Meanfieldinferens

Meanfieldinferens refers to mean-field inference, a common approach for approximating complex posterior distributions in probabilistic models. It is widely used in variational Bayesian methods and is sometimes encountered under variant spellings such as meanfieldinferens. The core idea is to replace an intractable posterior p(z|x) with a simpler distribution q(z) that factorizes over groups of latent variables, reducing a high-dimensional problem to a set of manageable updates.

The mean-field approximation assumes that the latent variables are approximately independent under q, so q(z) = ∏_i q_i(z_i). The objective is to minimize the Kullback–Leibler divergence D_KL(q(z) || p(z|x)), equivalently maximizing the evidence lower bound (ELBO). In practice, one performs coordinate ascent: each factor q_i is updated using q_i(z_i) ∝ exp(E_{q_{-i}}[log p(z, x)]), where the expectation is taken with respect to all other factors. When conjugacy holds, these updates have closed-form expressions; otherwise numerical or approximate updates are used. The procedure iterates until convergence, judged by small changes in the ELBO or the q factors.

Advantages of mean-field inference include scalability to large datasets and models, and the ability to provide approximate posterior marginals and uncertainty estimates. Limitations arise from the independence assumption, which can yield underestimation of uncertainty and biased inferences; the method may converge to local optima sensitive to initialization.

Applications span topic models (such as latent Dirichlet allocation), mixture and hidden Markov models, Bayesian neural networks, and other large-scale graphical models. Variants, such as structured mean-field, relax the independence assumption to capture some dependencies and improve approximation quality.