Berry–Esseen bounds for Chernoff-type nonstandard asymptotics in isotonic regression

Risk Bounds in Isotonic Regression

Nonasymptotic risk bounds are provided for maximum likelihood-type isotonic estimators of an unknown nondecreasing regression function, with general average loss at design points. These bounds are optimal up to scale constants, and they imply uniform n−1/3-consistency of the p risk for unknown regression functions of uniformly bounded variation, under mild assumptions on the joint probability d...

Improved Risk Bounds in Isotonic Regression

We consider the problem of estimating an unknown non-decreasing sequence θ from finitely many noisy observations. We give an improved global risk upper bound for the isotonic least squares estimator (LSE) in this problem. The obtained risk bound behaves differently depending on the form of the true sequence θ – one gets a whole range of rates from log n/n (when θ is constant) to n−2/3 (when θ i...

Chernoff Bounds

If m = 2, i.e., P = (p, 1 − p) and Q = (q, 1 − q), we also write DKL(p‖q). The Kullback-Leibler divergence provides a measure of distance between the distributions P and Q: it represents the expected loss of efficiency if we encode an m-letter alphabet with distribution P with a code that is optimal for distribution Q. We can now state the general form of the Chernoff Bound: Theorem 1.1. Let X1...

Chernoff - Bounds Wolfgang Mulzer

Variational Chernoff Bounds for Graphical Models

Recent research has made significant progress on the problem of bounding log partition functions for exponential family graphical models. Such bounds have associated dual parameters that are often used as heuristic estimates of the marginal probabilities required in inference and learning. However these variational estimates do not give rigorous bounds on marginal probabilities, nor do they giv...