We propose an analytical solution for approximating the gradient of the Evidence Lower Bound (ELBO) in variational inference problems where the statistical model is a Bayesian network consisting of observations drawn from a mixture of a Gaussian distribution embedded in unrelated clutter, known as the clutter problem. The method employs the reparameterization trick to move the gradient operator inside the expectation and relies on the assumption that, because the likelihood factorizes over the observed data, the variational distribution is generally more compactly supported than the Gaussian distribution in the likelihood factors. This allows efficient local approximation of the individual likelihood factors, which leads to an analytical solution for the integral defining the gradient expectation. We integrate the proposed gradient approximation as the expectation step in an EM (Expectation Maximization) algorithm for maximizing ELBO and test against classical deterministic approaches in Bayesian inference, such as the Laplace approximation, Expectation Propagation and Mean-Field Variational Inference. The proposed method demonstrates good accuracy and rate of convergence together with linear computational complexity.

翻译：我们提出了一种解析解，用于在变分推断问题中近似证据下界（ELBO）的梯度。该问题中的统计模型是一个贝叶斯网络，其观测数据来自嵌入无关杂波的高斯混合分布（即杂波问题）。该方法利用重参数化技巧将梯度算子移入期望内部，并基于一个假设：由于似然函数相对于观测数据可分解，变分分布通常比似然因子中的高斯分布具有更紧凑的支撑域。这一性质使得对各似然因子进行高效的局部近似成为可能，进而得到定义梯度期望的积分的解析解。我们将所提出的梯度近似作为期望最大化（EM）算法中的期望步骤，用于最大化ELBO，并与贝叶斯推断中的经典确定性方法（如拉普拉斯近似、期望传播和平均场变分推断）进行对比实验。结果表明，所提方法在保持线性计算复杂度的同时，具有良好的精度与收敛速度。