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：我们提出一种解析方法，用于近似变分推断中证据下界（ELBO）的梯度，其中统计模型为贝叶斯网络，其观测数据来自嵌入无关杂波中的高斯分布混合（即杂波问题）。该方法采用重参数化技巧将梯度算子移入期望内部，并基于以下假设：由于似然函数对观测数据具有因子化结构，变分分布的支撑集通常比似然因子中的高斯分布更紧凑。这一特性使得对单个似然因子进行高效局部近似成为可能，从而导出梯度期望积分式的解析解。我们将所提出的梯度近似方法作为期望最大化（EM）算法中最大化ELBO的期望步，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）进行对比测试。实验表明，本方法具有较高的精度与收敛速度，同时保持线性计算复杂度。