@blei16.

In recent years several novel types of algorithms applicable for a wider range of models have been proposed [@dsvi; @vae; @salimans; @bbvi], based on direct gradient-based optimization of the lower bound . The core idea behind these algorithms is in using Monte Carlo estimates for the loss and its gradient $$\nabla_{\lambda} \mathcal{L}(\lambda) = 
  \nabla_\lambda \mathbb{E}_{q_\lambda(z)}[\log p(x,z) - \log q_\lambda(z)].
  \label{eq:ELBOgrad}$$ Given such estimates, the inference problem can be solved by standard gradient descent algorithms. This enables inference for non-conjugate likelihoods and for complex models for which closed-form updates would be hard to derive, making variational inference a feasible inference strategy for probabilistic programming languages [@advi; @edward; @pymc3]. In the following, we briefly describe two alternative strategies of estimating the gradient. In Section \[sec:method\] we will then show how the proposed importance sampling technique is applied for both cases.

### REPARAMETERIZATION ESTIMATE

The reparameterization estimate for (and consequently ) is based on representing the approximation $q_\lambda(z)$ using a differentiable transformation $z = f(\epsilon,
\lambda)$ of an underlying standard distribution $\phi(\epsilon)$ that does not have any free parameters. The core idea of how this enables computing the gradient was developed simultaneously by @vae [@salimans] and @dsvi with many of the mathematical 