Unnormalized (or energy-based) models provide a flexible framework for capturing the characteristics of data with complex dependency structures. However, the application of standard Bayesian inference methods has been severely limited because the parameter-dependent normalizing constant is either analytically intractable or computationally prohibitive to evaluate. A promising approach is score-based generalized Bayesian inference, which avoids evaluating the normalizing constant by replacing the likelihood with a scoring rule. However, this approach requires careful tuning of the likelihood information, and it may fail to yield valid inference without appropriate control. To overcome this difficulty, we propose a fully Bayesian framework for inference on unnormalized models that does not require such tuning. We build on noise contrastive estimation, which recasts inference as a binary classification problem between observed and noise samples, and treat the normalizing constant as an additional unknown parameter within the resulting likelihood. For exponential families, the classification likelihood becomes conditionally Gaussian via Pólya-Gamma data augmentation, leading to a simple Gibbs sampler. We demonstrate the proposed approach through two models: time-varying density models for temporal point process data and sparse torus graph models for multivariate circular data. Through simulation studies and real-data analyses, the proposed method provides accurate point estimation and enables principled uncertainty quantification.

翻译：非归一化（或基于能量的）模型为捕捉具有复杂依赖结构的数据特征提供了一个灵活的框架。然而，标准贝叶斯推理方法的应用受到了严重限制，因为依赖于参数的归一化常数要么在解析上难以处理，要么在计算上评估成本过高。一种有前景的方法是基于分数的广义贝叶斯推理，它通过用评分规则替换似然函数来避免评估归一化常数。然而，这种方法需要仔细调整似然信息，并且如果没有适当的控制，可能无法产生有效的推理。为了克服这一困难，我们提出了一个用于非归一化模型推理的完全贝叶斯框架，该框架不需要此类调整。我们基于噪声对比估计，将推理重新构建为观测样本与噪声样本之间的二元分类问题，并将归一化常数视为所得似然函数中的一个额外未知参数。对于指数族，通过Pólya-Gamma数据增强，分类似然变为条件高斯分布，从而产生一个简单的吉布斯采样器。我们通过两个模型展示了所提出的方法：用于时间点过程数据的时变密度模型和用于多元循环数据的稀疏环面图模型。通过模拟研究和实际数据分析，所提出的方法提供了准确的点估计，并能够进行有原则的不确定性量化。