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统计代写|贝叶斯分析代写Bayesian Analysis代考|MAST90125

Posted on 2022年6月30日2022年6月30日 by statistics-lab

如果你也在怎样代写贝叶斯分析Bayesian Analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

贝叶斯分析，一种统计推断方法（以英国数学家托马斯-贝叶斯命名），允许人们将关于人口参数的先验信息与样本所含信息的证据相结合，以指导统计推断过程。

statistics-lab™ 为您的留学生涯保驾护航在代写贝叶斯分析Bayesian Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写贝叶斯分析Bayesian Analysis代写方面经验极为丰富，各种代写贝叶斯分析Bayesian Analysis相关的作业也就用不着说。

我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|MAST90125

统计代写|贝叶斯分析代写Bayesian Analysis代考|INDEPENDENT AND CONDITIONALLY INDEPENDENT

A pair of random variables $(X, Y)$ is said to be independent if for any $A$ and $B$,
$$
p(X \in A \mid Y \in B)=p(X \in A),
$$
or alternatively $p(Y \in B \mid X \in A)=p(Y \in B)$ (these two definitions are correct and equivalent under very mild conditions that prevent ill-formed conditioning on an event that has zero probability).

Using the chain rule, it can also be shown that the above two definitions are equivalent to the requirement that $p(X \in A, Y \in B)=p(X \in A) p(Y \in B)$ for all $A$ and $B$.

Independence between random variables implies that the random variables do not provide information about each other. This means that knowing the value of $X$ does not help us infer anything about the value of $Y$-in other words, it does not change the probability of $Y$. (Or vice-versa $-Y$ does not tell us anything about $X$.) While independence is an important concept in probability and statistics, in this book we will more frequently make use of a more refined notion of independence, called “conditional independence”-which is a generalization of the notion of independence described in the beginning of this section. A pair of random variables $(X, Y)$ is conditionally independent given a third random variable $Z$, if for any $A, B$ and $z$, it holds that $p(X \in A \mid Y \in B, Z=z)=p(X \in A \mid Z=z)$.

Conditional independence between two random variables (given a third one) implies that the two variables are not informative about each other, if the value of the third one is known. 3
Conditional independence (and independence) can be generalized to multiple random variables as well. We say that a set of random variables $X_{1}, \ldots, X_{n}$, are mutually conditionally independent given another set of random variables $Z_{1}, \ldots, Z_{m}$ if the following applies for any $A_{1}, \ldots, A_{n}$ and $z_{1}, \ldots, z_{m}:$
$$
\begin{gathered}
p\left(X_{1} \in A_{1}, \ldots, X_{n} \in A_{n} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right)= \
\prod_{i=1}^{n} p\left(X_{i} \in A_{i} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right) .
\end{gathered}
$$
This type of independence is weaker than pairwise independence for a set of random variables, in which only pairs of random variables are required to be independent. (Also see exercises.)

统计代写|贝叶斯分析代写Bayesian Analysis代考|EXCHANGEABLE RANDOM VARIABLES

Another type of relationship that can be present between random variables is that of exchangeability. A sequence of random variables $X_{1}, X_{2}, \ldots$ over $\Omega$ is said to be exchangeable, if for any finite subset, permuting the random variables in this finite subset, does not change their joint distribution. More formally, for any $S=\left{a_{1}, \ldots, a_{m}\right}$ where $a_{i} \geq 1$ is an integer, and for any permutation $\pi$ on ${1, \ldots, m}$, it holds that: ${ }^{4}$
$$
p\left(x_{a_{1}}, \ldots, x_{a_{m}}\right)=p\left(x_{a_{\pi(1)}}, \ldots, x_{\left.a_{\pi(m)}\right)}\right) .
$$
Due to a theorem by de Finetti (Finetti, 1980), exchangeability can be thought of as meaning “conditionally independent and identically distributed” in the following sense. De Finetti showed that if a sequence of random variables $X_{1}, X_{2}, \ldots$ is exchangeable, then under some regularity conditions, there exists a sample space $\Theta$ and a distribution over $\Theta, p(\theta)$, such that:
$$
p\left(X_{a_{1}}, \ldots, X_{a_{m}}\right)=\int_{\theta} \prod_{i=1}^{m} p\left(X_{a_{i}} \mid \theta\right) p(\theta) d \theta,
$$
for any set of $m$ integers, $\left{a_{1}, \ldots, a_{m}\right}$. The interpretation of this is that exchangeable random variables can be represented as a (potentially infinite) mixture distribution. This theorem is also called the “representation theorem.”

The frequentist approach assumes the existence of a fixed set of parameters from which the data were generated, while the Bayesian approach assumes that there is some prior distribution over the set of parameters that generated the data. (This will hecome clearer as the hook progresses.) De Finetti’s theorem provides another connection between the Bayesian approach and the frequentist one. The standard “independent and identically distributed” (i.i.d.) assumption in the frequentist setup can be asserted as a setup of exchangeability where $p(\theta)$ is a point-mass distribution over the unknown (but single) parameter from which the data are sampled. This leads to the observations being unconditionally independent and identically distributed. In the Bayesian setup, however, the observations are correlated, because $p(\theta)$ is not a point-mass distribution. The prior distribution plays the role of $p(\theta)$. For a detailed discussion of this similarity, see O’Neill (2009).

统计代写|贝叶斯分析代写Bayesian Analysis代考|EXPECTATIONS OF RANDOM VARIABLES

If we consider again the naive definition of random variables, as functions that map the sample space to real values, then it is also useful to consider various ways in which we can summarize these random variables. One way to get a summary of a random variable is by computing its expectation, which is its weighted mean value according to the underlying probability model.
It is easiest to first consider the expectation of a continuous random variable with a density function. Say $p(\theta)$ defines a distribution over the random variable $\theta$, then the expectation of $\theta$, denoted $E[\theta]$ would be defined as:
$$
E[\theta]=\int_{\theta} p(\theta) \theta d \theta .
$$
For the discrete random variables that we consider in this book, we usually consider expectations of functions over these random variables. As mentioned in Section 1.2, discrete random variable values often range over a set which is not numeric. In these cases, there is no “mean value” for the values that these random variables accept. Instead, we will compute the mean value of a real-function of these random variables.
With $f$ being such a function, the expectation $E[f(X)]$ is defined as:
$$
E[f(X)]=\sum_{x} p(x) f(x)
$$ For the linguistic structures that are used in this book, we will often use a function $f$ that indicates whether a certain property holds for the structure. For example, if the sample space of $X$ is a set of sentences, $f(x)$ can be an indicator function that states whether the word “spring” appears in the sentence $x$ or not; $f(x)=1$ if the word “spring” appears in $x$ and 0 , otherwise. In that case, $f(X)$ itself can be thought of as a Bernoulli random variable, i.e., a binary random variable that has a certain probability $\theta$ to be 1 , and probability $1-\theta$ to be 0 . The expectation $E[f(X)]$ gives the probability that this random variable is 1 . Alternatively, $f(x)$ can count how many times the word “spring” appears in the sentence $x$. In that case, it can be viewed as a sum of Bernoulli variables, each indicating whether a certain word in the sentence $x$ is “spring” or not.

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|INDEPENDENT AND CONDITIONALLY INDEPENDENT

一对随机变量 $(X, Y)$ 据说是独立的，如果有的话 $A$ 和 $B$ ，
$$
p(X \in A \mid Y \in B)=p(X \in A),
$$
或者 $p(Y \in B \mid X \in A)=p(Y \in B)$ (这两个定义在非常温和的条件下是正确且等效的，可以防止对概率为零的事件进行不良条件反射) 。
使用链式法则，也可以证明上述两个定义等价于: $p(X \in A, Y \in B)=p(X \in A) p(Y \in B)$ 对所有人 $A$ 和 $B$.
随机变量之间的独立性意味着随机变量不提供关于彼此的信息。这意味着知道 $X$ 并不能帮助我们推断出任何关于 $Y$ – 换句话说，它不会改变 $Y$. (或相反亦然 $-Y$ 没有告诉我们任何关于 $X$.) 虽然独立性是概率和统计学中的一个重要概念，但在本书中，我们将更频做地使用一个更精炼的独立性概念，称为”条件独立性”一一它是对本书开头描述的独立性概念的概括。本节。一对随机变量 $(X, Y)$ 给定第三个随机变量条件独立 $Z$, 如果有的话 $A, B$ 和 $z$, 它认为 $p(X \in A \mid Y \in B, Z=z)=p(X \in A \mid Z=z)$.
两个随机变量之间的条件独立性（给定第三个）意味着如果第三个变量的值已知，则这两个变量不会相互提供信息。 3
条件独立性（和独立性）也可以推广到多个随机变量。我们说一组随机变量 $X_{1}, \ldots, X_{n}$ ，在给定另一组随机变量的情况下相互条件独立 $Z_{1}, \ldots, Z_{m}$ 如果以下适用于任何 $A_{1}, \ldots, A_{n}$ 和 $z_{1}, \ldots, z_{m}$ :
$$
p\left(X_{1} \in A_{1}, \ldots, X_{n} \in A_{n} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right)=\prod_{i=1}^{n} p\left(X_{i} \in A_{i} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right) \text {. }
$$
这种类型的独立性弱于一组随机变量的成对独立性，其中只要求成对的随机变量是独立的。（另见练习。）

统计代写|贝叶斯分析代写Bayesian Analysis代考|EXCHANGEABLE RANDOM VARIABLES

随机变量之间可能存在的另一种关系是可交换性。一系列随机变量 $X_{1}, X_{2}, \ldots$ 超过 $\Omega$ 据说是可交换的，如果对于任何有限子集，置换该有限子集中的随机变量，不会改变它们的联合分布。更正式地说，对于任何
$\mathrm{S}=|$ left{a_{1}，Vdots, a_{m}|right} 在哪里 $a_{i} \geq 1$ 是一个整数，并且对于任何排列 $\pi$ 上 $1, \ldots, m$ ，它认为: ${ }^{4}$
$$
p\left(x_{a_{1}}, \ldots, x_{a_{m}}\right)=p\left(x_{a_{\pi(1)}}, \ldots, x_{\left.a_{\pi(m)}\right)}\right) .
$$
由于 de Finetti (Finetti, 1980) 的一个定理，可交换性可以被认为是以下意义上的“条件独立且同分布”。De Finetti 证明，如果一系列随机变量 $X_{1}, X_{2}, \ldots$ 是可交换的，那么在一定的规律性条件下，存在一个样本空间 $\Theta$ 和分布 $\Theta, p(\theta)$, 这样:
$$
p\left(X_{a_{1}}, \ldots, X_{a_{m i}}\right)=\int_{\theta} \prod_{i=1}^{m} p\left(X_{a_{i}} \mid \theta\right) p(\theta) d \theta
$$
对于任何一组 $m$ 整数， Ileft{a_{1}, Idots, a_{m}\right}. 对此的解释是可交换随机变量可以表示为（可能是无限的) 混合分布。该定理也称为“表示定理”。
常客方法假设存在一组固定的参数，从这些参数中生成数据，而贝叶斯方法假设在生成数据的参数集上存在一些先验分布。(随着钩子的进行，这将变得更加清晰。) 德菲内蒂定理提供了贝叶斯方法和常客方法之间的另一种联系。常客设置中的标准”独立同分布” (iid) 假设可以断言为可交换性设置，其中 $p(\theta)$ 是从其中采样数据的末知 (但单一) 参数上的点质量分布。这导致观察结果是无条件独立且同分布的。然而，在贝叶斯设置中，观察结果是相关的，因为 $p(\theta)$ 不是点质量分布。先验分布的作用是 $p(\theta)$. 有关这种相似性的详细讨论，请参见 O’Neill (2009)。

统计代写|贝叶斯分析代写Bayesian Analysis代考|EXPECTATIONS OF RANDOM VARIABLES

如果我们再次考虑随机变量的朴素定义，作为将样本空间映射到真实值的函数，那么考虑总结这些随机变量的各种方式也是有用的。获得随机变量摘要的一种方法是计算其期望值，即根据潜在概率模型的加权平均值。
首先考虑具有密度函数的连续随机变量的期望是最容易的。说 $p(\theta)$ 定义随机变量的分布 $\theta$ ，那么期望 $\theta$ ，表示 $E[\theta]$ 将被定义为:
$$
E[\theta]=\int_{\theta} p(\theta) \theta d \theta
$$
对于我们在本书中考虑的离散随机变量，我们通常会考虑函数对这些随机变量的期望。如 $1.2$ 节所述，离散随机变量值通常在一个非数字的集合上。在这些情况下，这些随机变量接受的值没有“平均值”。相反，我们将计算这些随机变量的实函数的平均值。
和 $f$ 作为这样的功能，期望 $E[f(X)]$ 定义为:
$$
E[f(X)]=\sum_{x} p(x) f(x)
$$
对于本书中使用的语言结构，我们通常会使用一个函数 $f$ 这表明某个属性是否适用于该结构。例如，如果样本空间 $X$ 是一组句子， $f(x)$ 可以是一个指示函数，表明句子中是否出现了”spring”这个词 $x$ 或不: $f(x)=1$ 如果“春天”这个词出现在 $x$ 和 0 ，否则。在这种情况下， $f(X)$ 本身可以认为是伯努利随机变量，即具有一定概率的二元随机变量 $\theta$ 为 1 和概率 $1-\theta$ 为 0 。期望 $E[f(X)]$ 给出此随机变量为 1 的概率. 或者， $f(x)$ 可以数出“spring”这个词在句子中出现了多少次 $x$. 在那种情况下，它可以看作是伯努利变量的总和，每个变量表示句子中的某个词是否 $x$ 是不是 “春天”。

统计代写|贝叶斯分析代写Bayesian Analysis代考请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习和应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|贝叶斯分析代写Bayesian Analysis代考|MSH3

Posted on 2022年6月30日2022年6月30日 by statistics-lab

如果你也在怎样代写贝叶斯分析Bayesian Analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|JOINT DISTRIBUTION OVER MULTIPLE RANDOM VARIABLES

It is possible to define several random variables on the same sample space. For example, for a discrete sample space, such as a set of words, we can define two random variables $X$ and $Y$ that take integer values-one could measure word length and the other could measure the count of vowels in a word. Given two such random variables, the joint distribution $P(X, Y)$ is a function that maps pairs of events $(A, B)$ as follows:
$$
p(X \in A, Y \in B)=p\left(X^{-1}(A) \cap Y^{-1}(B)\right)
$$
It is often the case that we take several sets $\left{\Omega_{1}, \ldots, \Omega_{m}\right}$ and combine them into a single sample space $\Omega=\Omega_{1} \times \ldots \times \Omega_{m}$. Each of the $\Omega_{i}$ is associated with a random variable. Based on this, a joint probability distribution can be defined for all of these random variables together. For example, consider $\Omega=V \times P$ where $V$ is a vocabulary of words and $P$ is a part-of-speech tag. This sample space enables us to define probabilities $p(x, y)$ where $X$ denotes a word associated with a part of speech $Y$. In this case, $x \in V$ and $y \in P$.

With any joint distribution, we can marginalize some of the random variables to get a distribution which is defined over a subset of the original random variables (so it could still be a joint distribution, only over a subset of the random variables). Marginalization is done using integration (for continuous variables) or summing (for discrete random variables). This operation of summation or integration eliminates the random variable from the joint distribution. The result is a joint distribution over the non-marginalized random variables.

For the simple part-of-speech example above, we could either get the marginal $p(x)=$ $\sum_{y \in P} p(x, y)$ or $p(y)=\sum_{x \in V} p(x, y)$. The marginals $p(X)$ and $p(Y)$ do not uniquely determine the joint distribution value $p(X, Y)$. Only the reverse is true. However, whenever $X$ and $Y$ are independent then the joint distribution can be determined using the marginals. More about this in Section 1.3.2.

统计代写|贝叶斯分析代写Bayesian Analysis代考|CONDITIONAL DISTRIBUTIONS

Joint probability distributions provide an answer to questions about the probability of several random variables to obtain specific values. Conditional distributions provide an answer to a different, but related question. They help to determine the values that a random variable can obtain, when other variables in the joint distribution are restricted to specific values (or when they are “clamped”).

Conditional distributions are derivable from joint distributions over the same set of random variables. Consider a pair of random variables $X$ and $Y$ (either continuous or discrete). If $A$ is an event from the sample space of $X$ and $y$ is a value in the sample space of $Y$, then:
$$
p(X \in A \mid Y=y)=\frac{p(X \in A, Y=y)}{p(Y=y)}
$$
is to be interpreted as a conditional distribution that determines the probability of $X \in A$ conditioned on $Y$ obtaining the value $y$. The bar denotes that we are clamping $Y$ to the value $y$ and identifying the distribution induced on $X$ in the restricted sample space. Informally, the conditional distribution takes the part of the sample space where $Y=y$ and re-normalizes the joint distribution such that the result is a probability distribution defined only over that part of the sample space.

When we consider the joint distribution in Equation $1.1$ to be a function that maps events to probabilities in the space of $X$, with $y$ being fixed, we note that the value of $p(Y=y)$ is actually a normalization constant that can be determined from the numerator $p(X \in A, Y=y)$. For example, if $X$ is discrete when using a PMF, then:
$$
p(Y=y)=\sum_{x} p(X=x, Y=y) .
$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|BAYES’ RULE

Bayes’ rule is a basic result in probability that describes a relationship between two conditional distributions $p(X \mid Y)$ and $p(Y \mid X)$ for a pair of random variables (these random variables can also be continuous). More specifically, Bayes’ rule states that for any such pair of random variables, the following identity holds:
$$
p(Y=y \mid X=x)=\frac{p(X-x \mid Y-y) p(Y-y)}{p(X=x)}
$$

This result also generally holds true for any two events $A$ and $B$ with the conditional probability $p(X \in A \mid Y \in B)$.

The main advantage that Bayes’ rule offers is inversion of the conditional relationship between two random variables – therefore, if one variable is known, then the other can be calculated as well, assuming the marginal distributions $p(X=x)$ and $p(Y=y)$ are also known.
Bayes’ rule can be proven in several ways. One way to derive it is simply by using the chain rule twice. More specifically, we know that the joint distribution values can be rewritten as follows, using the chain rule, either first separating $X$ or first separating $Y$ :
$$
\begin{aligned}
p(X&=x, Y=y) \
&=p(X=x) p(Y=y \mid X=x) \
&=p(Y=y) p(X=x \mid Y=y)
\end{aligned}
$$
Taking the last equality above, $p(X=x) p(Y=y \mid X=x)=p(Y=y) p(X=x \mid Y=$ $y)$, and dividing both sides by $p(X=x)$ results in Bayes’ rule as described in Equation 1.2.
Bayes’ rule is the main pillar in Bayesian statistics for reasoning and learning from data. Bayes’ rule can invert the relationship between “observations” (the data) and the random variables we are interested in predicting. This makes it possible to infer target predictions from such observations. A more detailed description of these ideas is provided in Section 1.5, where statistical modeling is discussed.

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|JOINT DISTRIBUTION OVER MULTIPLE RANDOM VARIABLES

可以在同一个样本空间上定义多个随机变量。例如，对于一个离散的样本空间，比如一组词，我们可以定义两个随机变量 $X$ 和 $Y$ 取整数值一一一个可以测量单词长度，另一个可以测量单词中元音的数量。给定两个这样的随机变量，联合分布 $P(X, Y)$ 是映射成对事件的函数 $(A, B)$ 如下:
$$
p(X \in A, Y \in B)=p\left(X^{-1}(A) \cap Y^{-1}(B)\right)
$$
$\mathrm{~ 我们经常会采取几组 ~ V e f t { 1 O m e g a _ { 1 } , ~ \ d o t s , ~ I O m e g a _ { m }}$ $\Omega=\Omega_{1} \times \ldots \times \Omega_{m}$. 每个 $\Omega_{i}$ 与随机变量相关联。基于此，可以为所有这些随机变量一起定义联合概率分布。例如，考虑 $\Omega=V \times P$ 在哪里 $V$ 是一个词汇表和 $P$ 是词性标签。这个样本空间使我们能够定义概率 $p(x, y)$ 在哪里 $X$ 表示与词性相关的词 $Y$. 在这种情况下， $x \in V$ 和 $y \in P$.
对于任何联合分布，我们可以边缘化一些随机变量以获得在原始随机变量的子集上定义的分布（因此它仍然可以是联合分布，仅在随机变量的子集上)。使用积分 (对于连续变量) 或求和 (对于离散随机变量) 进行边缘化。这种求和或积分操作从联合分布中消除了随机变量。结果是非边缘化随机变量的联合分布。
对于上面简单的词性示例，我们可以得到边缘 $p(x)=\sum_{y \in P} p(x, y)$ 或者 $p(y)=\sum_{x \in V} p(x, y)$. 边缘人 $p(X)$ 和 $p(Y)$ 不唯一确定联合分布值 $p(X, Y)$. 只有反过来才是正确的。然而，每当 $X$ 和 $Y$ 是独立的，则可以使用边际确定联合分布。在第 $1.3 .2$ 节中了解更多信息。

统计代写|贝叶斯分析代写Bayesian Analysis代考|CONDITIONAL DISTRIBUTIONS

联合概率分布为有关几个随机变量获得特定值的概率问题提供了答案。条件分布为不同但相关的问题提供了答案。当联合分布中的其他变量被限制为特定值时（或当它们被”钅制”时），它们有助于确定随机变量可以获得的值。
条件分布可从同一组随机变量上的联合分布推导出来。考虑一对随机变量 $X$ 和 $Y$ (连续的或离散的)。如果 $A$ 是来自样本空间的事件 $X$ 和 $y$ 是样本空间中的一个值 $Y$ ，然后：
$$
p(X \in A \mid Y=y)=\frac{p(X \in A, Y=y)}{p(Y=y)}
$$
将被解释为确定概率的条件分布 $X \in A$ 以 $Y$ 获取价值 $y$. 条形表示我们正在夹紧 $Y$ 到价值 $y$ 并确定引起的分布 $X$ 在有限的样本空间中。非正式地，条件分布占据样本空间的一部分，其中 $Y=y$ 并重新归一化联合分布，使得结果是仅在样本空间的该部分上定义的概率分布。
当我们考虑方程中的联合分布时 $1.1$ 是一个将事件映射到空间中的概率的函数 $X$ ，和 $y$ 是固定的，我们注意到，价值 $p(Y=y)$ 实际上是一个归一化常数，可以从分子中确定 $p(X \in A, Y=y)$. 例如，如果 $X$ 使用 PMF 时是离散的，则:
$$
p(Y=y)=\sum_{x} p(X=x, Y=y)
$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|BAYES’ RULE

贝叶斯规则是描述两个条件分布之间关系的概率的基本结果 $p(X \mid Y)$ 和 $p(Y \mid X)$ 对于一对随机变量 (这些随机变量也可以是连续的）。更具体地说，贝叶斯规则指出，对于任何这样的随机变量对，以下恒等式成立：
$$
p(Y=y \mid X=x)=\frac{p(X-x \mid Y-y) p(Y-y)}{p(X=x)}
$$
这个结果通常也适用于任何两个事件 $A$ 和 $B$ 有条件概率 $p(X \in A \mid Y \in B)$.
贝叶斯规则提供的主要优点是反转两个随机变量之间的条件关系一一因此，如果一个变量是已知的，那么假设边际分布也可以计算另一个变量 $p(X=x)$ 和 $p(Y=y)$ 也是众所周知的。
贝叶斯规则可以通过多种方式证明。推导它的一种方法是简单地使用链式法则两次。更具体地说，我们知道联合分布值可以改写如下，使用链式法则，或者首先分离 $X$ 或先分离 $Y:$
$$
p(X=x, Y=y) \quad=p(X=x) p(Y=y \mid X=x)=p(Y=y) p(X=x \mid Y=y)
$$
取上面最后一个等式， $p(X=x) p(Y=y \mid X=x)=p(Y=y) p(X=x \mid Y=y)$ ，并将两边除以 $p(X=x)$ 得出公式 $1.2$ 中描述的贝叶斯规则。
贝叶斯规则是贝叶斯统计中用于推理和从数据中学习的主要支柱。贝叶斯规则可以颠倒“观察”（数据）和我们有兴趣预测的随机变量之间的关系。这使得从这些观察中推断出目标预测成为可能。1.5节提供了对这些想法的更详细描述，其中讨论了统计建模。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|贝叶斯分析代写Bayesian Analysis代考|DATA5711

Posted on 2022年6月30日2022年6月30日 by statistics-lab

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我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|DATA5711

统计代写|贝叶斯分析代写Bayesian Analysis代考|PROBABILITY MEASURES

At the core of probabilistic theory (and probabilistic modeling) lies the idea of a “sample space.” The sample space is a set $\Omega$ that consists of all possible elements over which we construct a probability distribution. In this book, the sample space most often consists of objects relating to language, such as words, phrase-structure trees, sentences, documents or sequences. As we see later, in the Bayesian setting, the sample space is defined to be a Cartesian product between a set of such objects and a set of model parameters (Section 1.5.1).

Once a sample space is determined, we can define a probability measure for that sample space. A probability measure $p$ is a function which attaches a real number to events-subsets of the sample space.
A probability measure has to satisfy three axiomatic properties:

It has to be a non-negative function such that $p(A) \geq 0$ for any event $A$.
For any countable disjoint sequence of events $A_{i} \subseteq \Omega, i \in{1, \ldots}$, if $A_{i} \cap A_{j}=\emptyset$ for $i \neq$ $j$, it should hold that $p\left(\bigcup_{i} A_{i}\right)=\sum_{i} p\left(A_{i}\right)$. This means that the sum of probabilities of disjoint events should equal the probability of the union of the events.
The probability of $\Omega$ is $1: p(\Omega)=1$.

There are a few consequences from these three axiomatic properties. The first is that $p(\emptyset)=0$ (to see this, consider that $p(\Omega)+p(\emptyset)=p(\Omega \cup \emptyset)=p(\Omega)=1$ ). The second is that $p(A \cup B)=p(A)+p(B)-p(A \cap B)$ for any two events $A$ and $B$ (to see this, consider that $p(A \cup B)=p(A)+p(B \backslash(A \cap B))$ and that $p(B)=p(B \backslash(A \cap B))+p(A \cap B))$. And finally, the complement of an event $A, \Omega \backslash A$ is such that $p(\Omega \backslash A)=1-p(A)$ (to see this, consider that $1=p(\Omega)=p((\Omega \backslash A) \cup A)=p(\Omega \backslash A)+p(A)$ for any event $A)$.

In the general case, not every subset of the sample space should be considered an event.
From a measure-theoretic point of view for probability theory, an event must be a “measurable set.” The collection of measurable sets of a given sample space needs to satisfy some axiomatic properties. ${ }^{1}$ A discussion of measure theory is beyond the scope of this book, but see Ash and Doléans-Dade (2000) for a thorough investigation of this topic.

For our discrete sample spaces, consisting of linguistic structures or other language-related discrete objects, this distinction of measurable sets from arbitrary subsets of the sample space is not crucial. We will consider all subsets of the sample space to be measurable, which means they could be used as events. For continuous spaces, we will be using well-known probability measures that rely on Lebesgue’s measure. This means that the sample space will be a subset of a Euclidean space, and the set of events will be the subsets of this space that can be integrated over using Lebesgue’s integration.

统计代写|贝叶斯分析代写Bayesian Analysis代考|RANDOM VARIABLES

In their most basic form, random variables are functions that map each $w \in \Omega$ to a real value. They are often denoted by capital letters such as $X$ and $Z$. Once such a function is defined, under some regularity conditions, it induces a probability measure over the real numbers. More specifically, for any $A \subseteq \mathbb{R}$ such that the pre-image, $X^{-1}(A)$, defined as, ${\omega \in \Omega \mid X(\omega) \in A}$, is an event, its probability is:
$$
p_{X}(A)=p(X \in A)=p\left(X^{-1}(A)\right),
$$
where $p_{X}$ is the probability measure induced by the random variable $X$ and $p$ is a probability measure originally defined for $\Omega$. The sample space for $p_{X}$ is $\mathbb{R}$. The set of events for this sample space includes all $A \subseteq \mathbb{R}$ such that $X^{-1}(A)$ is an event in the original sample space $\Omega$ of $p$.
It is common to define a statistical model directly in terms of random variables, instead of explicitly defining a sample space and its corresponding real-value functions. In this case, random variables do not have to be interpreted as real-value functions and the sample space is understood to be a range of the random variable function. For example, if one wants to define a probability distribution over a language vocabulary, then one can define a random variable $X(\omega)=\omega$ with $\omega$ ranging over words in the vocabulary. Following this, the probability of a word in the vocabulary is denoted by $p(X \in{\omega})=p(X=\omega)$.

统计代写|贝叶斯分析代写Bayesian Analysis代考|CONTINUOUS AND DISCRETE RANDOM VARIABLES

This book uses the two most common kinds of random variables available in statistics: continuous and discrete. Continuous random variables take values in a continuous space, usually a subspace of $\mathbb{R}^{d}$ for $d \geq 1$. Discrete random variables, on the other hand, take values from a discrete, possibly countable set. In this book, discrete variables are usually denoted using capital letters such as $X, Y$ and $Z$, while continuous variables are denoted using greek letters, such as $\theta$ and $\mu$.

The continuous variables in this book are mostly used to define a prior over the parameters of a discrete distribution, as is usually done in the Bayesian setting. See Section $1.5 .2$ for a discussion of continuous variables. The discrete variables, on the other hand, are used to model structures that will be predicted (such as parse trees, part-of-speech tags, alignments, clusters) or structures which are observed (such as a sentence, a string over some language vocabulary or other such sequences).

The discrete variables discussed in this book are assumed to have an underlying probability mass function (PMF)-i.e., a function that attaches a weight to each element in the sample space, $p(x)$. This probability mass function induces the probability measure $p(X \in A)$, which satisfies:
$$
p(X \in A)=\sum_{x \in A} p(x),
$$
where $A$ is a subset of the possible values $X$ can take. Note that this equation is the result of the axiom of probability measures, where the probability of an event equals the sum of probabilities of disjoint events that precisely cover that event (singletons, in our case).

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|PROBABILITY MEASURES

概率论 (和概率建模) 的核心是“样本空间”的概念。样本空间是一个集合 $\Omega$ 它由我们构建概率分布的所有可能元素组成。在本书中，样本空间通常由与语言相关的对象组成，例如单词、短语结构树、句子、文档或序列。正如我们稍后看到的，在贝叶斯设置中，样本空间被定义为一组此类对象和一组模型参数之间的笛卡尔积 (第 $1.5 .1$ 节) 。
一旦确定了样本空间，我们就可以为该样本空间定义概率测度。概率测度 $p$ 是一个将实数附加到样本空间的事件子集的函数。
概率度量必须满足三个公理性质:

它必须是一个非负函数，使得 $p(A) \geq 0$ 对于任何事件 $A$.
对于任何可数的不相交的事件序列 $A_{i} \subseteq \Omega, i \in 1, \ldots$ ，如果 $A_{i} \cap A_{j}=\emptyset$ 为了 $i \neq j$ ，它应该认为 $p\left(\bigcup_{i} A_{i}\right)=\sum_{i} p\left(A_{i}\right)$. 这意味着不相交事件的概率之和应该等于事件联合的概率。
的概率 $\Omega$ 是 $1: p(\Omega)=1$.
这三个公理性质有一些结果。第一个是 $p(\emptyset)=0$ (要看到这一点，请考虑
$p(\Omega)+p(\emptyset)=p(\Omega \cup \emptyset)=p(\Omega)=1)$ 。第二个是 $p(A \cup B)=p(A)+p(B)-p(A \cap B)$ 对于任何两个事件 $A$ 和 $B$ (要看到这一点，请考虑 $p(A \cup B)=p(A)+p(B \backslash(A \cap B))$ 然后
$p(B)=p(B \backslash(A \cap B))+p(A \cap B))$. 最后，事件的补充 $A, \Omega \backslash A$ 是这样的 $p(\Omega \backslash A)=1-p(A)$ (要看到这一点，请考虑 $1=p(\Omega)=p((\Omega \backslash A) \cup A)=p(\Omega \backslash A)+p(A)$ 对于任何事件 $A)$.
在一般情况下，并非样本空间的每个子集都应被视为一个事件。
从概率论的测度论观点来看，一个事件必须是一个”可测集”。给定样本空间的可测量集的集合需要满足一些公理性质。 1 测度论的讨论超出了本书的范围，但请参阅 Ash 和 Doléans-Dade (2000) 对该主题的深入研究。
对于我们的离散样本空间，由语言结构或其他与语言相关的离散对象组成，可测量集与样本空间任意子集的区别并不重要。我们将认为样本空间的所有子集都是可测量的，这意味着它们可以用作事件。对于连续空间，我们将使用依赖于 Lebesgue 测度的众所周知的概率测度。这意味着样本空间将是欧几里得空间的子集，而事件集将是该空间的子集，可以使用 Lebesgue 积分进行积分。

统计代写|贝叶斯分析代写Bayesian Analysis代考|RANDOM VARIABLES

在最基本的形式中，随机变量是映射每个 $w \in \Omega$ 到一个真正的价值。它们通常用大写字母表示，例如 $X$ 和 $Z$. 一旦定义了这样的函数，在某些规律性条件下，它就会对实数进行概率测度。更具体地说，对于任何 $A \subseteq \mathbb{R}$ 使得原像， $X^{-1}(A)$ ，定义为， $\omega \in \Omega \mid X(\omega) \in A$, 是一个事件，它的概率是:
$$
p_{X}(A)=p(X \in A)=p\left(X^{-1}(A)\right),
$$
在哪里 $p_{X}$ 是由随机变量引起的概率测度 $X$ 和 $p$ 是最初定义为的概率测度 $\Omega$. 样本空间为 $p_{X}$ 是 $\mathbb{R}$. 该样本空间的事件集包括所有 $A \subseteq \mathbb{R}$ 这样 $X^{-1}(A)$ 是原始样本空间中的事件 $\Omega$ 的 $p$.
通常直接根据随机变量定义统计模型，而不是明确定义样本空间及其对应的实值函数。在这种情况下，随机变量不必被解释为实值函数，样本空间被理解为随机变量函数的范围。例如，如果想定义一个语言词汇表的概率分布，那么可以定义一个随机变量 $X(\omega)=\omega$ 和 $\omega$ 涵盖词汇表中的单词。此后，词汇表中单词的概率表示为
$$
p(X \in \omega)=p(X=\omega) \text {. }
$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|CONTINUOUS AND DISCRETE RANDOM VARIABLES

本书使用统计学中两种最常见的随机变量: 连续变量和离散变量。连续随机变量在连续空间中取值，通常是 $\mathbb{R}^{d}$ 为了 $d \geq 1$. 另一方面，离散随机变量从离散的、可能可数的集合中获取值。在本书中，离散变量通常用大写字母表示，例如 $X, Y$ 和 $Z$ ，而连续变量用希腊字母表示，例如 $\theta$ 和 $\mu$.
本书中的连续变量主要用于定义离散分布参数的先验，这通常在贝叶斯设置中完成。见部分 $1.5 .2$ 讨论连续变量。另一方面，离散变量用于对将要预测的结构（例如解析树、词性标签、对齐、聚类) 或观察到的结构 (例如句子、某个字符串语言词汇或其他此类序列)。
假设本书中讨论的离散变量有一个潜在的概率质量函数（PMF) 一一即一个赋予样本空间中每个元素权重的函数， $p(x)$. 这个概率质量函数引入了概率测度 $p(X \in A)$ ，满足:
$$
p(X \in A)=\sum_{x \in A} p(x)
$$
在哪里 $A$ 是可能值的子集 $X$ 可以采取。请注意，这个方程是概率度量公理的结果，其中事件的概率等于精确覆盖该事件的不相交事件的概率之和（在我们的例子中是单例）。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|贝叶斯分析代写Bayesian Analysis代考|STATS 3023

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写贝叶斯分析Bayesian Analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|STATS 3023

统计代写|贝叶斯分析代写Bayesian Analysis代考|Solving equations

In most of the Bayesian models so far examined, the calculations required could be done analytically. For example, the model given by:
$$
(Y \mid \theta) \sim \operatorname{Binomial}(5, \theta)
$$
$$
\theta \sim U(0,1) \text {, }
$$
together with data $y=5$, implies the posterior $(\theta \mid y) \sim \operatorname{Beta}(6,1)$. So $\theta$ has posterior pdf $f(\theta \mid y)=6 \theta^{5}$ and posterior cdf $F(\theta \mid y)=\theta^{6}$. Then, setting $F(\theta \mid y)=1 / 2$ yields the posterior median, $\theta=1 / 2^{1 / 6}=0.8909$.
But what if the equation $F(\theta \mid y)=1 / 2$ were not so easy to solve? In that case we could employ a number of strategies. One of these is trial and error, and another is via special functions in software packages, for example using the qbeta () function in $R$. This yields the correct answer. Yet another method is the Newton-Raphson algorithm, our next topic.
R Code for Section $4.1$
qbeta(0.5,6,1) #0.8908987

统计代写|贝叶斯分析代写Bayesian Analysis代考|The Newton-Raphson algorithm

The Newton-Raphson (NR) algorithm is a useful technique for solving equations of the form $g(x)=0$.

This algorithm involves choosing a suitable starting value $x_{0}$ and iteratively applying the equation
$$
x_{j+1}=x_{j}-g^{\prime}\left(x_{j}\right)^{-1} g\left(x_{j}\right)
$$
until convergence had been achieved to a desired degree of precision.
How does the NR algorithm work? Figure 4.1 illustrates the idea.

Here, $a$ is the desired solution of the equation $g(x)=0, c$ is a guess at that solution, and $b$ is a better estimate of $a$. Observe that the slope of the tangent at point $Q$ is equal to both $g^{\prime}(c)$ and $g(c) /(c-b)$. Equating these two expressions we get $b=c-g(c) / g^{\prime}(c)$.

Note: Sometimes the NR algorithm takes a long time to converge, and sometimes it converges to the wrong or even impossible value or gets ‘stuck’ and fails to converge at all. This is a general problem with the NR algorithm, namely its instability and the need to start it off with an initial guess that is sufficiently close to the desired solution.

统计代写|贝叶斯分析代写Bayesian Analysis代考|The multivariate Newton-Raphson algorithm

The Newton-Raphson algorithm can also be used to solve several equations simultaneously, say
$$
g_{k}\left(x_{1}, \ldots, x_{K}\right)=0, k=1, \ldots, K \text {. }
$$

Let: $x=\left(\begin{array}{c}x_{1} \ \vdots \ x_{K}\end{array}\right), g(x)=\left(\begin{array}{c}g_{1}(x) \ \vdots \ g_{K}(x)\end{array}\right), 0=\left(\begin{array}{c}0 \ \vdots \ 0\end{array}\right)$ (a column vector of length $K$ ).
Then the system of $K$ equations may be expressed as
$$
g(x)=0,
$$
and the NR algorithm involves iterating according to
$$
x^{(j+1)}=x^{(j)}-g^{\prime}\left(x^{(j)}\right)^{-1} g\left(x^{(j)}\right),
$$
where: $x^{(j)}=\left(\begin{array}{c}x_{1}^{(j)} \ \vdots \ x_{K}^{(j)}\end{array}\right)$ is the value of $x$ at the $j$ th iteration
$$
x^{(j+1)}=\left(\begin{array}{c}
x_{1}^{(j+1)} \
\vdots \
x_{K}^{(j+1)}
\end{array}\right), \quad g\left(x^{(j)}\right)=\left(\begin{array}{c}
g_{1}\left(x^{(j)}\right) \
\vdots \
g_{K}\left(x^{(j)}\right)
\end{array}\right)=\left[\left.\begin{array}{c}
g_{1}(x) \
\vdots \
\left.g_{K}(x)\right)
\end{array}\right|{x{x} x^{(j)}}\right]
$$
$$
\begin{aligned}
&g^{\prime}\left(x^{(j)}\right)=\left[\left.g^{\prime}(x)\right|{x=x^{\prime \prime \prime}}\right] \ &g^{\prime}(x)=\left(\begin{array}{ccc} \partial g{1}(x) / \partial x^{T} \
\vdots \
\partial g_{K}(x) / \partial x^{T}
\end{array}\right)=\left(\begin{array}{ccc}
\partial g_{1}(x) / \partial x_{1} & \cdots & \partial g_{1}(x) / \partial x_{K} \
\vdots & \ddots & \vdots \
\partial g_{K}(x) / \partial x_{1} & \cdots & \partial g_{K}(x) / \partial x_{K}
\end{array}\right) .
\end{aligned}
$$

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|Solving equations

在迄今为止检查的大多数贝叶斯模型中，所需的计算可以通过分析完成。例如，给出的模型：

(是∣θ)∼二项式⁡(5,θ)

θ∼在(0,1),
连同数据是=5, 意味着后验(θ∣是)∼贝塔⁡(6,1). 所以θ有后验pdfF(θ∣是)=6θ5和后 cdfF(θ∣是)=θ6. 然后，设置F(θ∣是)=1/2产生后中位数，θ=1/21/6=0.8909.
但是如果方程F(θ∣是)=1/2没那么容易解决？在这种情况下，我们可以采用多种策略。其中一种是反复试验，另一种是通过软件包中的特殊功能，例如使用 qbeta() 函数R. 这会产生正确的答案。另一种方法是牛顿-拉夫森算法，这是我们的下一个主题。
部分的 R 代码4.1
qbeta (0.5,6,1) #0.8908987

统计代写|贝叶斯分析代写Bayesian Analysis代考|The Newton-Raphson algorithm

Newton-Raphson (NR) 算法是求解以下方程的有用技术G(X)=0.

该算法涉及选择合适的起始值X0并迭代地应用方程

Xj+1=Xj−G′(Xj)−1G(Xj)
直到收敛到所需的精度。
NR 算法是如何工作的？图 4.1 说明了这个想法。

这里，一个是方程的期望解G(X)=0,C是对该解决方案的猜测，并且b是更好的估计一个. 观察点处切线的斜率问等于两者G′(C)和G(C)/(C−b). 使这两个表达式相等，我们得到b=C−G(C)/G′(C).

注意：有时 NR 算法需要很长时间才能收敛，有时它会收敛到错误甚至不可能的值，或者会“卡住”而根本无法收敛。这是 NR 算法的一个普遍问题，即它的不稳定性以及需要以足够接近所需解决方案的初始猜测来启动它。

统计代写|贝叶斯分析代写Bayesian Analysis代考|The multivariate Newton-Raphson algorithm

Newton-Raphson 算法也可用于同时求解多个方程，例如

Gķ(X1,…,Xķ)=0,ķ=1,…,ķ.

让：X=(X1 ⋮ Xķ),G(X)=(G1(X) ⋮ Gķ(X)),0=(0 ⋮ 0)（长度的列向量ķ）。
那么系统ķ方程可以表示为

G(X)=0,
NR算法涉及根据

X(j+1)=X(j)−G′(X(j))−1G(X(j)),
在哪里：X(j)=(X1(j) ⋮ Xķ(j))是的价值X在j第一次迭代

X(j+1)=(X1(j+1) ⋮ Xķ(j+1)),G(X(j))=(G1(X(j)) ⋮ Gķ(X(j)))=[G1(X) ⋮ Gķ(X))|XXX(j)]

G′(X(j))=[G′(X)|X=X′′′] G′(X)=(∂G1(X)/∂X吨 ⋮ ∂Gķ(X)/∂X吨)=(∂G1(X)/∂X1⋯∂G1(X)/∂Xķ ⋮⋱⋮ ∂Gķ(X)/∂X1⋯∂Gķ(X)/∂Xķ).

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|贝叶斯分析代写Bayesian Analysis代考|MAST90125

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写贝叶斯分析Bayesian Analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian predictive inference

In addition to estimating model parameters (and functions of those parameters) there is often interest in predicting some future data (or some other quantity which is not just a function of the model parameters).

Consider a Bayesian model specified by $f(y \mid \theta)$ and $f(\theta)$, with posterior as derived in ways already discussed and given by $f(\theta \mid y)$.
Now consider any other quantity $x$ whose distribution is defined by a density of the form $f(x \mid y, \theta)$.

The posterior predictive distribution of $x$ is given by the posterior predictive density $f(x \mid y)$. This can typically be derived using the following equation:
$$
\begin{aligned}
f(x \mid y) &=\int f(x, \theta \mid y) d \theta \
&=\int f(x \mid y, \theta) f(\theta \mid y) d \theta
\end{aligned}
$$
Note: For the case where $\theta$ is discrete, a summation needs to be performed rather than an integral.

The posterior predictive density $f(x \mid y)$ forms a basis for making probability statements about the quantity $x$ given the observed data $y$.
Point and interval estimation for future values $x$ can be performed in very much the same way as that for model parameters, except with a slightly different terminology.

Now, instead of referring to $\hat{x}=E(x \mid y)$ as the posterior mean of $x$, we may instead use the term predictive mean.

Also, the ‘ $\mathrm{P}$ ‘ in HPDR, and CPDR may be read as predictive rather than as posterior. For example, the CPDR for $x$ is now the central predictive density region for $x$.
As an example of point prediction, the predictive mean of $x$ is
$$
\hat{x}=E(x \mid y)=\int x f(x \mid y) d x .
$$

Often it is easier to obtain the predictive mean of $x$ using the equation
$$
\begin{aligned}
\hat{x}=E(x \mid y) &=E{E(x \mid y, \theta) \mid y} \
&=\int E(x \mid y, \theta) f(\theta \mid y) d \theta
\end{aligned}
$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|Posterior predictive p-values

Earlier, in Section 1.3, we discussed Bayes factors as a form of hypothesis testing within the Bayesian framework. An entirely different way to perform hypothesis testing in that framework is via the theory of posterior predictive $p$-values (Meng, 1994). As in the theory of Bayes factors, this involves first specifying a null hypothesis
$$
H_{0}: E_{0}
$$
and an alternative hypothesis
$$
H_{1}: E_{1} \text {, }
$$
where $E_{0}$ and $E_{1}$ are two events.
Note: As in Section $1.3, E_{0}$ and $E_{1}$ may or may not be disjoint. Also, $E_{0}$ and $E_{1}$ may instead represent two different models for the same data.
In the context of a single Bayesian model with data $y$ and parameter $\theta$, the theory of posterior predictive p-values involves the following steps:
(i) Define a suitable discrepancy measure (or test statistic), denoted $T(y, \theta)$,
following careful consideration of both $H_{0}$ and $H_{1}$ (see below).
(ii) Define $x$ as an independent future replicate of the data $y$.
(iii) Calculate the posterior predictive $p$-value (ppp-value), defined as
$$
p=P{T(x, \theta) \geq T(y, \theta) \mid y} .
$$
Note 1: The ppp-value is calculated under the implicit assumption that $H_{0}$ is true. Thus we could also write $p=P\left{T(x, \theta) \geq T(y, \theta) \mid y, H_{0}\right}$.
Note 2 : The discrepancy measure may or may not depend on the model parameter, $\theta$. Thus in some cases, $T(y, \theta)$ may also be written as $T(y)$.
The underlying idea behind the choice of discrepancy measure $T$ is that if the observed data $y$ is highly inconsistent with $H_{0}$ in favour of $H_{1}$ then $p$ should likely be small. This is the same idea as behind classical hypothesis testing. In fact, the classical theory may be viewed as a special case of the theory of ppp-values. The advantage of the ppp-value framework is that it is far more versatile and can be used in situations where it is not obvious how the classical theory should be applied.

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian models with multiple parameters

So far we have examined Bayesian models involving some data $y$ and a parameter $\theta$, where $\theta$ is a strictly scalar quantity. We now consider the case of Bayesian models with multiple parameters, starting with a focus on just two, say $\theta_{1}$ and $\theta_{2}$. In that case, the Bayesian model may be defined by specifying $f(y \mid \theta)$ and $f(\theta)$ in the same way as previously, but with an understanding that $\theta$ is a vector of the form $\theta=\left(\theta_{1}, \theta_{2}\right)$.
The first task now is to find the joint posterior density of $\theta_{1}$ and $\theta_{2}$, according to
$$
f(\theta \mid y) \propto f(\theta) f(y \mid \theta),
$$
or equivalently
$$
f\left(\theta_{1}, \theta_{2} \mid y\right) \propto f\left(\theta_{1}, \theta_{2}\right) f\left(y \mid \theta_{1}, \theta_{2}\right),
$$
where
$$
f(\theta)=f\left(\theta_{1}, \theta_{2}\right)
$$
is the joint prior density of the two parameters.
Often, this joint prior density is specified as an unconditional prior multiplied by a conditional prior, for example as
$$
f\left(\theta_{1}, \theta_{2}\right)=f\left(\theta_{1}\right) f\left(\theta_{2} \mid \theta_{1}\right) .
$$
Once a Bayesian model with two parameters has been defined, one task is to find the marginal posterior densities of $\theta_{1}$ and $\theta_{2}$, respectively, via the equations:
$$
\begin{aligned}
&f\left(\theta_{1} \mid y\right)=\int f\left(\theta_{1}, \theta_{2} \mid y\right) d \theta_{2} \
&f\left(\theta_{2} \mid y\right)=\int f\left(\theta_{1}, \theta_{2} \mid y\right) d \theta_{1}
\end{aligned}
$$

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian predictive inference

除了估计模型参数（和这些参数的函数）之外，通常还有兴趣预测一些未来数据（或一些其他数量，而不仅仅是模型参数的函数）。

考虑一个由下式指定的贝叶斯模型F(是∣θ)和F(θ), 后验的推导方式已经讨论过并由下式给出F(θ∣是).
现在考虑任何其他数量X其分布由以下形式的密度定义F(X∣是,θ).

的后验预测分布X由后验预测密度给出F(X∣是). 这通常可以使用以下等式得出：

F(X∣是)=∫F(X,θ∣是)dθ =∫F(X∣是,θ)F(θ∣是)dθ
注意：对于这种情况θ是离散的，需要进行求和而不是积分。

后验预测密度F(X∣是)构成对数量进行概率陈述的基础X给定观察到的数据是.
未来值的点和区间估计X可以以与模型参数非常相似的方式执行，除了术语略有不同。

现在，而不是指X^=和(X∣是)作为后验均值X，我们可以改为使用术语预测均值。

此外，’磷’ 在 HPDR 中，CPDR 可能被解读为预测性而非后验性。例如，CPDR 为X现在是中心预测密度区域X.
作为点预测的一个例子，X是

X^=和(X∣是)=∫XF(X∣是)dX.

通常更容易获得预测平均值X使用等式

X^=和(X∣是)=和和(X∣是,θ)∣是 =∫和(X∣是,θ)F(θ∣是)dθ

统计代写|贝叶斯分析代写Bayesian Analysis代考|Posterior predictive p-values

早些时候，在第 1.3 节中，我们将贝叶斯因子作为贝叶斯框架内的一种假设检验形式进行了讨论。在该框架中执行假设检验的一种完全不同的方法是通过后验预测理论p值（孟，1994）。与贝叶斯因子理论一样，这涉及首先指定零假设

H0:和0
和一个替代假设

H1:和1,
在哪里和0和和1是两个事件。
注意：如部分1.3,和0和和1可能会或可能不会脱节。还，和0和和1相反，它可能代表相同数据的两个不同模型。
在具有数据的单个贝叶斯模型的上下文中是和参数θ，后验预测 p 值理论涉及以下步骤：
(i) 定义合适的差异度量（或检验统计量），表示为吨(是,θ)，
经过仔细考虑两者H0和H1（见下文）。
(ii) 定义X作为数据的独立未来副本是.
(iii) 计算后验预测p-值（ppp值），定义为

p=磷吨(X,θ)≥吨(是,θ)∣是.
注 1：ppp 值是在隐含假设下计算的H0是真的。因此我们也可以写p=P\left{T(x,\theta)\geq T(y,\theta)\mid y, H_{0}\right}p=P\left{T(x,\theta)\geq T(y,\theta)\mid y, H_{0}\right}.
注 2：差异度量可能取决于模型参数，也可能不取决于模型参数，θ. 因此在某些情况下，吨(是,θ)也可以写成吨(是).
差异度量选择背后的基本思想吨是如果观察到的数据是高度不一致H0赞成H1然后p应该很小。这与经典假设检验背后的想法相同。事实上，经典理论可以看作是 ppp 值理论的一个特例。ppp-value 框架的优势在于它更加通用，并且可以在不明显应该如何应用经典理论的情况下使用。

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian models with multiple parameters

到目前为止，我们已经检查了涉及一些数据的贝叶斯模型是和一个参数θ，在哪里θ是一个严格的标量。我们现在考虑具有多个参数的贝叶斯模型的情况，从只关注两个开始，比如说θ1和θ2. 在这种情况下，贝叶斯模型可以通过指定F(是∣θ)和F(θ)以与以前相同的方式，但理解为θ是形式的向量θ=(θ1,θ2).
现在的第一个任务是找到联合后验密度θ1和θ2，根据

F(θ∣是)∝F(θ)F(是∣θ),
或等效地

F(θ1,θ2∣是)∝F(θ1,θ2)F(是∣θ1,θ2),
在哪里

F(θ)=F(θ1,θ2)
是两个参数的联合先验密度。
通常，这个联合先验密度被指定为无条件先验乘以条件先验，例如

F(θ1,θ2)=F(θ1)F(θ2∣θ1).
一旦定义了具有两个参数的贝叶斯模型，一项任务就是找到θ1和θ2，分别通过以下方程：

F(θ1∣是)=∫F(θ1,θ2∣是)dθ2 F(θ2∣是)=∫F(θ1,θ2∣是)dθ1

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|贝叶斯分析代写Bayesian Analysis代考|MSH3

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写贝叶斯分析Bayesian Analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|The posterior expected loss

We have defined the risk function as the expectation of the loss function given the parameter, namely
$$
R(\theta)=E(L(\hat{\theta}, \theta) \mid \theta)=\int L(\hat{\theta}(y), \theta) f(y \mid \theta) d y .
$$
Conversely, we now define the posterior expected loss (PEL) as the expectation of the loss function given the data, and we denote this function by
$$
P E L(y)=E{L(\hat{\theta}, \theta) \mid y}=\int L(\hat{\theta}(y), \theta) f(\theta \mid y) d \theta .
$$
Then, just as the risk function can be used to compute the Bayes risk according to
$$
r=E L(\hat{\theta}, \theta)=E E{L(\hat{\theta}, \theta) \mid \theta}=E R(\theta)=\int R(\theta) f(\theta) d \theta,
$$
so also can the PEL be used, but with the formula
$$
r=E L(\hat{\theta}, \theta)=E E{L(\hat{\theta}, \theta) \mid y}=E{P E L(y)}=\int \operatorname{PEL}(y) f(y) d y .
$$
Note: Both of these formulae for the Bayes risk use the law of iterated expectation, but with different conditionings.

统计代写|贝叶斯分析代写Bayesian Analysis代考|The Bayes estimate

The Bayes estimate (or estimator) is defined to be the choice of the function $\hat{\theta}=\hat{\theta}(y)$ for which the Bayes risk $r=E L(\hat{\theta}, \theta)$ is minimised. This estimator has the smallest overall expected loss over all estimators under the specified loss function $L(\hat{\theta}, \theta)$.

In many cases, the procedure for finding a Bayes estimate can be considerably simplified by considering which estimate minimises the posterior expected loss function, $P E L(y)=E{L(\hat{\theta}, \theta) \mid y}$.

If we can find an estimate $\hat{\theta}=\hat{\theta}(y)$ which minimises $\operatorname{PEL}(y)$ for all possible values of the data $y$, then that estimate must also minimise the Bayes risk.

This is because the Bayes risk may be written as a weighted average of the PEL, namely
$$
r=E L(\hat{\theta}, \theta)=E E{L(\hat{\theta}, \theta) \mid y}=E{P E L(y)}=\int \operatorname{PEL}(y) f(y) d y .
$$
Exercise 2.10 Bayes estimate under the QELF
Find the Bayes estimate under the quadratic error loss function.
Solution to Exercise 2.10
$$
\text { Observe that } \quad \begin{aligned}
P E L(y) &=E\left{(\hat{\theta}-\theta)^{2} \mid y\right}=E\left{\hat{\theta}^{2}-2 \hat{\theta} \theta+\theta^{2} \mid y\right} \
&=\hat{\theta}^{2}-2 \hat{\theta} E(\theta \mid y)+E\left(\theta^{2} \mid y\right) \
&=[\hat{\theta}-E(\theta \mid y)]^{2}-{E(\theta \mid y)}^{2}+E\left(\theta^{2} \mid y\right)
\end{aligned}
$$
Note: We have completed the square in $\hat{\theta}$.
We see that the PEL is a quadratic function of $\hat{\theta}$ which is clearly minimised at the posterior mean, $\hat{\theta}=E(\theta \mid y)$. So the Bayes estimate under the QELF is that posterior mean.

统计代写|贝叶斯分析代写Bayesian Analysis代考|Inference given functions of the data

Sometimes we observe a function of the data rather than the data itself. In such cases the function typically degrades the information available in some way. An example is censoring, where we observe a value only if that value is less than some cut-off point (right censoring) or greater than some cut-off value (left censoring). It is also possible to have censoring on the left and right simultaneously. Another example is rounding, where we only observe values to the nearest multiple of $0.1,1$ or 5 , etc.
Exercise 3.I Right censoring of exponential observations
Each light bulb of a certain type has a life which is conditionally exponential with mean $m=1 / c$, where $c$ has a prior distribution which is standard exponential. We observe $n=5$ light bulbs of this type for 6 units of time, and the lifetimes are:
$$
2.6,3.2, *, 1.2 \text {, * }
$$
where $*$ indicates a right-censored value which is greater than 6 . (Only values less than or equal to 6 could be observed.)

Find the posterior distribution and mean of the average light bulb lifetime, $m$.

The data here is
$$
D=\left{y_{1}=2.6, y_{2}=3.2, y_{3}>6, y_{4}=1.2, y_{5}>6\right} \text {, }
$$
and the probability of censoring is
$$
P\left(y_{i}>6 \mid c\right)=\int_{6}^{\infty} c e^{-c y_{i}} d y_{i}=e^{-6 c} .
$$
Therefore the posterior density of $c$ is
$$
\begin{aligned}
f(c \mid D) & \propto f(c) f(D \mid c) \
& \propto f(c) f\left(y_{1} \mid c\right) f\left(y_{2} \mid c\right) P\left(y_{3}>6 \mid c\right) f\left(y_{4} \mid c\right) P\left(y_{5}>6 \mid c\right)
\end{aligned}
$$ $$
\begin{aligned}
&\propto e^{-c}\left(c e^{-c c_{1}}\right)\left(c e^{-c y_{2}}\right)\left(e^{-6 c}\right)\left(c e^{-c y_{4}}\right)\left(e^{-6 c}\right) \
&=c^{3} \exp \left{-c\left(1+y_{1}+y_{2}+6+y_{4}+6\right)\right} \
&=c^{4-1} \exp {-c(1+2.6+3.2+6+1.2+6)} \
&=c^{4-1} \exp (-20 c)
\end{aligned}
$$
Hence:
$$
\begin{aligned}
&(c \mid D) \sim G(4,20) \
&(m \mid D) \sim I G(4,20) \
&f(m \mid D)=20^{4} m^{-(4+1)} e^{-20 / m} / \Gamma(4), m>0 \
&E(m \mid D)=20 /(4-1)=6.667
\end{aligned}
$$
It will be observed that this estimate of $m$ is appropriately higher than the estimate obtained by simply averaging the observed values, namely
$$
(1 / 3)(2.6+3.2+1.2)=2.333 \text {. }
$$
The estimate $6.667$ is also higher than the estimate obtained by simply replacing the censored values with 6 , namely
$$
(1 / 3)(2.6+3.2+6+1.2+6)=3.8 \text {. }
$$

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|The posterior expected loss

我们将风险函数定义为给定参数的损失函数的期望值，即

R(θ)=和(大号(θ^,θ)∣θ)=∫大号(θ^(是),θ)F(是∣θ)d是.
相反，我们现在将后验期望损失（PEL）定义为给定数据的损失函数的期望，我们将这个函数表示为

磷和大号(是)=和大号(θ^,θ)∣是=∫大号(θ^(是),θ)F(θ∣是)dθ.
然后，就像风险函数可以用来计算贝叶斯风险一样

r=和大号(θ^,θ)=和和大号(θ^,θ)∣θ=和R(θ)=∫R(θ)F(θ)dθ,
所以也可以使用 PEL，但使用公式

r=和大号(θ^,θ)=和和大号(θ^,θ)∣是=和磷和大号(是)=∫佩尔⁡(是)F(是)d是.
注意：这两个贝叶斯风险公式都使用了迭代期望定律，但条件不同。

统计代写|贝叶斯分析代写Bayesian Analysis代考|The Bayes estimate

贝叶斯估计（或估计器）被定义为函数的选择θ^=θ^(是)贝叶斯风险r=和大号(θ^,θ)被最小化。在指定损失函数下，该估计器在所有估计器中具有最小的总体预期损失大号(θ^,θ).

在许多情况下，通过考虑哪个估计最小化后验期望损失函数，可以大大简化寻找贝叶斯估计的过程，磷和大号(是)=和大号(θ^,θ)∣是.

如果我们能找到一个估计θ^=θ^(是)最小化佩尔⁡(是)对于数据的所有可能值是，那么该估计也必须最小化贝叶斯风险。

这是因为贝叶斯风险可以写成 PEL 的加权平均值，即

r=和大号(θ^,θ)=和和大号(θ^,θ)∣是=和磷和大号(是)=∫佩尔⁡(是)F(是)d是.
练习 2.10 QELF 下
的贝叶斯估计求二次误差损失函数下的贝叶斯估计。
习题 2.10 的解法

\text { 观察到 } \quad \begin{aligned} P E L(y) &=E\left{(\hat{\theta}-\theta)^{2} \mid y\right}=E\left{\帽子{\theta}^{2}-2 \hat{\theta} \theta+\theta^{2} \mid y\right} \ &=\hat{\theta}^{2}-2 \hat{\ theta} E(\theta \mid y)+E\left(\theta^{2} \mid y\right) \ &=[\hat{\theta}-E(\theta \mid y)]^{2 }-{E(\theta \mid y)}^{2}+E\left(\theta^{2} \mid y\right) \end{对齐}\text { 观察到 } \quad \begin{aligned} P E L(y) &=E\left{(\hat{\theta}-\theta)^{2} \mid y\right}=E\left{\帽子{\theta}^{2}-2 \hat{\theta} \theta+\theta^{2} \mid y\right} \ &=\hat{\theta}^{2}-2 \hat{\ theta} E(\theta \mid y)+E\left(\theta^{2} \mid y\right) \ &=[\hat{\theta}-E(\theta \mid y)]^{2 }-{E(\theta \mid y)}^{2}+E\left(\theta^{2} \mid y\right) \end{对齐}
注意：我们已经完成了广场θ^.
我们看到 PEL 是一个二次函数θ^在后验均值处明显最小化，θ^=和(θ∣是). 所以 QELF 下的贝叶斯估计就是后验均值。

统计代写|贝叶斯分析代写Bayesian Analysis代考|Inference given functions of the data

有时我们观察的是数据的函数而不是数据本身。在这种情况下，该功能通常会以某种方式降低可用信息的质量。一个例子是审查，只有当该值小于某个截止点（右删失）或大于某个截止值（左删失）时，我们才观察该值。也可以同时对左右进行审查。另一个例子是四舍五入，我们只观察到最接近的倍数的值0.1,1
or
5等米=1/C，在哪里C具有标准指数的先验分布。我们观察n=5此类灯泡可使用 6 个单位时间，其寿命为：

2.6,3.2,∗,1.2, *
在哪里∗表示大于 6 的右删失值。（只能观察到小于或等于 6 的值。）

求灯泡平均寿命的后验分布和均值，米.

这里的数据是

D=\left{y_{1}=2.6, y_{2}=3.2, y_{3}>6, y_{4}=1.2, y_{5}>6\right} \text {, }D=\left{y_{1}=2.6, y_{2}=3.2, y_{3}>6, y_{4}=1.2, y_{5}>6\right} \text {, }
审查的概率是

磷(是一世>6∣C)=∫6∞C和−C是一世d是一世=和−6C.
因此后验密度C是

F(C∣D)∝F(C)F(D∣C) ∝F(C)F(是1∣C)F(是2∣C)磷(是3>6∣C)F(是4∣C)磷(是5>6∣C)

\begin{aligned} &\propto e^{-c}\left(c e^{-c c_{1}}\right)\left(c e^{-c y_{2}}\right)\left(e ^{-6 c}\right)\left(c e^{-c y_{4}}\right)\left(e^{-6 c}\right) \ &=c^{3} \exp \left {-c\left(1+y_{1}+y_{2}+6+y_{4}+6\right)\right} \ &=c^{4-1} \exp {-c(1+ 2.6+3.2+6+1.2+6)} \ &=c^{4-1} \exp (-20 c) \end{对齐}\begin{aligned} &\propto e^{-c}\left(c e^{-c c_{1}}\right)\left(c e^{-c y_{2}}\right)\left(e ^{-6 c}\right)\left(c e^{-c y_{4}}\right)\left(e^{-6 c}\right) \ &=c^{3} \exp \left {-c\left(1+y_{1}+y_{2}+6+y_{4}+6\right)\right} \ &=c^{4-1} \exp {-c(1+ 2.6+3.2+6+1.2+6)} \ &=c^{4-1} \exp (-20 c) \end{对齐}
因此：

(C∣D)∼G(4,20) (米∣D)∼我G(4,20) F(米∣D)=204米−(4+1)和−20/米/Γ(4),米>0 和(米∣D)=20/(4−1)=6.667
可以看出，这个估计米适当地高于通过简单地平均观察值获得的估计值，即

(1/3)(2.6+3.2+1.2)=2.333.
估计6.667也高于通过简单地将删失值替换为 6 获得的估计值，即

(1/3)(2.6+3.2+6+1.2+6)=3.8.

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|贝叶斯分析代写Bayesian Analysis代考|DATA5711

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写贝叶斯分析Bayesian Analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|Dealing with a priori ignorance

The Bayesian approach requires a prior distribution to be specified even when there is complete (or total) a priori ignorance (meaning no prior information at all). This feature presents a general and philosophical problem with the Bayesian paradigm, one for which several theoretical solutions have been advanced but which does not yet have a universally accepted solution. We have already discussed finding an uninformative prior in relation to particular Bayesian models, as follows.

For the normal-normal model defined by $\left(y_{1}, \ldots, y_{n} \mid \mu\right) \sim$ iid $N\left(\mu, \sigma^{2}\right)$ and $\mu \sim N\left(\mu_{0}, \sigma_{0}^{2}\right)$, an uninformative prior is given by $\sigma_{0}=\infty$, that is, $f(\mu) \propto 1, \mu \in \Re$

For the normal-gamma model defined by $\left(y_{1}, \ldots, y_{n} \mid \mu\right) \sim$ iid $N(\mu, 1 / \lambda)$ and $\lambda \sim \operatorname{Gamma}(\alpha, \beta)$, an uninformative prior is given by $\alpha=\beta=0$, that is, $f(\lambda) \propto 1 / \lambda, \lambda>0$.

For the binomial-beta model defined by $(y \mid \theta) \sim \operatorname{Binomial}(n, \theta)$ and $\theta \sim \operatorname{Beta}(\alpha, \beta)$ (having the posterior $(\theta \mid y) \sim \operatorname{Beta}(\alpha+y, \beta+n-y)$ ), an uninformative prior is the Bayes prior given by $\alpha=\beta=1$, that is, $f(\theta)=1,0<\theta<1$. This is the prior that was originally advocated by Thomas Bayes.

Unlike for the normal-normal and normal-gamma models, more than one uninformative prior specification has been proposed as reasonable in the context of the binomial-beta model.
One of these is the improper Haldane prior, defined by $\alpha=\beta=0$, or
$$
f(\theta) \propto \frac{1}{\theta(1-\theta)}, \quad 0<\theta<1
$$
Under the prior $\theta \sim \operatorname{Beta}(\alpha, \beta)$ generally, the posterior mean of $\theta$ is
$$
\hat{\theta}=E(\theta \mid y)=\frac{(\alpha+y)}{(\alpha+y)+(\beta+n-y)}=\frac{\alpha+y}{\alpha+\beta+n}
$$
This reduces to the MLE $y / n$ under the Haldane prior but not under the Bayes prior. In contrast, the Bayes prior leads to a posterior mode which is equal to the MLE.

统计代写|贝叶斯分析代写Bayesian Analysis代考|The Jeffreys prior

The statistician Harold Jeffreys devised a rule for finding a suitable uninformative prior in a wide variety of situations. His idea was to construct a prior which is invariant under reparameterisation. For the case of a univariate model parameter $\theta$, the Jeffreys prior is given by the following equation (also known as Jeffreys’ rule):
$$
f(\theta) \propto \sqrt{I(\theta)},
$$
where $I(\theta)$ is the Fisher information defined by
$$
I(\theta)=E\left{\left(\frac{\partial}{\partial \theta} \log f(y \mid \theta)\right)^{2} \mid \theta\right} .
$$
Note 1: If $\log f(y \mid \theta)$ is twice differentiable with respect to $\theta$, and certain regularity conditions hold, then
$$
I(\theta)=-E\left{\frac{\partial^{2}}{\partial \theta^{2}} \log f(y \mid \theta) \mid \theta\right} .
$$
Note 2: Jeffreys’ rule also extends to the multi-parameter case (not considered here).

The significance of Jeffreys’ rule may be described as follows. Consider a prior given by $f(\theta) \propto \sqrt{I(\theta)}$ and the transformed parameter $\phi=g(\theta)$,where $g$ is a strictly increasing or decreasing function. (For simplicity, we only consider this case.) Then the prior density for $\phi$ is
$$
\begin{aligned}
f(\phi) & \propto f(\theta)\left|\frac{\partial \theta}{\partial \phi}\right| \text { by the transformation rule } \
& \propto \sqrt{I(\theta)\left(\frac{\partial \theta}{\partial \phi}\right)^{2}}=\sqrt{E\left{\left(\frac{\partial}{\partial \theta} \log f(y \mid \theta)\right)^{2} \mid \theta\right}\left(\frac{\partial \theta}{\partial \phi}\right)^{2}} \
&=\sqrt{E\left{\left(\frac{\partial}{\partial \theta} \log f(y \mid \theta) \frac{\partial \theta}{\partial \phi}\right)^{2} \mid \theta\right}} \
&=\sqrt{E\left{\left(\frac{\partial}{\partial \phi} \log f(y \mid \phi)\right)^{2} \mid \phi\right}} \
&=\sqrt{I(\phi)} .
\end{aligned}
$$
Thus, Jeffreys’ rule is ‘invariant under reparameterisation’, in the sense that if a prior is constructed according to
$$
f(\theta) \propto \sqrt{I(\theta)},
$$
then, for another parameter $\phi=g(\theta)$, it is also true that
$$
f(\phi) \propto \sqrt{I(\phi)} .
$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian decision theory

The posterior mean, mode and median, as well as other Bayesian point estimates, can all be derived and interpreted using the principles and theory of decision theory. Suppose we wish to choose an estimate of $\theta$ which minimises costs in some sense. To this end, let $L(\hat{\theta}, \theta)$ denote generally a loss function (LF) associated with an estimate $\hat{\theta}$.

Note: The estimator $\hat{\theta}$ is a function of the data $y$ and so could also be written $\hat{\theta}(y)$. For example, in the context where $(y \mid \theta) \sim \operatorname{Bin}(n, \theta)$, the sample proportion or MLE is the function given by $\hat{\theta}=\hat{\theta}(y)=y / n$.
The loss function $L$ represents the cost incurred when the true value $\theta$ is estimated by $\hat{\theta}$ and usually satisfies the property $L(\theta, \theta)=0$.
The three most commonly used loss functions are defined as follows: $L(\hat{\theta}, \theta)=|\hat{\theta}-\theta| \quad$ the absolute error loss function (AELF) $L(\hat{\theta}, \theta)=(\hat{\theta}-\theta)^{2} \quad \begin{aligned}&\text { the quadratic error loss function (QELF) }\end{aligned}$ $L(\hat{\theta}, \theta)=I(\hat{\theta} \neq \theta)=\left{\begin{array}{ll}0 & \text { if } \hat{\theta}=\theta \ 1 & \text { if } \hat{\theta} \neq \theta\end{array}\right} \quad$ the indicator error loss function (IELF), also known as the zero-one loss function (ZOLF) or the all-or-nothing error loss function (ANLF).
Figures $2.8$ and $2.9$ illustrate these three basic loss functions.

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|Dealing with a priori ignorance

即使存在完全（或全部）先验无知（意味着根本没有先验信息），贝叶斯方法也需要指定先验分布。这一特征提出了贝叶斯范式的一个普遍的哲学问题，已经提出了几种理论解决方案，但还没有一个普遍接受的解决方案。我们已经讨论过寻找与特定贝叶斯模型相关的无信息先验，如下所示。

对于由下式定义的正态-正态模型(是1,…,是n∣μ)∼独立同居ñ(μ,σ2)和μ∼ñ(μ0,σ02)，一个无信息的先验由下式给出σ0=∞，那是，F(μ)∝1,μ∈ℜ

对于由以下定义的正常伽马模型(是1,…,是n∣μ)∼独立同居ñ(μ,1/λ)和λ∼伽玛⁡(一个,b)，一个无信息的先验由下式给出一个=b=0，那是，F(λ)∝1/λ,λ>0.

对于由下式定义的二项式 beta 模型(是∣θ)∼二项式⁡(n,θ)和θ∼贝塔⁡(一个,b)（有后(θ∣是)∼贝塔⁡(一个+是,b+n−是))，无信息先验是由下式给出的贝叶斯先验一个=b=1，那是，F(θ)=1,0<θ<1. 这是托马斯·贝叶斯最初提倡的先验。

与正态正态和正态 gamma 模型不同，在二项式 beta 模型的背景下，已经提出了不止一个无信息的先验规范。
其中之一是不恰当的 Haldane 先验，定义为一个=b=0，或者

F(θ)∝1θ(1−θ),0<θ<1
根据之前θ∼贝塔⁡(一个,b)一般情况下，后验均值θ是

θ^=和(θ∣是)=(一个+是)(一个+是)+(b+n−是)=一个+是一个+b+n
这减少到 MLE是/n在 Haldane 先验之下，但不在 Bayes 先验之下。相反，贝叶斯先验导致等于 MLE 的后验模式。

统计代写|贝叶斯分析代写Bayesian Analysis代考|The Jeffreys prior

统计学家 Harold Jeffreys 设计了一个规则，用于在各种情况下找到合适的非信息先验。他的想法是构建一个在重新参数化下不变的先验。对于单变量模型参数的情况θ，杰弗里斯先验由以下等式（也称为杰弗里斯规则）给出：

F(θ)∝我(θ),
在哪里我(θ)是由以下定义的 Fisher 信息

I(\theta)=E\left{\left(\frac{\partial}{\partial \theta} \log f(y \mid \theta)\right)^{2} \mid \theta\right} 。I(\theta)=E\left{\left(\frac{\partial}{\partial \theta} \log f(y \mid \theta)\right)^{2} \mid \theta\right} 。
注 1：如果日志⁡F(是∣θ)是关于θ，并且某些正则条件成立，则

I(\theta)=-E\left{\frac{\partial^{2}}{\partial \theta^{2}} \log f(y \mid \theta) \mid \theta\right} 。I(\theta)=-E\left{\frac{\partial^{2}}{\partial \theta^{2}} \log f(y \mid \theta) \mid \theta\right} 。
注 2：Jeffreys 的规则也扩展到多参数情况（此处不考虑）。

杰弗里斯规则的意义可以描述如下。考虑先验F(θ)∝我(θ)和转换后的参数φ=G(θ)，在哪里G是严格递增或递减的函数。（为简单起见，我们只考虑这种情况。）那么先验密度φ是

\begin{对齐} f(\phi) & \propto f(\theta)\left|\frac{\partial \theta}{\partial \phi}\right| \text { 通过变换规则 } \ & \propto \sqrt{I(\theta)\left(\frac{\partial \theta}{\partial \phi}\right)^{2}}=\sqrt{E \left{\left(\frac{\partial}{\partial \theta} \log f(y \mid \theta)\right)^{2} \mid \theta\right}\left(\frac{\partial \theta}{\partial \phi}\right)^{2}} \ &=\sqrt{E\left{\left(\frac{\partial}{\partial \theta} \log f(y \mid \ theta) \frac{\partial \theta}{\partial \phi}\right)^{2} \mid \theta\right}} \ &=\sqrt{E\left{\left(\frac{\partial} {\partial \phi} \log f(y \mid \phi)\right)^{2} \mid \phi\right}} \ &=\sqrt{I(\phi)} 。\end{对齐}\begin{对齐} f(\phi) & \propto f(\theta)\left|\frac{\partial \theta}{\partial \phi}\right| \text { 通过变换规则 } \ & \propto \sqrt{I(\theta)\left(\frac{\partial \theta}{\partial \phi}\right)^{2}}=\sqrt{E \left{\left(\frac{\partial}{\partial \theta} \log f(y \mid \theta)\right)^{2} \mid \theta\right}\left(\frac{\partial \theta}{\partial \phi}\right)^{2}} \ &=\sqrt{E\left{\left(\frac{\partial}{\partial \theta} \log f(y \mid \ theta) \frac{\partial \theta}{\partial \phi}\right)^{2} \mid \theta\right}} \ &=\sqrt{E\left{\left(\frac{\partial} {\partial \phi} \log f(y \mid \phi)\right)^{2} \mid \phi\right}} \ &=\sqrt{I(\phi)} 。\end{对齐}
因此，Jeffreys 的规则是“在重新参数化下不变”，在某种意义上说，如果先验是根据

F(θ)∝我(θ),
然后，对于另一个参数φ=G(θ), 这也是真的

F(φ)∝我(φ).

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian decision theory

后验均值、众数和中位数，以及其他贝叶斯点估计，都可以使用决策理论的原理和理论来推导和解释。假设我们希望选择一个估计θ这在某种意义上可以最大限度地降低成本。为此，让大号(θ^,θ)通常表示与估计相关的损失函数（LF）θ^.

注：估计器θ^是数据的函数是所以也可以写成θ^(是). 例如，在上下文中(是∣θ)∼垃圾桶⁡(n,θ), 样本比例或 MLE 是由下式给出的函数θ^=θ^(是)=是/n.
损失函数大号表示当真实值发生时所发生的成本θ估计为θ^并且通常满足属性大号(θ,θ)=0.
三个最常用的损失函数定义如下：大号(θ^,θ)=|θ^−θ|绝对误差损失函数 (AELF)大号(θ^,θ)=(θ^−θ)2 二次误差损失函数 (QELF) L(\hat{\theta}, \theta)=I(\hat{\theta} \ne \theta)=\left{\begin{array}{ll}0 & \text { if } \hat{\theta }=\theta\1 & \text { if }\that{\theta}\q\theta\end{array}\right}\quadL(\hat{\theta}, \theta)=I(\hat{\theta} \ne \theta)=\left{\begin{array}{ll}0 & \text { if } \hat{\theta }=\theta\1 & \text { if }\that{\theta}\q\theta\end{array}\right}\quad指标误差损失函数（IELF），也称为零一损失函数（ZOLF）或全有或全无误差损失函数（ANLF）。
数字2.8和2.9说明这三个基本损失函数。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|贝叶斯分析代写Bayesian Analysis代考|Credibility estimates

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写贝叶斯分析Bayesian Analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|Credibility estimates

In actuarial studies, a credibility estimate is one which can be expressed as a weighted average of the form
$$
C=(1-k) A+k B \text {, }
$$
where:
$\begin{array}{ll}A & \text { is the subjective estimate (or the collateral data estimate) } \ B & \text { is the objective estimate (or the direct data estimate) } \ k \quad \text { is the credibility factor, a number that is between } 0 \text { and } 1 \ & \text { (inclusive) and represents the weight assigned to the } \ \text { objective estimate. }\end{array}$
A high value of $k$ implies $C \cong B$, representing a situation where the objective estimate is assigned ‘high credibility’. A primary aim of credibility theory is to determine an appropriate value or formula for $k$, as is done, for example, in the theory of the Bühlmann model (Bühlmann, 1967). Many Bayesian models lead to a point estimate which can be expressed as an intuitively appealing credibility estimate.

Earlier we showed that
$$
(\theta \mid y) \sim \operatorname{Beta}(\alpha+y, \beta+n-y),
$$
and hence that the posterior mean of $\theta$ is
$$
\hat{\theta}=E(\theta \mid y)=\frac{(\alpha+y)}{(\alpha+y)+(\beta+n-y)}=\frac{\alpha+y}{\alpha+\beta+n}
$$
Observe that the prior mean of $\theta$ is $E \theta=\alpha /(\alpha+\beta)$, and the maximum likelihood estimate (MLE) of $\theta$ is $y / n$. This suggests that we write
$$
\begin{aligned}
\hat{\theta} &=\frac{\alpha}{\alpha+\beta+n}+\frac{y}{\alpha+\beta+n} \
&=\frac{\alpha}{\alpha+\beta+n}\left(\frac{\alpha+\beta}{\alpha}\right)\left(\frac{\alpha}{\alpha+\beta}\right)+\frac{n}{\alpha+\beta+n}\left(\frac{y}{n}\right) \
&=\frac{\alpha+\beta}{\alpha+\beta+n}\left(\frac{\alpha}{\alpha+\beta}\right)+\frac{n}{\alpha+\beta+n}\left(\frac{y}{n}\right)
\end{aligned}
$$
Thus $\hat{\theta}=(1-k) A+k B$
where: $\quad A=\frac{\alpha}{\alpha+\beta}, \quad B=\frac{y}{n}, \quad k=\frac{n}{\alpha+\beta+n}$.
We see that the posterior mean $\hat{\theta}$ is a credibility estimate in the form of a weighted average of the prior mean $A=E \theta=\alpha /(\alpha+\beta)$ and the MLE $B=y / n$, where the weight assigned to the MLE is the credibility factor given by $k=n /(n+\alpha+\beta)$. Observe that as $n$ increases, the credibility factor $k$ approaches 1 . This makes sense: if there is a lot of data then the prior should not have much influence on the estimation.

统计代写|贝叶斯分析代写Bayesian Analysis代考|Frequentist characteristics of Bayesian estimators

Consider a Bayesian model defined by a likelihood $f(y \mid \theta)$ and a prior $f(\theta)$, leading to the posterior
$$
f(\theta \mid y)=\frac{f(\theta) f(y \mid \theta)}{f(y)}
$$
Suppose that we choose to perform inference on $\theta$ by constructing a point estimate $\hat{\theta}$ (such as the posterior mean, mode or median) and a $(1-\alpha)$-level interval estimate $I=(L, U)$ (such as the CPDR or HPDR).
Then $\hat{\theta}, I, L$ and $U$ are functions of the data $y$ and may be written $\hat{\theta}(y)$, $I(y), L(y)$ and $U(y)$. Once these functions are defined, the estimates which they define stand on their own, so to speak, and may be studied from many different perspectives.

Naturally, the characteristics of these estimates may be seen in the context of the Bayesian framework in which they were constructed. More will be said on this below when we come to discuss Bayesian decision theory.

However, another important use of Bayesian estimates is as a proxy for classical estimates. We have already mentioned this in relation to the normal-normal model:
$$
\begin{aligned}
&\left(y_{1}, \ldots, y_{n} \mid \mu\right) \sim \text { iid } N\left(\mu, \sigma^{2}\right) \
&\mu \sim N\left(\mu_{0}, \sigma_{0}^{2}\right)
\end{aligned}
$$
where the use of a particular prior, namely the one specified by $\sigma_{0}=\infty$, led to the point estimate $\hat{\mu}=\hat{\mu}(y)=\bar{y}$ and the interval estimate
$$
I(y)=(L(y), U(y))=\left(\bar{y} \pm z_{\alpha / 2} \sigma / \sqrt{n}\right) .
$$
As we noted earlier, these estimates are exactly the same as the usual estimates used in the context of the corresponding classical model, $$
y_{1}, \ldots, y_{n} \sim i i d N\left(\mu, \sigma^{2}\right),
$$
where $\mu$ is an unknown constant and $\sigma^{2}$ is given.

统计代写|贝叶斯分析代写Bayesian Analysis代考|Mixture prior distributions

So far we have considered Bayesian models with priors that are limited in the types of prior information that they can represent. For example, the normal-normal model does not allow a prior for the normal mean which has two or more modes. If a non-normal class of prior is used to represent one’s complicated prior beliefs regarding the normal mean, then that prior will not be conjugate, and this will lead to difficulties down the track when making inferences based on the nonstandard posterior distribution.

Fortunately, this problem can be addressed in any Bayesian model for which a conjugate class of prior exists by specifying the prior as a mixture of members of that class.

Generally, a random variable $X$ with a mixture distribution has a density of the form
$$
f(x)=\sum_{m=1}^{M} c_{m} f_{m}(x),
$$
where each $f_{m}(x)$ is a proper density and the $c_{m}$ values are positive and sum to 1 .

If our prior beliefs regarding a parameter $\theta$ do not follow any single well-known distribution, those beliefs can in that case be conveniently approximated to any degree of precision by a suitable mixture prior distribution with a density having the form
$$
f(\theta)=\sum_{m=1}^{M} c_{m} f_{m}(\theta) .
$$
It can be shown (see Exercise $2.3$ below) that if each component prior $f_{m}(\theta)$ is conjugate then $f(\theta)$ is also conjugate. This means that $\theta$ ‘s posterior distribution is also a mixture with density of the form
$$
f(\theta \mid y)=\sum_{m=1}^{M} c_{m}^{\prime} f_{m}(\theta \mid y),
$$
where $f_{m}(\theta \mid y)$ is the posterior implied by the $m$ th prior $f_{m}(\theta)$ and is from the same family of distributions as that prior.

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|Credibility estimates

在精算研究中，可信度估计是一种可以表示为形式的加权平均值的估计

C=(1−ķ)一个+ķ乙,
在哪里：
一个是主观估计（或抵押数据估计）乙是客观估计（或直接数据估计） ķ 是可信度因子，介于 0 和 1 （含）并表示分配给客观估计。
高价值ķ暗示C≅乙，表示客观估计被赋予“高可信度”的情况。可信度理论的主要目的是确定一个适当的值或公式ķ，例如，在 Bühlmann 模型的理论中所做的 (Bühlmann, 1967)。许多贝叶斯模型导致点估计，可以表示为直观吸引人的可信度估计。

早些时候我们展示了

(θ∣是)∼贝塔⁡(一个+是,b+n−是),
因此，后验均值θ是

θ^=和(θ∣是)=(一个+是)(一个+是)+(b+n−是)=一个+是一个+b+n
观察到先验均值θ是和θ=一个/(一个+b), 最大似然估计 (MLE)θ是是/n. 这表明我们写

θ^=一个一个+b+n+是一个+b+n =一个一个+b+n(一个+b一个)(一个一个+b)+n一个+b+n(是n) =一个+b一个+b+n(一个一个+b)+n一个+b+n(是n)
因此θ^=(1−ķ)一个+ķ乙
在哪里：一个=一个一个+b,乙=是n,ķ=n一个+b+n.
我们看到后验均值θ^是先前均值的加权平均形式的可信度估计一个=和θ=一个/(一个+b)和 MLE乙=是/n，其中分配给 MLE 的权重是由下式给出的可信度因子ķ=n/(n+一个+b). 观察到n增加，可信度因素ķ接近 1 . 这是有道理的：如果有很多数据，那么先验应该不会对估计产生太大影响。

统计代写|贝叶斯分析代写Bayesian Analysis代考|Frequentist characteristics of Bayesian estimators

考虑一个由可能性定义的贝叶斯模型F(是∣θ)和之前的F(θ), 导致后

F(θ∣是)=F(θ)F(是∣θ)F(是)
假设我们选择对θ通过构建点估计θ^（例如后验均值、众数或中位数）和(1−一个)水平区间估计我=(大号,在)（例如 CPDR 或 HPDR）。
然后θ^,我,大号和在是数据的函数是并且可以写成θ^(是), 我(是),大号(是)和在(是). 一旦定义了这些函数，它们定义的估计就可以说是独立存在的，并且可以从许多不同的角度进行研究。

自然地，这些估计的特征可以在构建它们的贝叶斯框架的背景下看到。当我们讨论贝叶斯决策理论时，下面将对此进行更多说明。

然而，贝叶斯估计的另一个重要用途是作为经典估计的代理。我们已经在法线-法线模型中提到了这一点：

(是1,…,是n∣μ)∼ 独立同居 ñ(μ,σ2) μ∼ñ(μ0,σ02)
其中使用特定的先验，即由σ0=∞，导致点估计μ^=μ^(是)=是¯和区间估计

我(是)=(大号(是),在(是))=(是¯±和一个/2σ/n).
正如我们之前提到的，这些估计与在相应经典模型的上下文中使用的通常估计完全相同，

是1,…,是n∼一世一世dñ(μ,σ2),
在哪里μ是一个未知常数并且σ2给出。

统计代写|贝叶斯分析代写Bayesian Analysis代考|Mixture prior distributions

到目前为止，我们已经考虑了具有先验的贝叶斯模型，这些模型在它们可以表示的先验信息类型方面受到限制。例如，正态-正态模型不允许具有两个或多个模态的正态平均值的先验。如果使用非正态先验类别来表示一个人对正态均值的复杂先验信念，则该先验将不是共轭的，这将导致在基于非标准后验分布进行推断时遇到困难。

幸运的是，这个问题可以在任何存在共轭先验类的贝叶斯模型中解决，方法是将先验指定为该类成员的混合。

一般来说，随机变量X具有混合分布的密度为

F(X)=∑米=1米C米F米(X),
其中每个F米(X)是一个适当的密度和C米值为正且总和为 1 。

如果我们对参数的先前信念θ不遵循任何单一的众所周知的分布，在那种情况下，这些信念可以方便地通过适当的混合先验分布以任何精度近似，其密度具有以下形式

F(θ)=∑米=1米C米F米(θ).
它可以显示（见练习2.3下面），如果每个组件之前F米(θ)那么是共轭的F(θ)也是共轭的。这意味着θ的后验分布也是与密度的混合形式

F(θ∣是)=∑米=1米C米′F米(θ∣是),
在哪里F米(θ∣是)是由米先验F米(θ)并且来自与先前相同的分布系列。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian point estimation

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写贝叶斯分析Bayesian Analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian point estimation

Once the posterior distribution or density $f(\theta \mid y)$ has been obtained, Bayesian point estimates of the model parameter $\theta$ can be calculated. The three most commonly used point estimates are as follows.

The posterior mean of $\theta$ is
$$
E(\theta \mid y)=\int \theta d F(\theta \mid y)= \begin{cases}\int \theta f(\theta \mid y) d \theta & \text { if } \theta \text { is continuous } \ \sum_{\theta} \theta f(\theta \mid y) & \text { if } \theta \text { is discrete. }\end{cases}
$$
The posterior mode of $\theta$ is
$\operatorname{Mode}(\theta \mid y)=$ any value $m \in \mathfrak{R}$ which satisfies
$$
f(\theta=m \mid x)=\max {\theta} f(\theta \mid x) $$ or $\lim {\theta \rightarrow m} f(\theta \mid x)=\sup f(\theta \mid x)$,
or the set of all such values.
The posterior median of $\theta$ is
$\operatorname{Median}(\theta \mid y)=$ any value $m$ of $\theta$ such that
$$
\begin{array}{r}
P(\theta \leq m \mid y) \geq 1 / 2 \
\text { and } P(\theta \geq m \mid y) \geq 1 / 2
\end{array}
$$
or the set of all such values.

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian interval estimation

There are many ways to construct a Bayesian interval estimate, but the two most common ways are defined as follows. The $1-\alpha$ (or $100(1-\alpha) \%$ ) highest posterior density region (HPDR) for $\theta$ is the smallest set $S$ such that:
$$
P(\theta \in S \mid y) \geq 1-\alpha
$$
and $f\left(\theta_{1} \mid y\right) \geq f\left(\theta_{2} \mid y\right)$ if $\theta_{1} \in S$ and $\theta_{2} \notin S$.
Figure $1.6$ illustrates the idea of the HPDR. In the very common situation where $\theta$ is scalar, continuous and has a posterior density which is unimodal with no local modes (i.e. has the form of a single ‘mound’), the 1- $\alpha$ HPDR takes on the form of a single interval defined by two points at which the posterior density has the same value. When the HPDR is a single interval, it is the shortest possible single interval over which the area under the posterior density is $1-\alpha$.

The $1-\alpha$ central posterior density region (CPDR) for a scalar parameter $\theta$ may be defined as the shortest single interval $[a, b]$ such that:
$P(\thetab \mid y) \leq \alpha / 2$.

In the common case of a continuous parameter with a posterior density in the form of a single ‘mound’ which is furthermore symmetric, the CPDR and HPDR are identical.

Note 1: The 1- $\alpha$ CPDR for $\theta$ may alternatively be defined as the shortest single open interval $(a, b)$ such that:
$$
P(\theta \leq a \mid y) \leq \alpha / 2
$$
and $P(\theta \geq b \mid y) \leq \alpha / 2$.
Other variations are possible (of the form $[a, b)$ and $(a, b])$; but when the parameter of interest $\theta$ is continuous these definitions are all equivalent. Yet another definition of the 1- $\alpha$ CPDR is any of the CPDRs as defined above but with all a posteriori impossible values of $\theta$ excluded.

Note 2: As regards terminology, whenever the HPDR is a single interval, it may also be called the highest posterior density interval (HPDI). Likewise, the CPDR, which is always a single interval, may also be called the central posterior density interval (CPDI).

统计代写|贝叶斯分析代写Bayesian Analysis代考|Inference on functions of the model parameter

So far we have examined Bayesian models with a single parameter $\theta$ and described how to perform posterior inference on that parameter. Sometimes there may also be interest in some function of the model parameter, denoted by (say)
$$
\psi=g(\theta)
$$
Then the posterior density of $\psi$ can be derived using distribution theory, for example by applying the transformation rule,
$$
f(\psi \mid y)=f(\theta \mid y)\left|\frac{d \theta}{d \psi}\right|
$$
in cases where $\psi=g(\theta)$ is strictly increasing or strictly decreasing.
Point and interval estimates of $\psi$ can then be calculated in the usual way, using $f(\psi \mid y)$. For example, the posterior mean of $\psi$ equals
$$
E(\psi \mid y)=\int \psi f(\psi \mid y) d \psi \text {. }
$$
Sometimes it is more practical to calculate point and interval estimates another way, without first deriving $f(\psi \mid y)$.
For example, another expression for the posterior mean is
$$
E(\psi \mid y)=E(g(\theta) \mid y)=\int g(\theta) f(\theta \mid y) d \theta .
$$

Also, the posterior median of $\psi$, call this $M$, can typically be obtained by simply calculating
$$
M=g(m) \text {, }
$$
where $m$ is the posterior median of $\theta$.
Note: To see why this works, we write
$$
\begin{aligned}
P(\psi<M \mid y) &=P(g(\theta)<M \mid y) \
&=P(g(\theta)<g(m) \mid y)=P(\theta<m \mid y)=1 / 2 .
\end{aligned}
$$

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian point estimation

一旦后验分布或密度F(θ∣是)已获得模型参数的贝叶斯点估计θ可以计算。三个最常用的点估计如下。

的后验均值θ是
和(θ∣是)=∫θdF(θ∣是)={∫θF(θ∣是)dθ 如果 θ 是连续的 ∑θθF(θ∣是) 如果 θ 是离散的。
后模态θ是
模式⁡(θ∣是)=任何值米∈R满足
F(θ=米∣X)=最大限度θF(θ∣X)或者林θ→米F(θ∣X)=支持F(θ∣X)，
或所有此类值的集合。
后中位数θ是
中位数⁡(θ∣是)=任何值米的θ这样
磷(θ≤米∣是)≥1/2 和磷(θ≥米∣是)≥1/2
或所有这些值的集合。

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian interval estimation

构造贝叶斯区间估计的方法有很多，但最常见的两种方法定义如下。这1−一个（或者100(1−一个)%) 最高后验密度区域 (HPDR)θ是最小的集合小号这样：

磷(θ∈小号∣是)≥1−一个
和F(θ1∣是)≥F(θ2∣是)如果θ1∈小号和θ2∉小号.
数字1.6说明了 HPDR 的想法。在非常常见的情况下θ是标量的，连续的并且具有单峰的后密度，没有局部模式（即具有单个“土堆”的形式），1-一个HPDR 采用由后验密度具有相同值的两个点定义的单个区间的形式。当 HPDR 为单个区间时，它是后验密度下面积为的最短可能单个区间1−一个.

这1−一个标量参数的中心后密度区域 (CPDR)θ可以定义为最短的单个区间[一个,b]这样：
$P(\theta b \mid y) \leq \alpha / 2$。

在具有单个“土丘”形式的后验密度的连续参数的常见情况下，并且对称，CPDR 和 HPDR 是相同的。

注1：1-一个CPDR 为θ也可以定义为最短的单个开区间(一个,b)这样：

磷(θ≤一个∣是)≤一个/2
和磷(θ≥b∣是)≤一个/2.
其他变化是可能的（形式[一个,b)和(一个,b]); 但是当感兴趣的参数θ是连续的，这些定义都是等价的。1的另一个定义-一个CPDR 是上面定义的任何 CPDR，但具有所有后验不可能的值θ排除。

注 2：关于术语，只要 HPDR 是单个区间，它也可以称为最高后验密度区间 (HPDI)。同样，始终为单个区间的 CPDR 也可称为中心后验密度区间 (CPDI)。

统计代写|贝叶斯分析代写Bayesian Analysis代考|Inference on functions of the model parameter

到目前为止，我们已经检查了具有单个参数的贝叶斯模型θ并描述了如何对该参数进行后验推断。有时也可能对模型参数的某些函数感兴趣，用 (say) 表示

ψ=G(θ)
那么后验密度ψ可以使用分布理论推导出来，例如通过应用变换规则，

F(ψ∣是)=F(θ∣是)|dθdψ|
在这种情况下ψ=G(θ)是严格增加或严格减少。
点和区间估计ψ然后可以以通常的方式计算，使用F(ψ∣是). 例如，后验均值ψ等于

和(ψ∣是)=∫ψF(ψ∣是)dψ.
有时以另一种方式计算点和区间估计更实际，无需先推导F(ψ∣是).
例如，后验均值的另一个表达式是

和(ψ∣是)=和(G(θ)∣是)=∫G(θ)F(θ∣是)dθ.

此外，后中位数ψ, 称之为米, 通常可以通过简单的计算得到

米=G(米),
在哪里米是的后中位数θ.
注意：要了解为什么会这样，我们写

磷(ψ<米∣是)=磷(G(θ)<米∣是) =磷(G(θ)<G(米)∣是)=磷(θ<米∣是)=1/2.

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金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|贝叶斯分析代写Bayesian Analysis代考|Finite and infinite population inference

Posted on 2022年6月21日2022年6月21日 by statistics-lab

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我们提供的贝叶斯分析Bayesian Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|Finite and infinite population inference

In the last example (Exercise 1.8), with the model:
$$
(y \mid \theta) \sim \operatorname{Binomial}(n, \theta)
$$
$$
\theta \sim \operatorname{Beta}(\alpha, \beta),
$$
the quantity of interest $\theta$ is the probability of success on a single Bernoulli trial.

This quantity may be thought of as the average of a hypothetically infinite number of Bernoulli trials. For that reason we may refer to derivation of the posterior distribution,
$$
(\theta \mid y) \sim \operatorname{Beta}(\alpha+y, \beta+n-y),
$$
as infinite population inference.
In contrast, for the ‘buses’ example further above (Exercise 1.6), which involves the model:
$$
\begin{aligned}
&f(y \mid \theta)=1 / \theta, y=1, \ldots, \theta \
&f(\theta)=1 / 5, \theta=1, \ldots, 5
\end{aligned}
$$
the quantity of interest $\theta$ represents the number of buses in a population of buses, which of course is finite.
Therefore derivation of the posterior,
$$
f(\theta \mid y)=\left{\begin{array}{l}
20 / 47, \theta=3 \
15 / 47, \theta=4 \
12 / 47, \theta=5
\end{array}\right.
$$
may be termed finite population inference.
Another example of finite population inference is the ‘balls in a box’ example (Exercise 1.7), where the model is:
$$
\begin{aligned}
&(y \mid \theta) \sim \operatorname{Hyp}(N, \theta, n) \
&\theta \sim D U(1, \ldots, N)
\end{aligned}
$$
and where the quantity of interest $\theta$ is the number of red balls initially in the selected box $(1,2, \ldots, 8$ or 9$)$.

And another example of infinite population inference is the ‘loaded dice’ example (Exercises $1.4$ and $1.5$ ), where the model is:
$$
\begin{aligned}
&f(y \mid \theta)=\left(\begin{array}{c}
2 \
y
\end{array}\right) \theta^{y}(1-\theta)^{2-y}, y=0,1,2 \
&f(\theta)=10 \theta / 6, \theta=0.1,0.2,0.3
\end{aligned}
$$ and where the quantity of interest $\theta$ is the probability of 6 coming up on a single roll of the chosen die (i.e. the average number of $6 s$ that come up on a hypothetically infinite number of rolls of that particular die).

统计代写|贝叶斯分析代写Bayesian Analysis代考|Continuous data

So far, all the Bayesian models considered have featured data which is modelled using a discrete distribution. (Some of these models have a discrete parameter and some have a continuous parameter.) The following is an example with data that follows a continuous probability distribution. (This example also has a continuous parameter.)
Exercise 1.9 The exponential-exponential model
Suppose $\theta$ has the standard exponential distribution, and the conditional distribution of $y$ given $\theta$ is exponential with mean $1 / \theta$. Find the posterior density of $\theta$ given $y$.
Solution to Exercise $1.9$
The Bayesian model here is: $f(y \mid \theta)=\theta e^{-\theta y}, y>0$
$$
f(\theta)=e^{-\theta}, \theta>0 .
$$
So $f(\theta \mid y) \propto f(\theta) f(y \mid \theta) \propto e^{-\theta} \times \theta e^{-\theta y}=\theta^{2-1} e^{-\theta(y+1)}, y>0$.
This is the kernel of a gamma distribution with parameters 2 and $y+1$, as per the definitions in Appendix B.2. Thus we may write
$$
(\theta \mid y) \sim \operatorname{Gamma}(2, y+1),
$$
from which it follows that the posterior density of $\theta$ is
$$
f(\theta \mid y)=\frac{(y+1)^{2} \theta^{2-1} e^{-\theta(y+1)}}{\Gamma(2)}, \theta>0 .
$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|Conjugacy

When the prior and posterior distributions are members of the same class of distributions, we say that they form a conjugate pair, or that the prior is conjugate. For example, consider the binomial-beta model:
$$
\begin{array}{rll}
& (y \mid \theta) \sim \operatorname{Binomial}(n, \theta) & \
& \theta \sim \operatorname{Beta}(\alpha, \beta) & \text { (prior) } \
\Rightarrow & (\theta \mid y) \sim \operatorname{Beta}(\alpha+y, \beta+n-y) & \text { (posterior). }
\end{array}
$$
Since both prior and posterior are beta, the prior is conjugate.
Likewise, consider the exponential-exponential model:
$$
\begin{array}{rll}
& f(y \mid \theta)=\theta e^{-\theta y}, y>0 & \
& \left.f(\theta)=e^{-\theta}, \theta>0 \quad \text { (i.e. } \theta \sim \operatorname{Gamma}(1,1)\right) & \text { (prior) } \
\Rightarrow & (\theta \mid y) \sim \operatorname{Gamma}(2, y+1) \quad \text { (posterior). }
\end{array}
$$

Since both prior and posterior are gamma, the prior is conjugate.
On the other hand, consider the model in the buses example:
$(y \mid \theta) \sim D U(1, \ldots, \theta)$
$\theta \sim D U(1, \ldots, 5)$
$\Rightarrow f(\theta \mid y=3)=\left{\begin{array}{l}20 / 47, \theta=3 \ 15 / 47, \theta=4 \ 12 / 47, \theta=5\end{array}\right.$
The prior is discrete uniform but the posterior is not. So in this case the prior is not conjugate.

Specifying a Bayesian model using a conjugate prior is generally desirable because it can simplify the calculations required.

贝叶斯分析代考

统计代写|贝叶斯分析代写Bayesian Analysis代考|Finite and infinite population inference

在最后一个示例（练习 1.8）中，使用模型：

(是∣θ)∼二项式⁡(n,θ)

θ∼贝塔⁡(一个,b),
感兴趣的数量θ是单次伯努利试验的成功概率。

这个量可以被认为是假设无限次伯努利试验的平均值。出于这个原因，我们可以参考后验分布的推导，

(θ∣是)∼贝塔⁡(一个+是,b+n−是),
作为无限人口推断。
相反，对于上面的“公共汽车”示例（练习 1.6），它涉及模型：

F(是∣θ)=1/θ,是=1,…,θ F(θ)=1/5,θ=1,…,5
感兴趣的数量θ表示公共汽车中的公共汽车数量，这当然是有限的。
因此推导后验，
$$
f(\theta \mid y)=\left{

20/47,θ=3 15/47,θ=4 12/47,θ=5\正确的。

米一个是b和吨和r米和dF一世n一世吨和p○p在l一个吨一世○n一世nF和r和nC和.一个n○吨H和r和X一个米pl和○FF一世n一世吨和p○p在l一个吨一世○n一世nF和r和nC和一世s吨H和‘b一个lls一世n一个b○X′和X一个米pl和(和X和rC一世s和1.7),在H和r和吨H和米○d和l一世s:

(是∣θ)∼炒作⁡(ñ,θ,n) θ∼D在(1,…,ñ)
$$
和感兴趣的数量θ是最初在选定框中的红球数(1,2,…,8或 9).

无限人口推断的另一个例子是“加载骰子”的例子（练习1.4和1.5)，其中模型为：

F(是∣θ)=(2 是)θ是(1−θ)2−是,是=0,1,2 F(θ)=10θ/6,θ=0.1,0.2,0.3以及感兴趣的数量θ是 6 出现在所选骰子的单个掷骰上的概率（即平均数量6s假设该特定骰子的掷骰数是无限的）。

统计代写|贝叶斯分析代写Bayesian Analysis代考|Continuous data

到目前为止，所有考虑的贝叶斯模型都具有使用离散分布建模的特征数据。（其中一些模型具有离散参数，而另一些具有连续参数。）以下是一个示例，其中的数据遵循连续概率分布。（这个例子也有一个连续参数。）
练习 1.9 指数-指数模型
假设θ具有标准指数分布和条件分布是给定θ是指数的平均值1/θ. 求后验密度θ给定是.
运动解决方案1.9
这里的贝叶斯模型是：F(是∣θ)=θ和−θ是,是>0

F(θ)=和−θ,θ>0.
所以F(θ∣是)∝F(θ)F(是∣θ)∝和−θ×θ和−θ是=θ2−1和−θ(是+1),是>0.
这是具有参数 2 和是+1，根据附录 B.2 中的定义。因此我们可以写

(θ∣是)∼伽玛⁡(2,是+1),
由此得出后验密度θ是

F(θ∣是)=(是+1)2θ2−1和−θ(是+1)Γ(2),θ>0.

统计代写|贝叶斯分析代写Bayesian Analysis代考|Conjugacy

当先验分布和后验分布是同一类分布的成员时，我们说它们形成了共轭对，或者说先验是共轭的。例如，考虑二项式 beta 模型：

(是∣θ)∼二项式⁡(n,θ) θ∼贝塔⁡(一个,b) （事先的） ⇒(θ∣是)∼贝塔⁡(一个+是,b+n−是) （后）。
由于先验和后验都是 beta，因此先验是共轭的。
同样，考虑指数-指数模型：

F(是∣θ)=θ和−θ是,是>0 F(θ)=和−θ,θ>0 （IE θ∼伽玛⁡(1,1)) （事先的） ⇒(θ∣是)∼伽玛⁡(2,是+1) （后）。

由于先验和后验都是伽马，因此先验是共轭的。
另一方面，考虑总线示例中的模型：
(是∣θ)∼D在(1,…,θ)
θ∼D在(1,…,5)
$\Rightarrow f(\theta \mid y=3)=\left{

20/47,θ=3 15/47,θ=4 12/47,θ=5\right.$
先验是离散均匀的，但后验不是。所以在这种情况下，先验不是共轭的。

通常需要使用共轭先验指定贝叶斯模型，因为它可以简化所需的计算。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写