many existing approaches, like for example bottom-up or top-down forecasting, therefore attempt to achieve this goal in a way that guarantees that the forecasts will also be aggregate consistent. We propose to split the problem of HTS into two independent steps: first one comes up with the best possible forecasts for the time series without worrying about aggregate consistency; and then a reconciliation procedure is used to make the forecasts aggregate consistent. We introduce a Game-Theoretically OPtimal (GTOP) reconciliation method, which is guaranteed to only improve any given set of forecasts. This opens up new possibilities for constructing the forecasts. For example, it is not necessary to assume that bottom-level forecasts are unbiased, and aggregate forecasts may be constructed by regressing both on bottom-level forecasts and on other covariates that may only be available at the aggregate level. We illustrate the benefits of our approach both on simulated data and on real electricity consumption data.