This paper focuses on the Bregman divergence defined by the reciprocal function, called the inverse divergence. It clarifies the combinations of the statistical model and function f that eliminate the bias correction term when using inverse divergence for estimation.
For the IGT distribution, the unbiased estimating equation holds without a bias correction term if and only if the integral of the product of the generating function g and the derivative of the function f is bounded. This condition is different from the condition for the GIGT mixture distribution, which is a generalization of the continuous Bregman distribution.
The paper also extends the results to the multi-dimensional case by expressing the inverse divergence as a linear sum over the dimensions. The corresponding statistical model, the multivariate IGT (MIGT) distribution, is newly defined, and the condition for the function f is provided as a double integral.
The IGT and GIGT mixture distributions are special cases of the regular exponential family, and the inverse divergence is a special case of the β-divergence. This suggests that the inverse divergence is a unique Bregman divergence that ensures the unbiased estimating equation without a bias correction term for certain statistical models.
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by Masahiro Kob... at arxiv.org 04-26-2024
https://arxiv.org/pdf/2404.16519.pdfDeeper Inquiries