Making valid statistical inferences from privatized data is a key challenge in modern analysis. In Bayesian settings, data augmentation MCMC (DAMCMC) methods impute unobserved confidential data given noisy privatized summaries, enabling principled uncertainty quantification. However, standard DAMCMC often suffers from slow mixing due to component-wise Metropolis-within-Gibbs updates. We propose the Single-Offer-Multiple-Attempts (SOMA) sampler. This novel algorithm improves acceptance rates by generating a single proposal and simultaneously evaluating its suitability to replace all components. By sharing proposals across components, SOMA rejects fewer proposal points. We prove lower bounds on SOMA's acceptance probability and establish convergence rates in the two-component case. Experiments on synthetic and real census data with linear regression and other models confirm SOMA's efficiency gains.

翻译：从私有化数据中做出有效的统计推断是现代分析中的关键挑战。在贝叶斯框架下，数据增强MCMC（DAMCMC）方法基于噪声化的私有化摘要数据对未观测的机密数据进行插补，从而实现原则性的不确定性量化。然而，标准的DAMCMC常因采用分量式Metropolis-within-Gibbs更新而导致混合速度缓慢。我们提出单提议多尝试（SOMA）采样器。该新颖算法通过生成单一提议并同时评估其替换所有分量的适用性，提高了接受率。通过跨分量共享提议，SOMA拒绝了更少的提议点。我们证明了SOMA接受概率的下界，并在二分量的情况下建立了收敛速率。在线性回归及其他模型上对合成数据和真实人口普查数据的实验验证了SOMA的效率提升。