Implement bijector for gamma and exponential distributions

[bayesianbrad]

Due to the support of these two distributions being defined only on the positive reals, we must ensure that we can transform the parameter space to the whole of $$ \mathbb{R}$$.

To do this we can use the $$\exp(Y) = X$$ transform and so the transforming to the unconstrained space $$Y$$ we have $$f_{y}(Y) = |\partial h(y) \ \partial y | f_{X}(h(y)) $$
which means that the transformed distribution is given as:
$$log_gamma(\alpha,\beta) = \frac{exp(y\beta - \exp(x)//\alpha}{\alpha^{\beta} \Gamma{\beta}}$$

[haohaiziround]

@bradleygramhansen
Bijector from Edward.
https://github.com/blei-lab/edward/blob/master/edward/inferences/hmc.py

Transformation:
$$\theta \in (0, + \infinity) -> \tilde{\theta} = \log \theta$$
$$\theta \in (0, 1) -> \tilde{\theta} = \log \frac{ \theta}{1-\theta}$$

[bayesianbrad]

• Exponential constrain between (0, \infty) using log(\theta)
• Gamma constrain between (0,\infty) using log(\theta)
• Dirichlet constrain between (0,1) transform using log(\theta / \theta -1)