Data from large surveys are often supplemented with sampling weights that are designed to reflect unequal probabilities of response and selection inherent in complex survey sampling methods. We propose two methods for Bayesian estimation of parametric models in a setting where the survey data and the weights are available, but where information on how the weights were constructed is unavailable. The first approach is to simply replace the likelihood with the pseudo likelihood in the formulation of Bayes theorem. This is proven to lead to a consistent estimator but also leads to credible intervals that suffer from systematic undercoverage. Our second approach involves using the weights to generate a representative sample which is integrated into a Markov chain Monte Carlo (MCMC) or other simulation algorithms designed to estimate the parameters of the model. In the extensive simulation studies, the latter methodology is shown to achieve performance comparable to the standard frequentist solution of pseudo maximum likelihood, with the added advantage of being applicable to models that require inference via MCMC. The methodology is demonstrated further by fitting a mixture of gamma densities to a sample of Australian household income.

中文翻译：

---

来自调查的贝叶斯加权推理

来自大型调查的数据通常补充有抽样权重，这些权重旨在反映复杂调查抽样方法中固有的响应和选择的不平等概率。我们提出了两种参数模型的贝叶斯估计方法，其中调查数据和权重可用，但有关如何构建权重的信息不可用。第一种方法是在贝叶斯定理的表述中简单地用伪似然替换似然。事实证明，这会导致一致的估计量，但也会导致可信区间受到系统覆盖不足的影响。我们的第二种方法涉及使用权重来生成一个有代表性的样本，该样本被集成到马尔可夫链蒙特卡罗 (MCMC) 或其他旨在估计模型参数的模拟算法中。在广泛的模拟研究中，后一种方法被证明可以实现与伪最大似然标准常客解决方案相当的性能，并且具有适用于需要通过 MCMC 进行推理的模型的额外优势。通过将混合伽马密度拟合到澳大利亚家庭收入样本中，进一步证明了该方法。具有适用于需要通过 MCMC 进行推理的模型的额外优势。通过将混合伽马密度拟合到澳大利亚家庭收入样本中，进一步证明了该方法。具有适用于需要通过 MCMC 进行推理的模型的额外优势。通过将混合伽马密度拟合到澳大利亚家庭收入样本中，进一步证明了该方法。