Unbiased Test Error Estimation in the Poisson Means Problem via Coupled Bootstrap Techniques

We propose a coupled bootstrap (CB) method for the test error of an arbitrary algorithm that estimates the mean in a Poisson sequence, often called the Poisson means problem. The idea behind our method is to generate two carefully-designed data vectors from the original data vector, by using synthetic binomial noise. One such vector acts as the training sample and the second acts as the test sample. To stabilize the test error estimate, we average this over multiple bootstrap B of the synthetic noise. A key property of the CB estimator is that it is unbiased for the test error in a Poisson problem where the original mean has been shrunken by a small factor, driven by the success probability $p$ in the binomial noise. Further, in the limit as $B \to \infty$ and $p \to 0$, we show that the CB estimator recovers a known unbiased estimator for test error based on Hudson's lemma, under no assumptions on the given algorithm for estimating the mean (in particular, no smoothness assumptions). Our methodology applies to two central loss functions that can be used to define test error: Poisson deviance and squared loss. Via a bias-variance decomposition, for each loss function, we analyze the effects of the binomial success probability and the number of bootstrap samples and on the accuracy of the estimator. We also investigate our method empirically across a variety of settings, using simulated as well as real data.