## 统计代写|贝叶斯分析代写Bayesian Analysis代考|STAT365

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。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Discrete probability examples: genetics and spell checking

We next demonstrate Bayes’ theorem with two examples in which the immediate goal is inference about a particular discrete quantity rather than with the estimation of a parameter that describes an entire population. These discrete examples allow us to see the prior, likelihood, and posterior probabilities directly.
Human males have one X-chromosome and one Y-chromosome, whereas females have two $\mathrm{X}$-chromosomes, each chromosome being inherited from one parent. Hemophilia is a disease that exhibits X-chromosome-linked recessive inheritance, meaning that a male who inherits the gene that causes the disease on the $\mathrm{X}$-chromosome is affected, whereas a female carrying the gene on only one of her two X-chromosomes is not affected. The disease is generally fatal for women who inherit two such genes, and this is rare, since the frequency of occurrence of the gene is low in human populations.

Prior distribution. Consider a woman who has an affected brother, which implies that her mother must be a carrier of the hemophilia gene with one ‘good’ and one ‘bad’ hemophilia gene. We are also told that her father is not affected; thus the woman herself has a fifty-fifty chance of having the gene. The unknown quantity of interest, the state of the woman, has just two values: the woman is either a carrier of the gene $(\theta=1)$ or not $(\theta=0)$. Based on the information provided thus far, the prior distribution for the unknown $\theta$ can be expressed simply as $\operatorname{Pr}(\theta=1)=\operatorname{Pr}(\theta=0)=\frac{1}{2}$.

Data model and likelihood. The data used to update the prior information consist of the affection status of the woman’s sons. Suppose she has two sons, neither of whom is affected. Let $y_i=1$ or 0 denote an affected or unaffected son, respectively. The outcomes of the two sons are exchangeable and, conditional on the unknown $\theta$, are independent; we assume the sons are not identical twins. The two items of independent data generate the following likelihood function:
\begin{aligned} & \operatorname{Pr}\left(y_1=0, y_2=0 \mid \theta=1\right)=(0.5)(0.5)=0.25 \ & \operatorname{Pr}\left(y_1=0, y_2=0 \mid \theta=0\right)=(1)(1)=1 . \end{aligned}

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Spelling correction

Classification of words is a problem of managing uncertainty. For example, suppose someone types ‘radom.’ How should that be read? It could be a misspelling or mistyping of ‘random’ or ‘radon’ or some other alternative, or it could be the intentional typing of ‘radom’ (as in its first use in this paragraph). What is the probability that ‘radom’ actually means random? If we label $y$ as the data and $\theta$ as the word that the person was intending to type, then
$$\operatorname{Pr}(\theta \mid y=\text { ‘radom’ }) \propto p(\theta) \operatorname{Pr}(y=\text { ‘radom’ } \mid \theta) .$$
This product is the unnormalized posterior density. In this case, if for simplicity we consider only three possibilities for the intended word, $\theta$ (random, radon, or radom), we can compute the posterior probability of interest by first computing the unnormalized density for all three values of theta and then normalizing:
$$p(\text { random } \mid \text { ‘radom’ })=\frac{p\left(\theta_1\right) p\left(\text { ‘radom’ } \mid \theta_1\right)}{\sum_{j=1}^3 p\left(\theta_j\right) p\left(\text { ‘radom’ } \mid \theta_j\right)},$$

where $\theta_1=$ random, $\theta_2=$ radon, and $\theta_3=$ radom. The prior probabilities $p\left(\theta_j\right)$ can most simply come from frequencies of these words in some large database, ideally one that is adapted to the problem at hand (for example, a database of recent student emails if the word in question is appearing in such a document). The likelihoods $p\left(y \mid \theta_j\right)$ can come from some modeling of spelling and typing errors, perhaps fit using some study in which people were followed up after writing emails to identify any questionable words.

Prior distribution. Without any other context, it makes sense to assign the prior probabilities $p\left(\theta_j\right)$ based on the relative frequencies of these three words in some databases. Here are probabilities supplied by researchers at Google:
\begin{tabular}{lc}
\multicolumn{1}{c}{$\theta$} & $p(\theta)$ \
\hline random & $7.60 \times 10^{-5}$ \
radon & $6.05 \times 10^{-6}$ \
radom & $3.12 \times 10^{-7}$
\end{tabular}
Since we are considering only these possibilities, we could renormalize the three numbers to sum to $1\left(p(\right.$ random $)=\frac{760}{760+60.5+3.12}$, etc. $)$ but there is no need, as the adjustment would merely be absorbed into the proportionality constant in (1.6).

# 贝叶斯分析代考

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Discrete probability examples: genetics and spell checking

\begin{aligned} & \operatorname{Pr}\left(y_1=0, y_2=0 \mid \theta=1\right)=(0.5)(0.5)=0.25 \ & \operatorname{Pr}\left(y_1=0, y_2=0 \mid \theta=0\right)=(1)(1)=1 . \end{aligned}

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Spelling correction

$$\operatorname{Pr}(\theta \mid y=\text { ‘radom’ }) \propto p(\theta) \operatorname{Pr}(y=\text { ‘radom’ } \mid \theta) .$$

$$p(\text { random } \mid \text { ‘radom’ })=\frac{p\left(\theta_1\right) p\left(\text { ‘radom’ } \mid \theta_1\right)}{\sum_{j=1}^3 p\left(\theta_j\right) p\left(\text { ‘radom’ } \mid \theta_j\right)},$$

\begin{tabular}{lc}
\multicolumn{1}{c}{$\theta$} & $p(\theta)$ \hline random & $7.60 \times 10^{-5}$ \radon &$6.05 \times 10^{-6}$ \radom &$3.12 \times 10^{-7}$
\end{tabular}

## 有限元方法代写

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。

## 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。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|PSYC750

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。

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

To make inferences about an unknown observable, often called predictive inferences, we follow a similar logic. Before the data $y$ are considered, the distribution of the unknown but observable $y$ is
$$p(y)=\int p(y, \theta) d \theta=\int p(\theta) p(y \mid \theta) d \theta$$
This is often called the marginal distribution of $y$, but a more informative name is the prior predictive distribution: prior because it is not conditional on a previous observation of the process, and predictive because it is the distribution for a quantity that is observable.
After the data $y$ have been observed, we can predict an unknown observable, $\tilde{y}$, from the same process. For example, $y=\left(y_1, \ldots, y_n\right)$ may be the vector of recorded weights of an object weighed $n$ times on a scale, $\theta=\left(\mu, \sigma^2\right)$ may be the unknown true weight of the object and the measurement variance of the scale, and $\tilde{y}$ may be the yet to be recorded weight of the object in a planned new weighing. The distribution of $\tilde{y}$ is called the posterior predictive distribution, posterior because it is conditional on the observed $y$ and predictive because it is a prediction for an observable $\tilde{y}$ :
\begin{aligned} p(\tilde{y} \mid y) & =\int p(\tilde{y}, \theta \mid y) d \theta \ & =\int p(\tilde{y} \mid \theta, y) p(\theta \mid y) d \theta \ & =\int p(\tilde{y} \mid \theta) p(\theta \mid y) d \theta . \end{aligned}
The second and third lines display the posterior predictive distribution as an average of conditional predictions over the posterior distribution of $\theta$. The last step follows from the assumed conditional independence of $y$ and $\tilde{y}$ given $\theta$.

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

Using Bayes’ rule with a chosen probability model means that the data $y$ affect the posterior inference (1.2) only through $p(y \mid \theta)$, which, when regarded as a function of $\theta$, for fixed $y$, is called the likelihood function. In this way Bayesian inference obeys what is sometimes called the likelihood principle, which states that for a given sample of data, any two probability models $p(y \mid \theta)$ that have the same likelihood function yield the same inference for $\theta$.

The likelihood principle is reasonable, but only within the framework of the model or family of models adopted for a particular analysis. In practice, one can rarely be confident that the chosen model is correct. We shall see in Chapter 6 that sampling distributions (imagining repeated realizations of our data) can play an important role in checking model assumptions. In fact, our view of an applied Bayesian statistician is one who is willing to apply Bayes’ rule under a variety of possible models.
Likelihood and odds ratios
The ratio of the posterior density $p(\theta \mid y)$ evaluated at the points $\theta_1$ and $\theta_2$ under a given model is called the posterior odds for $\theta_1$ compared to $\theta_2$. The most familiar application of this concept is with discrete parameters, with $\theta_2$ taken to be the complement of $\theta_1$. Odds provide an alternative representation of probabilities and have the attractive property that Bayes’ rule takes a particularly simple form when expressed in terms of them:
$$\frac{p\left(\theta_1 \mid y\right)}{p\left(\theta_2 \mid y\right)}=\frac{p\left(\theta_1\right) p\left(y \mid \theta_1\right) / p(y)}{p\left(\theta_2\right) p\left(y \mid \theta_2\right) / p(y)}=\frac{p\left(\theta_1\right)}{p\left(\theta_2\right)} \frac{p\left(y \mid \theta_1\right)}{p\left(y \mid \theta_2\right)} .$$
In words, the posterior odds are equal to the prior odds multiplied by the likelihood ratio, $p\left(y \mid \theta_1\right) / p\left(y \mid \theta_2\right)$.

# 贝叶斯分析代考

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

$$p(y)=\int p(y, \theta) d \theta=\int p(\theta) p(y \mid \theta) d \theta$$

\begin{aligned} p(\tilde{y} \mid y) & =\int p(\tilde{y}, \theta \mid y) d \theta \ & =\int p(\tilde{y} \mid \theta, y) p(\theta \mid y) d \theta \ & =\int p(\tilde{y} \mid \theta) p(\theta \mid y) d \theta . \end{aligned}

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

$$\frac{p\left(\theta_1 \mid y\right)}{p\left(\theta_2 \mid y\right)}=\frac{p\left(\theta_1\right) p\left(y \mid \theta_1\right) / p(y)}{p\left(\theta_2\right) p\left(y \mid \theta_2\right) / p(y)}=\frac{p\left(\theta_1\right)}{p\left(\theta_2\right)} \frac{p\left(y \mid \theta_1\right)}{p\left(y \mid \theta_2\right)} .$$

## 有限元方法代写

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。

## 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。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|CS-E5710

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。

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

Bayesian statistical conclusions about a parameter $\theta$, or unobserved data $\tilde{y}$, are made in terms of probability statements. These probability statements are conditional on the observed value of $y$, and in our notation are written simply as $p(\theta \mid y)$ or $p(\tilde{y} \mid y)$. We also implicitly condition on the known values of any covariates, $x$. It is at the fundamental level of conditioning on observed data that Bayesian inference departs from the approach to statistical inference described in many textbooks, which is based on a retrospective evaluation of the procedure used to estimate $\theta$ (or $\tilde{y}$ ) over the distribution of possible $y$ values conditional on the true unknown value of $\theta$. Despite this difference, it will be seen that in many simple analyses, superficially similar conclusions result from the two approaches to statistical inference. However, analyses obtained using Bayesian methods can be easily extended to more complex problems. In this section, we present the basic mathematics and notation of Bayesian inference, followed in the next section by an example from genetics.
Probability notation
Some comments on notation are needed at this point. First, $p(\cdot \mid \cdot)$ denotes a conditional probability density with the arguments determined by the context, and similarly for $p(\cdot)$, which denotes a marginal distribution. We use the terms ‘distribution’ and ‘density’ interchangeably. The same notation is used for continuous density functions and discrete probability mass functions. Different distributions in the same equation (or expression) will each be denoted by $p(\cdot)$, as in (1.1) below, for example. Although an abuse of standard mathematical notation, this method is compact and similar to the standard practice of using $p(\cdot)$ for the probability of any discrete event, where the sample space is also suppressed in the notation. Depending on context, to avoid confusion, we may use the notation $\operatorname{Pr}(\cdot)$ for the probability of an event; for example, $\operatorname{Pr}(\theta>2)=\int_{\theta>2} p(\theta) d \theta$. When using a standard distribution, we use a notation based on the name of the distribution; for example, if $\theta$ has a normal distribution with mean $\mu$ and variance $\sigma^2$, we write $\theta \sim \mathrm{N}\left(\mu, \sigma^2\right)$ or $p(\theta)=\mathrm{N}\left(\theta \mid \mu, \sigma^2\right)$ or, to be even more explicit, $p\left(\theta \mid \mu, \sigma^2\right)=\mathrm{N}\left(\theta \mid \mu, \sigma^2\right)$. Throughout, we use notation such as $\mathrm{N}\left(\mu, \sigma^2\right)$ for random variables and $\mathrm{N}\left(\theta \mid \mu, \sigma^2\right)$ for density functions. Notation and formulas for several standard distributions appear in Appendix A.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayes’ rule

In order to make probability statements about $\theta$ given $y$, we must begin with a model providing a joint probability distribution for $\theta$ and $y$. The joint probability mass or density function can be written as a product of two densities that are often referred to as the prior distribution $p(\theta)$ and the sampling distribution (or data distribution) $p(y \mid \theta)$, respectively:
$$p(\theta, y)=p(\theta) p(y \mid \theta) .$$

Simply conditioning on the known value of the data $y$, using the basic property of conditional probability known as Bayes’ rule, yields the posterior density:
$$p(\theta \mid y)=\frac{p(\theta, y)}{p(y)}=\frac{p(\theta) p(y \mid \theta)}{p(y)},$$
where $p(y)=\sum_\theta p(\theta) p(y \mid \theta)$, and the sum is over all possible values of $\theta$ (or $p(y)=$ $\int p(\theta) p(y \mid \theta) d \theta$ in the case of continuous $\theta$ ). An equivalent form of (1.1) omits the factor $p(y)$, which does not depend on $\theta$ and, with fixed $y$, can thus be considered a constant, yielding the unnormalized posterior density, which is the right side of (1.2):
$$p(\theta \mid y) \propto p(\theta) p(y \mid \theta) .$$
The second term in this expression, $p(y \mid \theta)$, is taken here as a function of $\theta$, not of $y$. These simple formulas encapsulate the technical core of Bayesian inference: the primary task of any specific application is to develop the model $p(\theta, y)$ and perform the computations to summarize $p(\theta \mid y)$ in appropriate ways.

# 贝叶斯分析代考

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayes’ rule

$$p(\theta, y)=p(\theta) p(y \mid \theta) .$$

$$p(\theta \mid y)=\frac{p(\theta, y)}{p(y)}=\frac{p(\theta) p(y \mid \theta)}{p(y)},$$

## 有限元方法代写

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。

## 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。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Example: estimating the accuracy of record linkage

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。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Example: estimating the accuracy of record linkage

We emphasize the essentially empirical (not ‘subjective’ or ‘personal’) nature of probabilities with another example in which they are estimated from data.

Record linkage refers to the use of an algorithmic technique to identify records from different databases that correspond to the same individual. Record-linkage techniques are used in a variety of settings. The work described here was formulated and first applied in the context of record linkage between the U.S. Census and a large-scale post-enumeration survey, which is the first step of an extensive matching operation conducted to evaluate census coverage for subgroups of the population. The goal of this first step is to declare as many records as possible ‘matched’ by computer without an excessive rate of error, thereby avoiding the cost of the resulting manual processing for all records not declared ‘matched.’
Existing methods for assigning scores to potential matches
Much attention has been paid in the record-linkage literature to the problem of assigning ‘weights’ to individual fields of information in a multivariate record and obtaining a composite ‘score,’ which we call $y$, that summarizes the closeness of agreement between two records. Here, we assume that this step is complete in the sense that these rules have been chosen. The next step is the assignment of candidate matched pairs, where each pair of records consists of the best potential match for each other from the respective databases. The specified weighting rules then order the candidate matched pairs. In the motivating problem at the Census Bureau, a binary choice is made between the alternatives ‘declare matched’ vs. ‘send to followup,’ where a cutoff score is needed above which records are declared matched. The false-match rate is then defined as the number of falsely matched pairs divided by the number of declared matched pairs.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Estimating match probabilities empirically

We obtain accurate match probabilities using mixture modeling, a topic we discuss in detail in Chapter 22. The distribution of previously obtained scores for the candidate matches is considered a ‘mixture’ of a distribution of scores for true matches and a distribution for non-matches. The parameters of the mixture model are estimated from the data. The estimated parameters allow us to calculate an estimate of the probability of a false match (a pair declared matched that is not a true match) for any given decision threshold on the scores. In the procedure that was actually used, some elements of the mixture model (for example, the optimal transformation required to allow a mixture of normal distributions to apply) were fit using ‘training’ data with known match status (separate from the data to which we apply our calibration procedure), but we do not describe those details here. Instead we focus on how the method would be used with a set of data with unknown match status.

Support for this approach is provided in Figure 1.3, which displays the distribution of scores for the matches and non-matches in a particular dataset obtained from 2300 records from a ‘test Census’ survey conducted in a single local area two years before the 1990 Census. The two distributions, $p(y \mid$ match $)$ and $p(y \mid$ non-match $)$, are mostly distinct-meaning that in most cases it is possible to identify a candidate as a match or not given the score alonebut with some overlap.

In our application dataset, we do not know the match status. Thus we are faced with a single combined histogram from which we estimate the two component distributions and the proportion of the population of scores that belong to each component. Under the mixture model, the distribution of scores can be written as,
$$p(y)=\operatorname{Pr}(\text { match }) p(y \mid \text { match })+\operatorname{Pr}(\text { non-match }) p(y \mid \text { non-match })$$
The mixture probability ( $\operatorname{Pr}($ match $))$ and the parameters of the distributions of matches $(p(y \mid$ match $))$ and non-matches $(p(y \mid$ non-match $))$ are estimated using the mixture model approach (as described in Chapter 22) applied to the combined histogram from the data with unknown match status.

To use the method to make record-linkage decisions, we construct a curve giving the false-match rate as a function of the decision threshold, the score above which pairs will be ‘declared’ a match. For a given decision threshold, the probability distributions in (1.7) can be used to estimate the probability of a false match, a score $y$ above the threshold originating from the distribution $p(y \mid$ non-match $)$. The lower the threshold, the more pairs we will declare as matches. As we declare more matches, the proportion of errors increases. The approach described here should provide an objective error estimate for each threshold. (See the validation in the next paragraph.) Then a decision maker can determine the threshold that provides an acceptable balance between the goals of declaring more matches automatically (thus reducing the clerical labor) and making fewer mistakes.

# 贝叶斯分析代考

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Estimating match probabilities empirically

$$p(y)=\operatorname{Pr}(\text { match }) p(y \mid \text { match })+\operatorname{Pr}(\text { non-match }) p(y \mid \text { non-match })$$

## 有限元方法代写

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。

## 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。

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

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。

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

Bayesian statistical conclusions about a parameter $\theta$, or unobserved data $\tilde{y}$, are made in terms of probability statements. These probability statements are conditional on the observed value of $y$, and in our notation are written simply as $p(\theta \mid y)$ or $p(\tilde{y} \mid y)$. We also implicitly condition on the known values of any covariates, $x$. It is at the fundamental level of conditioning on observed data that Bayesian inference departs from the approach to statistical inference described in many textbooks, which is based on a retrospective evaluation of the procedure used to estimate $\theta$ (or $\tilde{y}$ ) over the distribution of possible $y$ values conditional on the true unknown value of $\theta$. Despite this difference, it will be seen that in many simple analyses, superficially similar conclusions result from the two approaches to statistical inference. However, analyses obtained using Bayesian methods can be easily extended to more complex problems. In this section, we present the basic mathematics and notation of Bayesian inference, followed in the next section by an example from genetics.

Probability notation
Some comments on notation are needed at this point. First, $p(\cdot \mid \cdot)$ denotes a conditional probability density with the arguments determined by the context, and similarly for $p(\cdot)$, which denotes a marginal distribution. We use the terms ‘distribution’ and ‘density’ interchangeably. The same notation is used for continuous density functions and discrete probability mass functions. Different distributions in the same equation (or expression) will each be denoted by $p(\cdot)$, as in (1.1) below, for example. Although an abuse of standard mathematical notation, this method is compact and similar to the standard practice of using $p(\cdot)$ for the probability of any discrete event, where the sample space is also suppressed in the notation. Depending on context, to avoid confusion, we may use the notation $\operatorname{Pr}(\cdot)$ for the probability of an event; for example, $\operatorname{Pr}(\theta>2)=\int_{\theta>2} p(\theta) d \theta$. When using a standard distribution, we use a notation based on the name of the distribution; for example, if $\theta$ has a normal distribution with mean $\mu$ and variance $\sigma^2$, we write $\theta \sim \mathrm{N}\left(\mu, \sigma^2\right)$ or $p(\theta)=\mathrm{N}\left(\theta \mid \mu, \sigma^2\right)$ or, to be even more explicit, $p\left(\theta \mid \mu, \sigma^2\right)=\mathrm{N}\left(\theta \mid \mu, \sigma^2\right)$. Throughout, we use notation such as $\mathrm{N}\left(\mu, \sigma^2\right)$ for random variables and $\mathrm{N}\left(\theta \mid \mu, \sigma^2\right)$ for density functions. Notation and formulas for several standard distributions appear in Appendix A.
We also occasionally use the following expressions for all-positive random variables $\theta$ : the coefficient of variation is defined as $\operatorname{sd}(\theta) / \mathrm{E}(\theta)$, the geometric mean is $\exp (\mathrm{E}[\log (\theta)])$, and the geometric standard deviation is $\exp (\operatorname{sd}[\log (\theta)])$.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayes’ rule

In order to make probability statements about $\theta$ given $y$, we must begin with a model providing a joint probability distribution for $\theta$ and $y$. The joint probability mass or density function can be written as a product of two densities that are often referred to as the prior distribution $p(\theta)$ and the sampling distribution (or data distribution) $p(y \mid \theta)$, respectively:
$$p(\theta, y)=p(\theta) p(y \mid \theta)$$

Simply conditioning on the known value of the data $y$, using the basic property of conditional probability known as Bayes’ rule, yields the posterior density:
$$p(\theta \mid y)=\frac{p(\theta, y)}{p(y)}=\frac{p(\theta) p(y \mid \theta)}{p(y)}$$
where $p(y)=\sum_\theta p(\theta) p(y \mid \theta)$, and the sum is over all possible values of $\theta$ (or $p(y)=$ $\int p(\theta) p(y \mid \theta) d \theta$ in the case of continuous $\left.\theta\right)$. An equivalent form of (1.1) omits the factor $p(y)$, which does not depend on $\theta$ and, with fixed $y$, can thus be considered a constant, yielding the unnormalized posterior density, which is the right side of (1.2):
$$p(\theta \mid y) \propto p(\theta) p(y \mid \theta)$$
The second term in this expression, $p(y \mid \theta)$, is taken here as a function of $\theta$, not of $y$. These simple formulas encapsulate the technical core of Bayesian inference: the primary task of any specific application is to develop the model $p(\theta, y)$ and perform the computations to summarize $p(\theta \mid y)$ in appropriate ways.

# 贝叶斯分析代考

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayes’ rule

$$p(\theta, y)=p(\theta) p(y \mid \theta)$$

$$p(\theta \mid y)=\frac{p(\theta, y)}{p(y)}=\frac{p(\theta) p(y \mid \theta)}{p(y)}$$

## 有限元方法代写

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。

## 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。