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.
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:
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.
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. $X = \sigma Z + \mu$.
A wrapped probability distribution $p_w(\theta) = \sum_{k \in \mathbb{Z}} p(\theta + 2\pi k)$ on a unit n-sphere is the distribution of random variable $Z = \arg e^{i \phi}$, where $\phi \sim p$ with support $\mathbb{R}$.
The set $M_{m, n}(\mathbb{R})$ of m-by-n real matrices is a smooth $(m n)$-manifold, which we may call the generic matrix manifold, because it is an $(m n)$-dimensional real vector space: $M_{m, n}(\mathbb{R}) \cong \mathbb{R}^{mn}$.
The set $\mathcal{M}(k, m \times n)$ of m-by-n real matrices of rank k, $k \le \min(m, n)$, is an embedded $k (m + n - k)$-submanifold of the m-by-n matrix manifold $M_{m,n}(\mathbb{R})$, which we may call rank-k manifold. Besides the metric induced from the Euclidean metric of the generic matrix manifold, other Riemannian metrics can be induced on fixed-rank matrix manifolds via Riemannian submersion $\pi(M, N) = M N^T$, given Riemannian metrics on the total space [@Absil2014]. The set $M^∗_{m, n}(\mathbb{R})$ or $\mathbb{R}_∗^{mn}$ of m-by-n full-rank real matrices is a smooth $(m n)$-manifold, which we may call full-rank manifold, because it is an open subset of the smooth manifold of m-by-n real matrices: $M^∗_{m, n}(\mathbb{R}) = \{X \in M_{m, n}(\mathbb{R}) : \text{rank}~X = \min\{m, n\}\}$, $M^∗_{m, n}(\mathbb{R}) \in \mathcal{T}_{M^∗_{m, n}(\mathbb{R})}$. In particular, the general linear group $\text{GL}(n, \mathbb{R})$ or $\text{GL}_n$ of nonsingular order-n real matrices is a smooth $n^2$-manifold.
Stiefel manifold $V_{k, n}$ is the space of orthonormal k-frames in the Euclidean n-space: $V_{k, n} = \{X \in M_{n,k}(\mathbb{R}) : X^T X = I\}$, $k \le n$. The Stiefel manifold is an embedded submanifold of the m-by-k matrix manifold $M_{n,k}(\mathbb{R})$, with dimension $k (2n - k - 1) / 2$. If $k = 1$, the Stiefel manifold coincides with the $(n-1)$-sphere: $V_{1,n} = \mathbb{S}^{n-1}$, with dimension $(n - 1)$. If $k = n$, the Stiefel manifold coincides with the orthogonal group: $V_{n,n} = O(n)$, with dimension $n (n - 1) / 2$.
Grassmann manifold or Grassmannian $G_{k, n}$ or $G_k(\mathbb{R}^n)$ is the space of rank-k projection matrices, or equivalently, the quotient manifold of k-subspaces in the Euclidean n-space: $G_{k, n} = M^∗_{n, k} / \text{GL}_k$, $G_{k, n} = \{\text{Span}(X) : X \in M^∗_{n, k}\}$; or $G_{k, n} = \{X X^T : X \in M^∗_{n, k}\}$. The Grassmann manifold is a compact smooth manifold of dimension $k (n - k)$. If $k = 1$, the Grassmann manifold coincides with the $(n-1)$-dimensional projective space, i.e. the quotient manifold of lines in the Euclidean n-space: $G_{1,n} = \mathbb{RP}^{n-1}$.
The set $S_+(k, n)$ of rank-k order-n positive-semidefinite matrices is a smooth manifold of dimension $k (2n - k + 1) / 2$, which we may call rank-k positive-semidefinite manifold: $S_+(k, n) = \{V \Lambda V^T : V \in V_{k, n}, \lambda \in \mathbb{R}^k_+, \Lambda = \text{diag}(\lambda)\}$; alternatively, it can be seen as the product manifold of the Stiefel manifold and the non-increasing positive Euclidean k-space, $S_+(k, n) = V_{k, n} \times \mathbb{R}^k_{+\downarrow}$, where $\mathbb{R}^k_{+\downarrow} = \{(x_i)_{i=1}^n \in \mathbb{R}^k_+ : \forall i < j, x_i \ge x_j\}$. In particular, the set $S_+(n)$ of order-n positive-definite matrices is a smooth manifold of dimension $n (n + 1) / 2$, which we may call positive-definite manifold, and can be seen as the product manifold of the orthogonal group and the non-increasing positive Euclidean n-space: $S_+(n) = O(n) \times \mathbb{R}^n_{+\downarrow}$.
Oblique manifold $\text{OB}_{m,n}$ is the set of m-tuples of points on the $(n-1)$-sphere that span the Euclidean n-space: $\text{OB}_{m,n} = \{X \in M^∗_{m, n}: \text{diag}(X X^T) = I\}$, or equivalently $\text{OB}_{m,n} = \{X \in \prod_{i=1}^m \mathbb{S}^{n-1} : \text{Span}~X = \mathbb{R}^n\}$.
Flag manifold $F_K(\mathbb{R}^n)$ is the set of all flags of type $K$ in the Euclidean n-space, i.e. a nested sequence of linear subspaces: given $K = (k_i)_{i=1}^m$, $F_K(\mathbb{R}^n) = \{(S_i)_{i=1}^m : S_i \in G_{k_i,n}, \forall i < j, S_i \subset S_j \}$.
Essential manifold is the set of essential matrices, i.e. the product of a skew-symmetric matrix and a rotation matrix: $E_n = \{\Omega Q : \Omega \in M_n(\mathbb{R}), \Omega = - \Omega^T, Q \in O(n)\}$.
Euclidean group $\text{SE}(3)$ is the Cartesian product of the order-3 special orthogonal group and the Euclidean 3-space: $\text{SE}(3) = \text{SO}(3) \times \mathbb{R}^3$.