### 统计代写|统计推断作业代写statistical inference代考|Elements of Bayesianism

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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|统计推断作业代写statistical inference代考|Elements of Bayesianism

Bayesian inference involves placing a probability distribution on all unknown quantities in a statistical problem. So in addition to a probability model for the data, probability is also specified for any unknown parameters associated with it. If future observables are to be predicted, probability is posed for these as well. The Bayesian inference is thus the conditional distribution of unknown parameters (and/or future observables) given the data. This can be quite simple if everything being modeled is discrete, or quite complex if the class of models considered for the data is broad.
This chapter presents the fundamental elements of Bayesian testing for simple versus simple, composite versus composite and for point null versus composite alternative, hypotheses. Several applications are given. The use of Jeffreys “noninformative” priors for making general Bayesian inferences in binomial and negative binomial sampling, and methods for hypergeometric and negative hypergeometric sampling, are also discussed. This discussion leads to a presentation and proof of de Finetti’s theorem for Bernoulli trials. The chapter then gives a presentation of another de Finetti result that Bayesian assignment of probabilities is both necessary and sufficient for “coherence,” in a particular setting. The chapter concludes with a discussion and illustration of model selection.

## 统计代写|统计推断作业代写statistical inference代考|TESTING A COMPOSITE VS. A COMPOSITE

Suppose we can assume that the parameter or set of parameters $\theta \in \Theta$ is assigned a prior distribution (subjective or objective) that is, we may be sampling $\theta$ from a hypothetical population specifying a probability function $g(\theta)$, or our beliefs about $\theta$ can be summarized by a $g(\theta)$ or we may assume a $g(\theta)$ that purports to reflect our prior ignorance. We are interested in deciding whether
$$H_{0}: \theta \in \Theta_{0} \quad \text { or } \quad H_{1}: \theta \in \Theta_{1} \quad \text { for } \quad \Theta_{0} \cap \Theta_{1}=\emptyset$$

Now the posterior probability function is
$$p(\theta \mid D) \propto L(\theta \mid D) g(\theta) \quad \text { or } \quad p(\theta \mid D)=\frac{L(\theta \mid D) g(\theta)}{\int_{\Theta} L(\theta \mid D) d G(\theta)}$$
using the Lebesgue-Stieltjes integral representation in the denominator above.
Usually $\Theta_{0} \cup \Theta_{1}=\Theta$ but this is not necessary. We calculate
\begin{aligned} &P\left(\theta \in \Theta_{0} \mid D\right)=\int_{\Theta_{0}} d P(\theta \mid D) \ &P\left(\theta \in \Theta_{1} \mid D\right)=\int_{\Theta_{1}} d P(\theta \mid D) \end{aligned}
and calculate the posterior odds
$$\frac{P\left(\theta \in \Theta_{0} \mid D\right)}{P\left(\theta \in \Theta_{1} \mid D\right)}$$
and if this is greater than some predetermined value $k$ choose $H_{0}$, and if less choose $H_{1}$, and if equal to $k$ be indifferent.

## 统计代写|统计推断作业代写statistical inference代考|SOME REMARKS ON PRIORS FOR THE BINOMIAL

1. Personal Priors. If one can elicit a personal subjective prior for $\theta$ then his posterior for $\theta$ is personal as well and depending on the reasoning that went into it may or may not convince anyone else about the posterior on $\theta$. A convenient prior that is often used when subjective opinion can be molded into this prior is the beta prior
$$g(\theta \mid a, b) \propto \theta^{a-1}(1-\theta)^{b-1}$$
when this is combined with the likelihood to yield
$$g(\theta \mid a, b, r) \propto \theta^{a+r-1}(1-\theta)^{b+n-r-1}$$
2. So-Called Ignorance or Informationless or Reference Priors. It appears that in absence of information regarding $\theta$, it was interpreted by Laplace that Bayes used a uniform prior in his “Scholium”. An objection raised by Fisher to this is essentially on the grounds of a lack of invariance. He argued that setting a parameter $\theta$ to be uniform resulted in, say $\tau=\theta^{3}$ (or $\tau=\tau(\theta))$ and then why not set $\tau$ to be uniform so that $g(\tau)=1,0<\tau<1$ then implies that $g(\theta)=3 \theta^{2}$ instead $g(\theta)=1$. Hence one will get different answers depending on what function of the parameter is assumed uniform.
Jeffreys countered this lack of invariance with the following:
The Fisher Information quantity of a probability function $f(x \mid \theta)$ is
$$I(\theta)=E\left(\frac{d \log f}{d \theta}\right)^{2}$$
assuming it exists. Then set
$$g(\theta)=I^{\frac{1}{2}}(\theta)$$
Now suppose $\tau=\tau(\theta)$. Then
\begin{aligned} I(\tau) &=E\left(\frac{d \log f}{d \tau}\right)^{2}=E\left(\frac{d \log f}{d \theta} \times \frac{d \theta}{d \tau}\right)^{2} \ &=E\left(\frac{d \log f}{d \theta}\right)^{2} \times\left(\frac{d \theta}{d \tau}\right)^{2} \end{aligned}

## 统计代写|统计推断作业代写statistical inference代考|TESTING A COMPOSITE VS. A COMPOSITE

H0:θ∈θ0 或者 H1:θ∈θ1 为了 θ0∩θ1=∅

p(θ∣D)∝大号(θ∣D)G(θ) 或者 p(θ∣D)=大号(θ∣D)G(θ)∫θ大号(θ∣D)dG(θ)

## 统计代写|统计推断作业代写statistical inference代考|SOME REMARKS ON PRIORS FOR THE BINOMIAL

1. 个人先验。如果可以引出个人主观先验θ然后他的后部为θ也是个人的，并且取决于进入它的推理可能会或可能不会说服其他人关于后验θ. 当主观意见可以被塑造成这个先验时，经常使用的一个方便的先验是 beta 先验
G(θ∣一种,b)∝θ一种−1(1−θ)b−1
当这与产生的可能性相结合时
G(θ∣一种,b,r)∝θ一种+r−1(1−θ)b+n−r−1
2. 所谓的无知或无信息或参考先验。似乎在缺乏有关信息的情况下θ，拉普拉斯解释说，贝叶斯在他的“Scholium”中使用了统一先验。费舍尔对此提出的反对意见主要是基于缺乏不变性。他认为设置参数θ统一导致，说τ=θ3（或者τ=τ(θ))然后为什么不设置τ是统一的，这样G(τ)=1,0<τ<1那么意味着G(θ)=3θ2反而G(θ)=1. 因此，根据假设参数的函数是一致的，人们将得到不同的答案。
Jeffreys 用以下方法反驳了这种缺乏不变性：
概率函数的 Fisher 信息量F(X∣θ)是
一世(θ)=和(d日志⁡Fdθ)2
假设它存在。然后设置
G(θ)=一世12(θ)
现在假设τ=τ(θ). 然后
一世(τ)=和(d日志⁡Fdτ)2=和(d日志⁡Fdθ×dθdτ)2 =和(d日志⁡Fdθ)2×(dθdτ)2

## 广义线性模型代考

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## MATLAB代写

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