Sufficient digits and density estimation: A Bayesian nonparametric approach using generalized finite P'olya trees

2025-06-11

AI Summary

提出基于广义有限Pólya树的贝叶斯非参数方法，通过数字表征解决连续随机变量密度估计问题，避免MCMC计算，并验证了后验一致性。

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper proposes a novel approach for statistical modelling of a continuous random variable $X$ on $[0, 1)$, based on its digit representation $X=.X_1X_2ldots$. In general, $X$ can be coupled with a random variable $N$ so that if a prior of $N$ is imposed, $(X_1,ldots,X_N)$ becomes a sufficient statistics and $.X_{N+1}X_{N+2}ldots$ is uniformly distributed. In line with this fact, and focusing on binary digits for simplicity, we propose a family of generalized finite P{'o}lya trees that induces a random density for a sample, which becomes a flexible tool for density estimation. Here, the digit system may be random and learned from the data. We provide a detailed Bayesian analysis, including closed form expression for the posterior distribution which sidesteps the need of MCMC methods for posterior inference. We analyse the frequentist properties as the sample size increases, and provide sufficient conditions for consistency of the posterior distributions of the random density and $N$. We consider an extension to data spanning multiple orders of magnitude, and propose a prior distribution that encodes the so-called extended Newcomb-Benford law. Such a model shows promising results for density estimation of human-activity data. Our methodology is illustrated on several synthetic and real datasets.

Problem

• Estimating continuous variable density via digit representation
• Developing Bayesian nonparametric method using Pólya trees
• Ensuring posterior consistency for random density and digit length

Innovation

• Bayesian nonparametric approach with Pólya trees
• Closed-form posterior avoids MCMC methods
• Learns digit system from data