### 统计代写|统计推断作业代写statistics interference代考| Bayesian discussion

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|统计推断作业代写statistics interference代考|Bayesian discussion

In the second approach to the problem we treat $\mu$ as having a probability distribution both with and without the data. This raises two questions: what is the meaning of probability in such a context, some extended or modified notion of probability usually being involved, and how do we obtain numerical values for the relevant probabilities? This is discussed further later, especially in Chapter $5 .$ For the moment we assume some such notion of probability concerned with measuring uncertainty is available.

If indeed we can treat $\mu$ as the realized but unobserved value of a random variable $M$, all is in principle straightforward. By Bayes’ theorem, i.e., by simple laws of probability,
$$f_{M \mid Y}(\mu \mid y)=f_{Y \mid M}(y \mid \mu) f_{M}(\mu) / \int f_{Y \mid M}(y \mid \phi) f_{M}(\phi) d \phi .$$
The left-hand side is called the posterior density of $M$ and of the two terms in the numerator the first is determined by the model and the other, $f_{M}(\mu)$, forms the prior distribution summarizing information about $M$ not arising from $y$. Any method of inference treating the unknown parameter as having a probability distribution is called Bayesian or, in an older terminology, an argument of inverse probability. The latter name arises from the inversion of the order of target and conditioning events as between the model and the posterior density.
The intuitive idea is that in such cases all relevant information about $\mu$ is then contained in the conditional distribution of the parameter given the data, that this is determined by the elementary formulae of probability theory and that remaining problems are solely computational.

In our example suppose that the prior for $\mu$ is normal with known mean $m$ and variance $v$. Then the posterior density for $\mu$ is proportional to
$$\exp \left{-\Sigma\left(y_{k}-\mu\right)^{2} /\left(2 \sigma_{0}^{2}\right)-(\mu-m)^{2} /(2 v)\right}$$
considered as a function of $\mu$. On completing the square as a function of $\mu$, there results a normal distribution of mean and variance respectively
$$\begin{gathered} \frac{\bar{y} /\left(\sigma_{0}^{2} / n\right)+m / v}{1 /\left(\sigma_{0}^{2} / n\right)+1 / v} \ \frac{1}{1 /\left(\sigma_{0}^{2} / n\right)+1 / v} \end{gathered}$$
for more details of the argument, see Note 1.5. Thus an upper limit for $\mu$ satisfied with posterior probability $1-c$ is
$$\frac{\bar{y} /\left(\sigma_{0}^{2} / n\right)+m / v}{1 /\left(\sigma_{0}^{2} / n\right)+1 / v}+k_{c}^{*} \sqrt{\frac{1}{1 /\left(\sigma_{0}^{2} / n\right)+1 / v}}$$

## 统计代写|统计推断作业代写statistics interference代考|Some further discussion

We now give some more detailed discussion especially of Example $1.4$ and outline a number of special models that illustrate important issues.

The linear model of Example $1.4$ and methods of analysis of it stemming from the method of least squares are of much direct importance and also are the base of many generalizations. The central results can be expressed in matrix form centring on the least squares estimating equations
$$z^{T} z \hat{\beta}=z^{T} Y,$$
the vector of fitted values
$$\hat{Y}=z \hat{\beta},$$
and the residual sum of squares
$$\text { RSS }=(Y-\hat{Y})^{T}(Y-\hat{Y})=Y^{T} Y-\hat{\beta}^{T}\left(z^{T} z\right) \hat{\beta} .$$
Insight into the form of these results is obtained by noting that were it not for random error the vector $Y$ would lie in the space spanned by the columns of $z$, that $\hat{Y}$ is the orthogonal projection of $Y$ onto that space, defined thus by
$$z^{T}(Y-\hat{Y})=z^{T}(Y-z \hat{\beta})=0$$
and that the residual sum of squares is the squared norm of the component of $Y$ orthogonal to the columns of $z$. See Figure 1.1.

There is a fairly direct generalization of these results to the nonlinear regression model of Example 1.5. Here if there were no error the observations would lie on the surface defined by the vector $\mu(\beta)$ as $\beta$ varies. Orthogonal projection involves finding the point $\mu(\hat{\beta})$ closest to $Y$ in the least squares sense, i.e., minimizing the sum of squares of deviations ${Y-\mu(\beta)}^{T}{Y-\mu(\beta)}$. The resulting equations defining $\hat{\beta}$ are best expressed by defining
$$z^{T}(\beta)=\nabla \mu^{T}(\beta),$$
where $\nabla$ is the $q \times 1$ gradient operator with respect to $\beta$, i.e., $\nabla^{T}=$ $\left(\partial / \partial \beta_{1}, \ldots, \partial / \partial \beta_{q}\right)$. Thus $z(\beta)$ is an $n \times q$ matrix, reducing to the previous $z$

in the linear case. Just as the columns of $z$ define the linear model, the columns of $z(\beta)$ define the tangent space to the model surface evaluated at $\beta$. The least squares estimating equation is thus
$$z^{T}(\hat{\beta}){Y-\mu(\hat{\beta})}=0 .$$
The local linearization implicit in this is valuable for numerical iteration. One of the simplest special cases arises when $E\left(Y_{k}\right)=\beta_{0} \exp \left(-\beta_{1} z_{k}\right)$ and the geometry underlying the nonlinear least squares equations is summarized in Figure 1.2.

The simple examples used here in illustration have one component random variable attached to each observation and all random variables are mutually independent. In many situations random variation comes from several sources and random components attached to different component observations may not be independent, showing for example temporal or spatial dependence.

## 统计代写|统计推断作业代写statistics interference代考|Parameters

A central role is played throughout the book by the notion of a parameter vector, $\theta$. Initially this serves to index the different probability distributions making up the full model. If interest were exclusively in these probability distributions as such, any $(1,1)$ transformation of $\theta$ would serve equally well and the choice of a particular version would be essentially one of convenience. For most of the applications in mind here, however, the interpretation is via specific parameters and this raises the need both to separate parameters of interest, $\psi$, from nuisance parameters, $\lambda$, and to choose specific representations. In relatively complicated problems where several different research questions are under study different parameterizations may be needed for different purposes.

There are a number of criteria that may be used to define the individual component parameters. These include the following:

• the components should have clear subject-matter interpretations, for example as differences, rates of change or as properties such as in a physical context mass, energy and so on. If not dimensionless they should be measured on a scale unlikely to produce very large or very small values;
• it is desirable that this interpretation is retained under reasonable perturbations of the model;
• different components should not have highly correlated errors of estimation;
• statistical theory for estimation should be simple;
• if iterative methods of computation are needed then speedy and assured convergence is desirable.
The first criterion is of primary importance for parameters of interest, at least in the presentation of conclusions, but for nuisance parameters the other criteria are of main interest. There are considerable advantages in formulations leading to simple methods of analysis and judicious simplicity is a powerful aid
14
Preliminaries
to understanding, but for parameters of interest subject-matter meaning must have priority.

## 统计代写|统计推断作业代写statistics interference代考|Bayesian discussion

F米∣是(μ∣是)=F是∣米(是∣μ)F米(μ)/∫F是∣米(是∣φ)F米(φ)dφ.

\exp \left{-\Sigma\left(y_{k}-\mu\right)^{2} /\left(2 \sigma_{0}^{2}\right)-(\mu-m)^ {2} /(2 v)\right}\exp \left{-\Sigma\left(y_{k}-\mu\right)^{2} /\left(2 \sigma_{0}^{2}\right)-(\mu-m)^ {2} /(2 v)\right}

## 统计代写|统计推断作业代写statistics interference代考|Some further discussion

Example 的线性模型1.4以及源自最小二乘法的分析方法具有直接的重要性，也是许多概括的基础。中心结果可以表示为以最小二乘估计方程为中心的矩阵形式

## 统计代写|统计推断作业代写statistics interference代考|Parameters

• 组件应具有明确的主题解释，例如差异、变化率或物理环境中的质量、能量等属性。如果不是无量纲的，它们应该在一个不可能产生非常大或非常小的值的尺度上进行测量；
• 在模型的合理扰动下保留这种解释是可取的；
• 不同组成部分不应有高度相关的估计误差；
• 估计的统计理论应该简单；
• 如果需要迭代计算方法，则需要快速且可靠的收敛。
第一个标准对于感兴趣的参数至关重要，至少在结论的呈现中，但对于有害参数，其他标准是主要关注的。有相当大的优势，导致简单的分析方法和明智的简单性是一个强大的帮助
14
初步
理解，但对于感兴趣的参数，主题意义必须优先。

## 广义线性模型代考

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## MATLAB代写

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