Appendix Conditional Independence Dependence in 10H and

We investigate the degree to which our conditional independence assumption is satisfied empirically in the datasets used in the paper. Specifically, of interest is the assumption of conditional independence of m(x) and h(x), given y. Assessing conditional independence is not straightforward given that m(x) is a K-dimensional real-valued vector and h(x) and yeach take one of K categorical values, with K = 10 for CIFAR-10H and K = 16 for ImageNet-16H. While there exist statistical tests for assessing conditional independence for categorical random variables, with real-valued variables the situation is less straightforward and there are multiple options such as different non-parametric tests involving different tradeoffs [Runge, 2018, Marx and Vreeken, 2019, Mukherjee et al., 2020, Berrett et al., 2020]. Given these issues we investigate the degree of conditional dependence using two relatively simple approaches. The first approach looks at the conditional mutual information (CMI) between the predicted label from the model and the predicted label from the human, conditioned on the true label.