分配家庭的定义？

分布族对统计的定义是否不同于其他学科？

通常，曲线族是一组曲线，每条曲线由一个函数或参数化给定，其中一个或多个参数发生变化。这样的族例如用于表征电子部件。

为了进行统计，根据形状来源的一个族是改变形状参数的结果。那么，我们如何才能理解伽玛分布具有形状和比例参数，并且只有广义伽玛分布才具有位置参数？这是否会使家庭成为改变位置参数的结果？根据@whuber一个家庭的意义是隐式A中的家庭的“参数化”是从ℝ的一个子集的连续映射Ñ，以其平常的拓扑结构，为分布的空间，其图像是家庭。$^n$

用简单的语言来说，统计分布族是什么？

关于同一个家庭的分布的统计属性之间的关系的一个问题已经为另一个问题引起了很大的争议，因此似乎值得探讨其含义。

不一定是一个简单的问题，是因为它在指数族这一短语中的使用而产生的，它与曲线族无关，但与通过重新参数化（不仅是参数）改变分布的PDF的形式有关。 ，还可以替换独立随机变量的功能。

统计和数学概念完全相同，请理解“家庭”是一个通用数学术语，其技术变化适用于不同情况：

参数族是所有分布空间中的曲线（或其表面或其他有限维概括）。

这篇文章的其余部分解释了这意味着什么。顺便说一句，我不认为这在数学上或统计学上都是有争议的（除了下面提到的一个小问题）。为了支持这种观点，我提供了许多参考（主要是Wikipedia文章）。

当将函数的类C Y研究成集合Y或“映射” 时，往往会使用“家庭”这一术语。给定一个域X，由某个集合Θ（“参数”）参数化的X上的地图族F是一个函数$\mathcal C_Y$$Y$$X$ $\mathcal F$$X$ $\Theta$

对于其中（1）为每个θ ＆Element; Θ，函数˚F θ：X → ý由下式给出˚F θ（X ）= ˚F（X ，θ ）是在Ç ÿ和（2）˚F本身具有一定的“好”的性质。$\theta\in\Theta$$\mathcal{F}_\theta:X\to Y$$\mathcal{F}_\theta(x)=\mathcal{F}(x,\theta)$$\mathcal{C}_Y$$\mathcal F$

这个想法是我们希望以“平滑”或受控的方式将功能从X更改为Y。属性（1）的装置，每个θ候任这样的功能，而属性（2）的细节将捕获，其中，在“小”的变化感测θ诱导足够“小”变化˚F θ。$X$$Y$$\theta$$\theta$$\mathcal{F}_\theta$

接近问题中提到的一个标准数学例子是同伦。在这种情况下，C Y是从拓扑空间X到拓扑空间Y的连续映射的类别；Θ = [ 0 ，1 ] ⊂ - [R是具有其通常的拓扑单元间隔，并且我们要求˚F是一个连续的从拓扑产物地图X × Θ到ÿ。可以认为是“图F的连续变形$\mathcal{C}_Y$ $X$$Y$$\Theta=[0,1]\subset\mathbb{R}$$\mathcal{F}$$X \times \Theta$$Y$ 0至 ˚F 1 “。当$\mathcal{F}_0$$\mathcal{F}_1$= [ 0 ，1 ]本身是一个间隔，例如地图是曲线在$X=[0,1]$ ÿ和同伦是平滑变形从一条曲线到另一个。$Y$

对于统计应用， C Y是R上所有分布的集合（或者，实际上，是R n上某个n的分布，但是为了使说明简单，我将集中在n = 1上）。我们可能会与该组所有非递减的识别它右连左极函数功能- [R → [ 0 ，1 ]其中它们的范围的闭合包括0和1：这些是累积分布函数，或简单地分布函数。因此，X = R和$\mathcal{C}_Y$$\mathbb{R}$$\mathbb{R}^n$$n$$n=1$$\mathbb{R}\to [0,1]$$0$$1$$X=\mathbb R$ Y = [0 ，1 ]。$Y=[0,1]$

一个家庭的分布是任何子集Ç ÿ。$\mathcal{C}_Y$ 家庭的另一个名字是统计模型。 它由我们假定的所有分布组成，但我们不知道哪个分布是实际分布。

• 一个家庭可以是空的。
• C Y本身是一个家庭。$\mathcal{C}_Y$
• 一个家庭可以只包含一个分布，也可以仅包含有限个分布。

这些抽象的集合论特征很少受到关注或使用。 仅当我们考虑C Y上的其他（相关）数学结构时，此概念才有用。但是，C Y的哪些属性具有统计意义？经常出现的一些是：$\mathcal{C}_Y$$\mathcal{C}_Y$

1. Ç ÿ是一个凸集：给出的任何两个分布 ˚F， ģ ∈ Ç Ý，我们可形成混合物分配（1-吨） ˚F +吨 ģ ∈ý所有吨∈[0，1]。这是从F到G的“同伦”。$\mathcal{C}_Y$${F}, {G}\in \mathcal{C}_Y$ $(1-t){F}+t{G}\in Y$$t\in[0,1]$$F$$G$

2. C Y的大部分支持各种伪度量，例如Kullback-Leibler散度或密切相关的Fisher信息度量。$\mathcal{C}_Y$

3. $\mathcal{C}_Y$ has an additive structure: corresponding to any two distributions $F$ and $G$ is their sum, ${F}\star {G}$.

4. $\mathcal{C}_Y$ supports many useful, natural functions, often termed "properties." These include any fixed quantile (such as the median) as well as the cumulants.

5. $\mathcal{C}_Y$ is a subset of a function space. As such, it inherits many useful metrics, such as the sup norm ($L^\infty$ norm) given by

   $$||F-G||_\infty = \sup_{x\in\mathbb{R}}|F(x)-G(x)|.$$

6. Natural group actions on $\mathbb R$ induce actions on $\mathcal{C}_Y$. The commonest actions are translations $T_\mu:x \to x+\mu$ and scalings $S_\sigma:x\to x\sigma$ for $\sigma\gt 0$. The effect these have on a distribution is to send $F$ to the distribution given by $F^{\mu,\sigma}(x) = F((x-\mu)/\sigma)$. These lead to the concepts of location-scale families and their generalizations. (I don't supply a reference, because extensive Web searches turn up a variety of different definitions: here, at least, may be a tiny bit of controversy.)

The properties that matter depend on the statistical problem and on how you intend to analyze the data. Addressing all the variations suggested by the preceding characteristics would take too much space for this medium. Let's focus on one common important application.

Take, for instance, Maximum Likelihood. In most applications you will want to be able to use Calculus to obtain an estimate. For this to work, you must be able to "take derivatives" in the family.

(Technical aside: The usual way in which this is accomplished is to select a domain $\Theta\subset \mathbb{R}^d$ for $d\ge 0$ and specify a continuous, locally invertible function $p$ from $\Theta$ into $\mathcal{C}_Y$. (This means that for every $\theta\in\Theta$ there exists a ball $B(\theta, \epsilon)$, with $\epsilon\gt 0$ for which $p\mid_{B(\theta,\epsilon)}: B(\theta,\epsilon)\cap \Theta \to \mathcal{C}_Y$ is one-to-one. In other words, if we alter $\theta$ by a sufficiently small amount we will always get a different distribution.))

Consequently, in most ML applications we require that $p$ be continuous (and hopefully, almost everywhere differentiable) in the $\Theta$ component. (Without continuity, maximizing the likelihood generally becomes an intractable problem.) This leads to the following likelihood-oriented definition of a parametric family:

A parametric family of (univariate) distributions is a locally invertible map

$$\mathcal{F}:\mathbb{R}\times\Theta \to [0,1],$$ with $\Theta\subset \mathbb{R}^n$, for which (a) each $\mathcal{F}_\theta$ is a distribution function and (b) for each $x\in\mathbb R$, the function $\mathcal{L}_x: \theta\to [0,1]$ given by $\mathcal{L}_x(\theta) = \mathcal{F}(x,\theta)$ is continuous and almost everywhere differentiable.

Note that a parametric family $\mathcal F$ is more than just the collection of $\mathcal{F}_\theta$: it also includes the specific way in which parameter values $\theta$ correspond to distributions.

Let's end up with some illustrative examples.

• Let $\mathcal{C}_Y$ be the set of all Normal distributions. As given, this is not a parametric family: it's just a family. To be parametric, we have to choose a parameterization. One way is to choose $\Theta = \{(\mu,\sigma)\in\mathbb{R}^2\mid \sigma \gt 0\}$ and to map $(\mu,\sigma)$ to the Normal distribution with mean $\mu$ and variance $\sigma^2$.

• The set of Poisson$(\lambda)$ distributions is a parametric family with $\lambda\in\Theta=(0,\infty)\subset\mathbb{R}^1$.

• The set of Uniform$(\theta, \theta+1)$ distributions (which features prominently in many textbook exercises) is a parametric family with $\theta\in\mathbb{R}^1$. In this case, $F_\theta(x) = \max(0, \min(1, x-\theta))$ is differentiable in $\theta$ except for $\theta\in\{x, x-1\}$.

• Let $F$ and $G$ be any two distributions. Then $\mathcal{F}(x,\theta)=(1-\theta)F(x)+\theta G(x)$ is a parametric family for $\theta\in[0,1]$. (Proof: the image of $\mathcal F$ is a set of distributions and its partial derivative in $\theta$ equals $-F(x)+G(x)$ which is defined everywhere.)

• The Pearson family is a four-dimensional family, $\Theta\subset\mathbb{R}^4$, which includes (among others) the Normal distributions, Beta distributions, and Inverse Gamma distributions. This illustrates the fact that any one given distribution may belong to many different distribution families. This is perfectly analogous to observing that any point in a (sufficiently large) space may belong to many paths that intersect there. This, together with the previous construction, shows us that no distribution uniquely determines a family to which it belongs.

• The family $\mathcal{C}_Y$ of all finite-variance absolutely continuous distributions is not parametric. The proof requires a deep theorem of topology: if we endow $\mathcal{C}_Y$ with any topology (whether statistically useful or not) and $p: \Theta\to\mathcal{C}_Y$ is continuous and locally has a continuous inverse, then locally $\mathcal{C}_Y$ must have the same dimension as that of $\Theta$. However, in all statistically meaningful topologies, $\mathcal{C}_Y$ is infinite dimensional.

To address a specific point brought up in the question: "exponential family" does not denote a set of distributions. (The standard, say, exponential distribution is a member of the family of exponential distributions, an exponential family; of the family of gamma distributions, also an exponential family; of the family of Weibull distributions, not an exponential family; & of any number of other families you might dream up.) Rather, "exponential" here refers to a property possessed by a family of distributions. So we shouldn't talk of "distributions in the exponential family" but of "exponential families of distributions"—the former is an abuse of terminology, as @JuhoKokkala points out. For some reason no-one commits this abuse when talking of location–scale families.

Thanks to @whuber there is enough information to summarize in what I hope is a simpler form relating to the question from which this post arose. "Another name for a family [Sic, statistical family] is [a] statistical model."

From that Wikipedia entry: A statistical model consists of all distributions that we suppose govern our observations, but we do not otherwise know which distribution is the actual one. What distinguishes a statistical model from other mathematical models is that a statistical model is non-deterministic. Thus, in a statistical model specified via mathematical equations, some of the variables do not have specific values, but instead have probability distributions; i.e., some of the variables are stochastic. A statistical model is usually thought of as a pair $( S , P )$, where $S$ is the set of possible observations, i.e., the sample space, and $P$ is a set of probability distributions on $S$.

Suppose that we have a statistical model $(S, \mathcal{P})$ with $\mathcal{P}=\{P_{\theta} : \theta \in \Theta\}$. The model is said to be a Parametric model if $\Theta$ has a finite dimension. In notation, we write that $\Theta \subseteq \mathbb{R}^d$ where $d$ is a positive integer ($\mathbb{R}$ denotes the real numbers; other sets can be used, in principle). Here, $d$ is called the dimension of the model.

As an example, if we assume that data arise from a univariate Gaussian distribution, then we are assuming that

Thus, if we reduce the dimensionality by assigning, for the example above, $\mu=0$, we can show a family of curves by plotting $\sigma=1,2,3,4,5$ or whatever choices for $\sigma$.