Variational inference is an umbrella term for algorithms which cast Bayesian inference as optimization. Classically, variational inference uses the Kullback-Leibler divergence to define the optimization. Though this divergence has been widely used, the resultant posterior approximation can suffer from undesirable statistical properties. To address this, we reexamine variational inference from its roots as an optimization problem. We use operators, or functions of functions, to design variational objectives.

We develop a Coordinate Ascent Variational Inference (CAVI) algorithm for Bayesian Mixed Data Sampling (MIDAS) regression with linear weight parameterizations. The model separates impact coeffcients from weighting function parameters through a normalization constraint, creating a bilinear structure that renders generic Hamiltonian Monte Carlo samplers unreliable while preserving conditional conjugacy exploitable by CAVI. Each variational update admits a closed-form solution: Gaussian for regression coefficients and weight parameters, Inverse-Gamma for the error variance. The algorithm propagates uncertainty across blocks through second moments, distinguishing it from naive plug-in approximations. In a Monte Carlo study spanning 21 data-generating configurations with up to 50 predictors, CAVI produces posterior means nearly identical to a block Gibbs sampler benchmark while achieving speedups of 107x to 1,772x (Table 9). Generic automatic differentiation VI (ADVI), by contrast, produces bias 714 times larger while being orders of magnitude slower, confirming the value of model-specific derivations. Weight function parameters maintain excellent calibration (coverage above 92%) across all configurations. Impact coefficient credible intervals exhibit the underdispersion characteristic of mean-field approximations, with coverage declining from 89% to 55% as the number of predictors grows a documented trade-off between speed and interval calibration that structured variational methods can address. An empirical application to realized volatility forecasting on S&P 500 daily returns cofirms that CAVI and Gibbs sampling yield virtually identical point forecasts, with CAVI completing each monthly estimation in under 10 milliseconds.

To do so, we first form a family of unnormalized densities (or a "path") parameterized by 2 [0, 1] between the two distributions of interest

Much of the recent progress has been achieved by taking advantage of continuous relaxations of the system, which are not always available or even possible.

We use the discrete variablezd,t to represent the topic that thetth word is generated from.

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We perform experiments on high-dimensional Gaussians, Lotka-V olterra dynamics, and a Pickover attractor.

Symmetries have proven useful in machine learning models, improving generalisation and overall performance.