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.

本文提出一种用于逼近变分推断问题中 Evidence Lower Bound（ELBO）梯度的解析解，其中统计模型是由从一个高斯分布嵌入在无关干扰中得到的观测值构成的贝叶斯网络，该方法采用重新参数化技巧将梯度操作符移至期望内部，并依赖于似然函数在观测数据上分解的假设，从而导致变分分布通常比似然函数中的高斯分布更紧支撑，这使得对各个似然因子的高效局部近似成为可能，从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集成为 EM（期望最大化）算法中的期望步骤，并在贝叶斯推断中与经典的确定性方法，如拉普拉斯逼近、期望传播和均场变分推断进行了测试，所提出的方法表现出良好的准确性和收敛速度，并具有线性计算复杂度。