The Exponential Family and Natural Parameters

The normal, Bernoulli, Poisson, exponential, and gamma distributions look unrelated at first glance, but they're all special cases of one shared template called the exponential family, written as $p(x \mid \theta) = h(x)\exp(\eta(\theta) \cdot T(x) - A(\theta))$. Once a distribution is written this way, $T(x)$ is called the sufficient statistic (everything the data tells you about $\theta$ is packed into it), $\eta(\theta)$ is the natural parameter (a reparameterization of $\theta$ that makes the formula clean), and $A(\theta)$ is a normalizing term ensuring the distribution integrates to one.

The payoff of writing distributions this way isn't cosmetic: a wide range of results — that the maximum likelihood estimator has a closed form in terms of the sufficient statistic, that conjugate priors exist for Bayesian updating, that generalized linear models (logistic regression, Poisson regression) all share the same fitting machinery — can be proven once for the whole exponential family instead of separately, distribution by distribution. That's precisely why logistic regression (Bernoulli) and Poisson regression share nearly identical fitting code under the hood: both are instances of a generalized linear model built on an exponential-family distribution, differing only in which member of the family and which link function connect predictors to the natural parameter.

For a quant, the practical payoff shows up whenever a model needs a distributional assumption on returns, counts of trades, or default indicators — recognizing which exponential family member fits the data means immediately inheriting a known, well-studied estimation and inference toolkit rather than building one from scratch.