A common approach to analyze a covariate-sample count matrix, an element of
which represents how many times a covariate appears in a sample, is to
factorize it under the Poisson likelihood. We show its limitation in capturing
the tendency for a covariate present in a sample to both repeat itself and
excite related ones. To address this limitation, we construct negative binomial
factor analysis (NBFA) to factorize the matrix under the negative binomial
likelihood, and relate it to a Dirichlet-multinomial distribution based
mixed-membership model. To support countably infinite factors, we propose the
hierarchical gamma-negative binomial process. By exploiting newly proved
connections between discrete distributions, we construct two blocked and a
collapsed Gibbs sampler that all adaptively truncate their number of factors,
and demonstrate that the blocked Gibbs sampler developed under a compound
Poisson representation converges fast and has low computational complexity.
Example results show that NBFA has a distinct mechanism in adjusting its number
of inferred factors according to the sample lengths, and provides clear
advantages in parsimonious representation, predictive power, and computational
complexity over previously proposed discrete latent variable models, which
either completely ignore burstiness, or model only the burstiness of the
covariates but not that of the factors.

通过构造负二项分析（NBFA）来解决泊松分布捕捉协变量出现在样本中自我重复及与其他协变量之间激发关系的局限性，并利用分层伽马负二项过程支持数不尽的因素。设计了两种基于多项式分布的混合成员模型，实现快速收敛和低计算复杂度的阻塞吉布斯采样器，提供比以前的方法更具优势的紧凑表示，预测能力和计算复杂性。

非参数贝叶斯负二项式因子分析

Nonparametric Bayesian Negative Binomial Factor Analysis

Beta process is the standard nonparametric Bayesian prior for latent factor
model. In this paper, we derive a structured mean-field variational inference
algorithm for a beta process non-negative matrix factorization (NMF) model with
Poisson likelihood. Unlike the linear Gaussian model, which is well-studied in
the nonparametric Bayesian literature, NMF model with beta process prior does
not enjoy the conjugacy. We leverage the recently developed stochastic
structured mean-field variational inference to relax the conjugacy constraint
and restore the dependencies among the latent variables in the approximating
variational distribution. Preliminary results on both synthetic and real
examples demonstrate that the proposed inference algorithm can reasonably
recover the hidden structure of the data.

本文从贝塔过程的角度，推导了一种针对具有 Poisson 分布似然函数的非负矩阵分解 (NMF) 模型的结构均场变分推理算法，并利用最近开发的随机结构均场变分推理方法来恢复潜在变量之间的依赖关系，并在合成和真实数据上进行了初步的实验验证。

具有随机结构均值场变分推理的 Beta 流程非负矩阵分解

Beta Process Non-negative Matrix Factorization with Stochastic  Structured Mean-Field Variational Inference

We develop a Bayesian Poisson matrix factorization model for forming
recommendations from sparse user behavior data. These data are large user/item
matrices where each user has provided feedback on only a small subset of items,
either explicitly (e.g., through star ratings) or implicitly (e.g., through
views or purchases). In contrast to traditional matrix factorization
approaches, Poisson factorization implicitly models each user's limited
attention to consume items. Moreover, because of the mathematical form of the
Poisson likelihood, the model needs only to explicitly consider the observed
entries in the matrix, leading to both scalable computation and good predictive
performance. We develop a variational inference algorithm for approximate
posterior inference that scales up to massive data sets. This is an efficient
algorithm that iterates over the observed entries and adjusts an approximate
posterior over the user/item representations. We apply our method to large
real-world user data containing users rating movies, users listening to songs,
and users reading scientific papers. In all these settings, Bayesian Poisson
factorization outperforms state-of-the-art matrix factorization methods.

研究使用贝叶斯泊松矩阵因式分解模型对大量的稀疏用户行为数据进行推荐，并应用于电影评分、歌曲收听和科学论文阅读等领域，结果表明其优于传统矩阵因式分解方法。