A parametric probabilistic model $\mathcal{P} = \{ P_{\boldsymbol \theta}\mid\boldsymbol \theta \in \Theta \}$ is a family of probability distributions that can be specified using a finite number of parameters $\theta$. The parameter space $\Theta \subseteq \mathbb{R}^k$ is the set of all allowable values for the parameter.

## Categories of Parametric Models

### Exponential Family

A parametric model is called an exponential family, if it has the form $f( \mathbf{x} \mid \boldsymbol \theta ) = c(\boldsymbol \theta) h(\mathbf{x}) \exp \{ \sum_{i=1}^l w_i(\boldsymbol \theta) t_i(\mathbf{x}) \}$, where $c$ and $h$ are nonnegative, and all functions are real-valued. Additionally, if $k < l$, the parametric model is called a curved exponential family; if $k = l$, it is called a full exponential family.

Examples of exponential families:

1. Discrete distributions: Binomial, Poisson, negative binomial;
2. Continuous distributions: Normal, Gamma, Beta.

Theorem (Natural Parameterization): For an exponential family, denote natural parameters $\eta_i = w_i(\boldsymbol \theta)$, then the parametric model can be rewritten as $f(\mathbf{x} \mid \boldsymbol \eta) = c^∗(\boldsymbol \eta) h(\mathbf{x}) \exp \{ \sum_{i=1}^l \eta_i t_i(\mathbf{x}) \}$. And the natural parameter space $\{ \boldsymbol \eta \mid \int f(\mathbf{x} \mid \boldsymbol \eta)~\mathrm{d}\mathbf{x} < \infty \}$ is convex.

### Location-Scale Family

The location-scale family with standard PDF $f(x)$ is the parametric model $\mathcal{P} = \{ f_{(\mu,\sigma)} = \frac{1}{\sigma} f(\frac{x-\mu}{\sigma}) \mid (\mu,\sigma) \in \mathbb{R}\times\mathbb{R}^+ \}$, with location parameter $\mu$ and scale parameter $\sigma$.

Theorem: $X \sim f_{(\mu,\sigma)} \Leftrightarrow \exists Z \sim f(z)$, s.t. $X = \sigma Z + \mu$.

### Wrapped Distributions

Wrapped probability distribution on the n-sphere is the distribution of random variable $\mathbf{z} = \arg e^{i \boldsymbol{\phi}}$ with density $p_w(\theta) = \sum_{k \in \mathbb{Z}} p(\theta + 2\pi k)$, where $p$ is the density of the real random variable $\boldsymbol{\phi}$.

## Parametric Families by Support

1. Discrete:
1. finite: Bernoulli, binomial, categorical, multinomial, hypergeometric, Zipf;
2. semi-infinite: Poisson, geometric, negative binomial;
2. Continuous (Euclidean):
1. bounded: uniform, triangular, semicircle, Beta, logit-normal;
2. semi-infinite: exponential, Erlang, Gamma, inverse Gamma, generalized inverse Gaussian; Pareto, Fréchet, Weibull; Chi-squared, Chi, F; log-normal, log-Cauchy, log-Laplace, log-logistic;
3. infinite: Gaussian, Cauchy, Student's t, Fisher's z; Laplace, logistic, Gumbel;
3. Manifold:
• simplex: Dirichlet;
• circle: von Mises;
• sphere: von Mises–Fisher [@Fisher1953], Bingham [@Bingham1974], Kent [@Kent1982];
• torus: bivariate von Mises [@Mardia1975];
• Stiefel manifolds: matrix Langevin (aka matrix von Mises-Fisher), matrix Bingham, matrix angular central Gaussian;
• Grassmann manifolds: matrix Langevin, matrix angular central Gaussian;
• Other matrix manifolds