Summary
This paper discusses, with measure-theoretical rigor, some basic aspects of the theory of separate inference. To analyze densities of marginal and conditional submodels, certain operators are introduced. First a general concept of decomposition of a model is proposed, and the corresponding factorization of densities of the model is established. Next it is shown that the property of smoothness of a family of densities is retained in the operation of conditioning, and therefore it yields the differentiability of the conditional expectation of a real-valued statistic in a certain sense. On the basis of this result, two measures of the effectiveness of a submodel in separate inference are investigated.
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Kuboki, H. Analysis of marginal and conditional density functions for separate inference. Ann Inst Stat Math 39, 1–23 (1987). https://doi.org/10.1007/BF02491445
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DOI: https://doi.org/10.1007/BF02491445