Bernoulli序列的MDL收敛速度
MDL Convergence Speed for Bernoulli Sequences
摘要：
研究了在线序列估计/预测的最小描述长度原则。如果底层模型类是离散的，那么总期望平方损失是一个特别有趣的性能度量：（a）这个量是有界的，意味着以概率1收敛，（b）它还指定了收敛速度。对于MDL，在一般情况下，只能有损失的界限是有限的，但指数大于贝叶斯混合。我们表明，即使是这种情况下，如果模型类只包含伯努利分布。我们得到了可数Bernoulli类预测误差的一个新的上界。这意味着对于某些重要的模型类，有一个小的界限（与贝叶斯混合模型的界限相当）。我们讨论了机器学习任务的应用，如分类和假设检验，并推广到可数类的独立同分布。模型
英文摘要：
The Minimum Description Length principle for online sequence estimation/prediction in a proper learning setup is studied. If the underlying model class is discrete, then the total expected square loss is a particularly interesting performance measure: (a) this quantity is finitely bounded, implying convergence with probability one, and (b) it additionally specifies the convergence speed. For MDL, in general one can only have loss bounds which are finite but exponentially larger than those for Bayes mixtures. We show that this is even the case if the model class contains only Bernoulli distributions. We derive a new upper bound on the prediction error for countable Bernoulli classes. This implies a small bound (comparable to the one for Bayes mixtures) for certain important model classes. We discuss the application to Machine Learning tasks such as classification and hypothesis testing, and generalization to countable classes of i.i.d. models.
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