Arranging a Two-Parameter Gamma Distribution into Exponential Family Form

Note: MathJax 3, Hugo, and Netlify just aren’t playing nice for some reason, so pardon any $LATEX$ spillage.

A probability distribution function (PDF) is part of the exponential family if it can be arranged into the form

$$f(x|\theta )=h(x)c(\theta )\exp (\sum _{i=1}^k{w}_i(\theta )t_i(x))$$

where $\theta$ is the vector of parameters. The shape-scale parameterization of the Gamma distribution PDF takes the form

$$f_X(x)=\frac{1}{\mathrm{\Gamma }(k)\beta (k)}{x}^{k-1}{e}^{-\frac{x}{\beta }}$$

We use $\beta$ to notate the rate parameter in the shape-rate parameterization of the Gamma PDF, while $k$ is the symbol for the scale parameter.

Can we arrange that PDF into an exponential family form? Spoiler: Yes. Here, we demonstrate that a Gamma PDF given two unknown parameters, $\beta$ and $k$, belongs to the exponential family.

We start by re-arranging the Gamma distribution:

$$\begin{array}{rl}f(x|k,\beta )& =\frac{1}{\mathrm{\Gamma }(k)\beta (k)}{x}^{k-1}{e}^{-\frac{x}{\beta }}\\ & =\frac{1}{\mathrm{\Gamma }(k)\beta (k)}{e}^{(k-1)\ln (x)}{e}^{-\frac{x}{\beta }}\\ & =\frac{1}{\mathrm{\Gamma }(k)\beta (k)}{e}^{(k-1)\ln (x)-\frac{x}{\beta }}\end{array}$$

The log identity $x^b=e^{b\ln (x)}$ is a very useful logarithmic identity to remember when trying to arrange PDFs into exponential family form.

Subsequently, let $h(x)=I_{x>0}(x)$ (If you see that $h(x)=1$, that is a cue to use an indicator function that ranges through the support of $x$ when shaping functions into exponential family form.) Hence, we assign pieces of the re-arranged Gamma PDF to their corresponding exponential family sections:

$$\begin{array}{rl}h(x)& =I_{x>0}(x)\\ c(k,\beta )& =\frac{1}{\mathrm{\Gamma }(k)\beta (k)}\\ w_1(k,\beta )& =k-1\\ w_2(k,\beta )& =-\frac{1}{\beta }\\ t_1(x)& =\ln (x)\\ t_2(x)& =x\end{array}$$

Hence, the Gamma distribution given unknown parameters $\beta$ and $k$ constituting a two-dimensional parameter vector $\theta$ can be shown to be part of the exponential family. A similar process will apply for showing that a Gamma PDF with one unknown parameter, $\beta$ or $k$ is also part of the exponential family.