Can the Beta-regression be written in the GLM form?

Question

The Beta distribution is: $$p(y)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}y^{\alpha-1}(1-y)^{\beta-1} $$

It's part of the exponential family.

We can reparametrize this with using mean and dispersion $\mu=\frac{\alpha}{\alpha+\beta}, \phi = \alpha+\beta$ and get:

$$ \mu=\frac{\alpha}{\alpha+\beta}, \phi = \alpha+\beta \\ \Rightarrow \mu = \frac{\alpha}{\phi} \Rightarrow \alpha = \mu\phi \\ \beta = \phi-\alpha=\phi-\mu\phi=\phi(1-\mu) \\ p(y)=\frac{\Gamma(\phi)}{\Gamma(\mu\phi)\Gamma(\phi(1-\mu))}y^{\mu\phi-1}(1-y)^{\phi(1-\mu)-1} \\ = \frac{\Gamma(\phi)}{y(1-y)^{\phi-1}}\exp\{\mu\phi\log y-\phi\mu\log (1-y)-[\log \Gamma(\mu\phi)+\log\Gamma(\phi(1-\mu))]\} \\ = \frac{\Gamma(\phi)}{y(1-y)^{\phi-1}}\exp\{\phi(\mu\log \frac{y}{1-y}-\frac{1}{\phi}[\log \Gamma(\mu\phi)+\log\Gamma(\phi(1-\mu))])\} $$

This suggests that it's part of the (GLM) exponential family, with

$$ a(\phi) = \frac{1}{\phi} \\ t(y) = \log \frac{y}{1-y} \\ \theta = \mu \\ b(\theta) =\frac{1}{\phi}[\log \Gamma(\mu\phi)+\log\Gamma(\phi(1-\mu))] \\ h(y,\phi) = \frac{\Gamma(\phi)}{y(1-y)^{\phi-1}} $$

There are a few problems with this:

1. The log-normalizer $b(\theta)$ seems to be dependent on the dispersion parameter (which usually doesn't happen for other distributions)

2. The mean-variance relationship in GLM's satisfy: $\mathbb V[y] = a(\phi)V(\mu)$, yet here we get that $\mathbb V[y]=\frac{1}{\phi+1}\mu(1-\mu)$, that is that $a(\phi) = \frac{1}{\phi+1}$ and not $\frac{1}{\phi}$

3. Taking the derivative of the log-normalizer w.r.t. the natural parameter doesn't seem to give the mean (we get $\psi(\alpha)-\psi(\beta)$, where $\psi$ is the digamma function, but I don't see how this is equal to $\frac{\alpha}{\alpha+\beta}$)

Here they claim "that the distribution of the response is not a member of the exponential family"...

So who's right? Wikipedia, or the paper guys? And how can we write the Beta distribution in GLM Expo-Family form? Can one parameterization of a distribution be part of the Expo-Family, and another not???

Answer 1

GLMs assume that the response distribution is an Exponential Dispersion Model (EDM): $$y_i \sim \mbox{ED}(\mu_i,\phi/w_i)$$ where $\phi$ is the dispersion parameter and $w_i$ is a known weight.

EDMs are both more special and more general than exponential families. If the dispersion $\phi$ is known, then an EDM is a single-parameter linear exponential family (LEF). LEFs are a particular simple type of exponential family so, in this sense, EDMs are a subset of exponential families. On the other hand, an EDM with unknown $\phi$ does not need to be a two-parameter exponential family, so EDMs are in this sense more general than exponential families.

An EDM ED$(\mu,\phi)$ density is writeable in the following form: $$f(y;\mu,\phi) = a(y,\phi) \exp(\frac{d(y,\mu)}{-2\phi})$$ where $d(y,\mu)$ is an (asymmetric) distance measure between $y$ and $\mu$. This form for the density clarifies that $\mu$ is a location parameter and $\phi$ is a dispersion parameter.

The Beta distribution is a two-parameter exponential family but not an EDM.

If $\phi=\alpha+\beta$ is known, then the Beta distribution can be transformed to a LEF, which is what you implicitly discovered in your calculations. The logit$(y)$ random variable $$z = \log\left(\frac{y}{1-y}\right)$$ follows a logistic-beta distribution, see Is there a "beta distribution" over the entire real line?. The logistic-beta distribution with known $\phi$ is a LEF, so it could be used to generate a GLM family, although I have never known anyone to do so. Note that this derivation only works when $\alpha+\beta$ is known; $\alpha+\beta$ would not be the dispersion parameter for a logistic-beta GLM.