信息几何中的拉普拉斯连续法则
Laplace's rule of succession in information geometry
摘要：
拉普拉斯的“加一”连续规则通过在观察到的计数上加一来修改头部和尾部序列中观察到的频率。这通过避免零概率来改进预测，并且对应于参数的均匀贝叶斯先验。典型的Jeffreys先验对应于“加一半”规则。我们证明了，指数分布的家庭，这样的贝叶斯预测可以近似的最大似然预测和\emph{sequential normalized maximum likelihood}预测从信息论的平均值。因此，在这种情况下，可以近似贝叶斯预测，而无需在参数空间中进行积分或采样。
英文摘要：
Laplace's "add-one" rule of succession modifies the observed frequencies in a sequence of heads and tails by adding one to the observed counts. This improves prediction by avoiding zero probabilities and corresponds to a uniform Bayesian prior on the parameter. The canonical Jeffreys prior corresponds to the "add-one-half" rule. We prove that, for exponential families of distributions, such Bayesian predictors can be approximated by taking the average of the maximum likelihood predictor and the \emph{sequential normalized maximum likelihood} predictor from information theory. Thus in this case it is possible to approximate Bayesian predictors without the cost of integrating or sampling in parameter space.
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