can approximate parameters constrained for positive values. Alternatively, we can directly reparameterize other common distributions such as Gamma or Dirichlet – see @naesseth [@grep] for details.

### SCORE FUNCTION ESTIMATE

An alternative estimate for can be derived based on manipulation of the log-derivatives; the use of the estimate for simulation of models was originally presented by @score and its use for variational approximation by @bbvi. The estimate for $\nabla_\lambda \mathcal{L}(\lambda) $ is provided by $$\mathbb{E}_{q_\lambda(z)}[(\log p(x,z) - \log q_\lambda(z)) \nabla_\lambda \log q_\lambda(z)],$$ which again leads into a straightforward Monte Carlo approximation of $\nabla_\lambda \mathcal{L}(\lambda)$ as $$\frac 1M \sum_{m = 1}^M(\log p(x,z_m) - \log q_\lambda(z_m)) \nabla_\lambda \log q_\lambda(z_m),
  \label{eq:score}$$ where $z_m \sim {q_\lambda(z)}$. The notable property of this technique is that it does not require derivatives of the model (that is, $\log p(x, z)$) itself, but instead relies solely on derivatives of the approximation. This is both a pro and a con; the model does not need to be differentiable, but at the same time the estimate is not using the valuable information the model gradient provides. This is shown to result in considerably higher variance compared to the reparameterization estimate, often by orders of magnitude. Variance reduction techniques [@bbvi] help, 