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# Exponential Family of Distributions

May 15, 2020
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Exponential family is a large class of probabilistic distributions, both discrete and continuous. Some of these distributions include Gaussian and Bernouli distributions. As the name suggested, distributions in this famility are in a generic exponential form.

Consider $X$ a random variable from a exponential family distribution. Its probability mass function (if $X$ is discrete) or probability density function (if it continuous) is written as

$p(x|\theta) = f(x) \exp( \eta(\theta) \cdot \phi(x) + g(\theta)),$

where

• $\phi(x)$ is $X$'s sufficient statistic(s);
• $\eta(\theta)$ is the natural parameter(s) of the distribution;
• $\theta$ is the parameter(s) of the distribution;
• $g(\theta)$ is the log-partition function, which act as a normalizer;
• $f(x)$ is a function that depends on $x$.
\begin{aligned} p(x|\theta) &= \frac{p(x| \theta)}{ \int p(x| \theta) \text{dx}} \\ &= \frac{ \cancel{g(\theta)} \exp(\eta(\theta)\cdot x) }{ \cancel{g(\theta)} \int f(x) \exp(\eta(\theta)\cdot x) }. \end{aligned}

## Some Distributions in Exponential Family

### Bernoulli Distribution

Bernoulli distribution has one paramerter, called $p \in [0, 1]$. Its sample space $\Omega = \{0, 1\}$, e.g. coin tossing. Its probablity mass function is usally written in the following form:

$p(x|p) = p^x (1-p)^{1-x}.$

We can rewrite the above equation using the exponential-logarithm trick:

\begin{aligned} p(x|p) &= \exp(\log p^x + \log (1-p)^{(1-x)}) \\ &= \exp(x\log p + (1-x)\log (1-p)) \\ &= \exp(x\log p -x\log (1-p) + \log (1-p))\\ &= \exp\bigg(x\log \frac{p}{(1-p)} + \log (1-p)\bigg). \end{aligned}

So, we can conclude the followings:

• $f(x) = 1$;
• $\phi(x) = x$;
• $\eta(p) = \log \bigg( \frac{p}{1-p} \bigg)$;
• $g(p) = \log (1-p)$.

### Gaussian Distribution

Let's turn to an exponential family distribution for continuous random variables. The most important one is the Gaussian distribution. For univariate settings, i.e. $x \in \Reals$, the density is

\begin{aligned} p(x| \mu, \sigma^2) &= \frac{1}{\sqrt{2\pi\sigma^2}}\exp \bigg(-\frac{(x-\mu)^2}{2\sigma^2}\bigg) \\ &= \frac{1}{\sqrt{2\pi\sigma^2}} \exp \bigg(-\frac{(x^2 -2x\mu+\mu^2)}{2\sigma^2}\bigg) \\ &= \frac{1}{\sqrt{2\pi}} \exp \bigg( \frac{x\mu}{\sigma^2} -\frac{x^2}{2\sigma^2} - \frac{\mu^2}{2\sigma^2 }- \log \sigma \bigg), \end{aligned}

where

• $f(x) = \frac{1}{\sqrt{2\pi}}$;
• $\phi(x) = (x, x^2)^T$
• $\eta(\bf \theta) = (\frac{\mu}{\sigma^2}, -\frac{1}{\sigma^2} )^T$;
• $g(\bf \theta) = - \frac{\mu^2}{2\sigma^2} - \log \sigma$.

## Cumulant: Moment Generating Function

Let $\eta = \eta(\theta)$. The cumulant $A(\eta) \equiv -g(\theta)$. In the following, we are going to show that we can get the moment parameter of Bernoulli and Gaussian distributions from $A(\eta)$.

### Bernoulli Distribution

Let recall that $g(\theta) = \log(1-p)$ for Bernouli distributions. We have

$A(\eta) = -\log (1-p).$

After rearranging the equation, it yields

$p = \frac{1}{1+e^{-\eta}} \implies A(\eta) = \log(1+e^\eta).$

Taking the first and second derivative, we have

\begin{aligned} A'(\eta) &= \frac{1}{1+e^{-\eta}} = p \\ A''(\eta) &= \underbrace{\bigg(\frac{1}{1+e^{-\eta}}\bigg)}_{p} \underbrace{\bigg( \frac{e^{-\eta}}{1+e^{-\eta}} \bigg)}_{1-p}. \end{aligned}

Therefore, we recover the mean $p$ and the variance $p(1-p)$ of Bernoulli distributions.

Noting, you might notice that the function transformating $\eta$ to $p$ looks familiar; indeed, this is the sigmoid function! In generalized linear models, it is the link function.

### Gaussian Distribution

Recall $g(\theta)$ of the Gaussian distribution. Let $\mathbf (\eta_1, \eta_2)^T \equiv \eta(\mathbf \theta)$ and $A(\mathbf{\eta_1, \eta_2}) = -g(\theta)$. Solving the equation, we have

$A(\eta_1, \eta_2) = \frac{\eta^2_1}{4\eta_2} + \frac{1}{2}\log(-2\eta_2).$

We know that $\eta_1$ corresponds to $\phi(x)_1$, i.e. $x$. If we compute the partial derivative $\frac{\partial}{\partial \eta_1} A(\eta)$ and $\frac{\partial^2}{\partial^2 \eta_1} A(\eta)$, we get

\begin{aligned} \frac{\partial}{\partial \eta_1}A(\eta_1, \eta_2) &= \frac{\eta_1}{2\eta_2} \\ &= \mu \\ \frac{\partial^2}{\partial^2 \eta_1}A(\eta_1, \eta_2) &= \frac{1}{2\eta_2} \\ &= \sigma^2. \end{aligned}

That means we discover $X$'s true mean (first moment) and variance (second moment) of the guassian distribution by differentiating its cumulant $A(\cdot)$.

## References

While writing this article, I was relying on Prof. M. Opper & Théo's lecture slides for Probabilistic Bayesian Modelling course (Summer 2020) and Prof. M. Jordan's reading matertial for his Bayesian Modeling and Inference (2010).

The first figure was made with Google Colab.