KLqp

KLqp is a variational inference algorithm used to approximate posterior distributions in probabilistic models. It denotes the minimization of the Kullback-Leibler divergence D_KL(q(z; phi) || p(z|x)) between a chosen variational distribution q over latent variables z and the true posterior p(z|x), with x representing observed data. The variational distribution q is parameterized by phi, and the goal is to find phi that minimizes this divergence.

This objective is equivalent to maximizing the Evidence Lower Bound (ELBO): L(phi) = E_{q(z;phi)}[log p(x,z) - log q(z;phi)], up to an additive constant. In practice, the ELBO provides a tractable objective for optimization, since it replaces the unknown posterior with an expectation under q. The gradient of the ELBO with respect to phi is estimated and used to update the variational parameters.

Optimization is typically performed with stochastic gradient methods. For continuous latent variables, the reparameterization trick is commonly used to obtain low-variance gradient estimates. For discrete or non-reparameterizable variables, score-function (REINFORCE) estimators or alternative approaches may be employed. The expressiveness of the variational family q influences the quality of the approximation; common choices include mean-field Gaussian, full-rank Gaussian, and structured factorized forms.

KLqp is implemented in several probabilistic programming libraries, such as Edward and TensorFlow Probability, as part of automatic variational inference workflows. It offers speed and scalability advantages over exact inference but can introduce bias, particularly when the variational family is limited or when the model exhibits multi-modality.

Notes: KLqp stands for minimizing the KL divergence from q to p (the posterior). It is conceptually distinct from KL(p||q), which has different optimization characteristics and consequences for the resulting approximation.