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

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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|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}

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

• 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

$P(\theta b \mid y) \leq \alpha / 2$。

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

ψ=G(θ)

F(ψ∣是)=F(θ∣是)|dθdψ|

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## MATLAB代写

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