We are interested in estimating the uncertainties of deep neural networks, which play an important role in many scientific and engineering problems. In this paper, we present a striking new finding that an ensemble of neural networks with the same weight initialization, trained on datasets that are shifted by a constant bias gives rise to slightly inconsistent trained models, where the differences in predictions are a strong indicator of epistemic uncertainties. Using the neural tangent kernel (NTK), we demonstrate that this phenomena occurs in part because the NTK is not shift-invariant. Since this is achieved via a trivial input transformation, we show that this behavior can therefore be approximated by training a single neural network -- using a technique that we call $Δ-$UQ -- that estimates uncertainty around prediction by marginalizing out the effect of the biases during inference. We show that $Δ-$UQ's uncertainty estimates are superior to many of the current methods on a variety of benchmarks -- outlier rejection, calibration under distribution shift, and sequential design optimization of black box functions. Code for $Δ-$UQ can be accessed at https://github.com/LLNL/DeltaUQ