Abstract
A credible band is the set of all functions between a lower and an upper bound that are constructed so that the set has prescribed mass under the posterior distribution. In a Bayesian analysis such a band is used to quantify the remaining uncertainty on the unknown function in a similar manner as a confidence band. We investigate the validity of a credible band in the nonparametric regression model with the prior distribution on the function given by a Gaussian process. We show that there are many true regression functions for which the credible band has the correct order of magnitude to be used as a confidence set. We also exhibit functions for which the credible band is misleading.
Highlights
Introduction and main resultsSuppose that we observe a vector Yn := (Y1,n, . . . , Yn,n)T with coordinates distributed according toYi,n = f (xi,n) + εi,n, i ∈ {1, . . . , n}. (1.1)Here the parameter is a function f : [0, 1] → R, the design points (xi,n) are a known sequence of points in [0, 1], and the errors εi,n are independent standard normal random variables
In this paper we investigate a nonparametric Bayesian method to estimate the regression function f, based on a Gaussian process prior
We are interested in the usefulness of the resulting posterior distribution for quantifying the remaining uncertainty about the function
Summary
Suppose that we observe a vector Yn := (Y1,n, . . . , Yn,n)T with coordinates distributed according to. The theorem shows that empirical or hierarchical Bayes credible bands cover the true function if the Holder smoothness α of the function f is at least the order of self-similarity β. The preceding theorems show that the credible bands cover the true regression function in some generality, and justify the use of the posterior distribution as an expression of remaining uncertainty. If the credible band covers the true function, its width gives the rate of estimation by the posterior mean for the (discrete) uniform norm. For a true function that is regular of order β in an appropriate L2-sense they choose a value of c that balances squared bias and variance in the form This yields a rate of contraction relative to the L2-norm of (c/n)1/4 = n−β/(2β+1). Its nontrivial quantiles must be in this range
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