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

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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|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.

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

C=(1−ķ)一个+ķ乙,

(θ∣是)∼贝塔⁡(一个+是,b+n−是),

θ^=和(θ∣是)=(一个+是)(一个+是)+(b+n−是)=一个+是一个+b+n

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)

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

F(θ∣是)=F(θ)F(是∣θ)F(是)

(是1,…,是n∣μ)∼ 独立同居 ñ(μ,σ2) μ∼ñ(μ0,σ02)

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

F(X)=∑米=1米C米F米(X),

F(θ)=∑米=1米C米F米(θ).

F(θ∣是)=∑米=1米C米′F米(θ∣是),

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