Inference
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INTRODUCTION
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Variational inference considers parametric approximations for posterior densities of probabilistic models. Following @Jordan1999 the classical variational approximation algorithms are based on coordinate descent algorithms for which individual steps of the algorithm are often carried out analytically. This limits the use of variational approximation to models with conjugate priors (or simple extensions of those) and restricts the family of potential approximating distributions based on analytic tractability.

Recent advances in variational approximation have lead to a phase transition; instead of closed-form updates, the approximation is nowadays often fit using generic gradient descent techniques instead – for a good overview see, [*e.g.*]{}, @gal. The key behind these advances is in using Monte Carlo approximation to estimate the gradient of the objective function that is an integral over the approximating distribution. This can be done in two alternative ways. The *reparameterization estimate* [@dsvi; @vae; @salimans] allows expressing the gradient of the objective function using the gradients of the model itself, whereas the *score function estimate* [@bbvi] is based on gradients of the approximation. Given the new family of algorithms we can apply variational approximation for a considerably wider range of probabilistic models, enabling for example use of variational inference as 