有根树的概率分布
Probability Distribution on Rooted Trees
摘要：
有根树的分层和递归表达能力适用于表示各种领域中的统计模型，例如数据压缩、图像处理和机器学习。另一方面，这种层次化的表达能力在树的选择中引起问题，以避免过拟合。解决这个问题的一种统一方法是贝叶斯方法，在该方法中，有根树被视为随机变量，并且可以在所选模型或新数据点的预测值上假设直接损失函数。然而，据我们所知，以前所有关于这种方法的研究都是基于满树上的概率分布。在本文中，我们提出了一个广义概率分布的任何根树，其中只有最大数量的子节点和最大深度是固定的。此外，我们推导出递归方法来评估没有任何近似的概率分布的特性。
英文摘要：
The hierarchical and recursive expressive capability of rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. On the other hand, such hierarchical expressive capability causes a problem in tree selection to avoid overfitting. One unified approach to solve this is a Bayesian approach, on which the rooted tree is regarded as a random variable and a direct loss function can be assumed on the selected model or the predicted value for a new data point. However, all the previous studies on this approach are based on the probability distribution on full trees, to the best of our knowledge. In this paper, we propose a generalized probability distribution for any rooted trees in which only the maximum number of child nodes and the maximum depth are fixed. Furthermore, we derive recursive methods to evaluate the characteristics of the probability distribution without any approximations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
