We propose two novel approaches to address a critical problem of reach measurement across multiple media -- how to estimate the reach of an unobserved subset of buying groups (BGs) based on the observed reach of other subsets of BGs. Specifically, we propose a model-free approach and a model-based approach. The former provides a coarse estimate for the reach of any subset by leveraging the consistency among the reach of different subsets. Linear programming is used to capture the constraints of the reach consistency. This produces an upper and a lower bound for the reach of any subset. The latter provides a point estimate for the reach of any subset. The key idea behind the latter is to exploit the conditional independence model. In particular, the groups of the model are created by assuming each BG has either high or low reach probability in a group, and the weights of each group are determined through solving a non-negative least squares (NNLS) problem. In addition, we also provide a framework to give both confidence interval and point estimates by integrating these two approaches with training points selection and parameter fine-tuning through cross-validation. Finally, we evaluate the two approaches through experiments on synthetic data.

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