统计代写|贝叶斯分析代写Bayesian Analysis代考|MAST90125

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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 predictive inference

In addition to estimating model parameters (and functions of those parameters) there is often interest in predicting some future data (or some other quantity which is not just a function of the model parameters).

Consider a Bayesian model specified by $f(y \mid \theta)$ and $f(\theta)$, with posterior as derived in ways already discussed and given by $f(\theta \mid y)$.
Now consider any other quantity $x$ whose distribution is defined by a density of the form $f(x \mid y, \theta)$.

The posterior predictive distribution of $x$ is given by the posterior predictive density $f(x \mid y)$. This can typically be derived using the following equation:
\begin{aligned} f(x \mid y) &=\int f(x, \theta \mid y) d \theta \ &=\int f(x \mid y, \theta) f(\theta \mid y) d \theta \end{aligned}
Note: For the case where $\theta$ is discrete, a summation needs to be performed rather than an integral.

The posterior predictive density $f(x \mid y)$ forms a basis for making probability statements about the quantity $x$ given the observed data $y$.
Point and interval estimation for future values $x$ can be performed in very much the same way as that for model parameters, except with a slightly different terminology.

Now, instead of referring to $\hat{x}=E(x \mid y)$ as the posterior mean of $x$, we may instead use the term predictive mean.

Also, the ‘ $\mathrm{P}$ ‘ in HPDR, and CPDR may be read as predictive rather than as posterior. For example, the CPDR for $x$ is now the central predictive density region for $x$.
As an example of point prediction, the predictive mean of $x$ is
$$\hat{x}=E(x \mid y)=\int x f(x \mid y) d x .$$

Often it is easier to obtain the predictive mean of $x$ using the equation
\begin{aligned} \hat{x}=E(x \mid y) &=E{E(x \mid y, \theta) \mid y} \ &=\int E(x \mid y, \theta) f(\theta \mid y) d \theta \end{aligned}

统计代写|贝叶斯分析代写Bayesian Analysis代考|Posterior predictive p-values

Earlier, in Section 1.3, we discussed Bayes factors as a form of hypothesis testing within the Bayesian framework. An entirely different way to perform hypothesis testing in that framework is via the theory of posterior predictive $p$-values (Meng, 1994). As in the theory of Bayes factors, this involves first specifying a null hypothesis
$$H_{0}: E_{0}$$
and an alternative hypothesis
$$H_{1}: E_{1} \text {, }$$
where $E_{0}$ and $E_{1}$ are two events.
Note: As in Section $1.3, E_{0}$ and $E_{1}$ may or may not be disjoint. Also, $E_{0}$ and $E_{1}$ may instead represent two different models for the same data.
In the context of a single Bayesian model with data $y$ and parameter $\theta$, the theory of posterior predictive p-values involves the following steps:
(i) Define a suitable discrepancy measure (or test statistic), denoted $T(y, \theta)$,
following careful consideration of both $H_{0}$ and $H_{1}$ (see below).
(ii) Define $x$ as an independent future replicate of the data $y$.
(iii) Calculate the posterior predictive $p$-value (ppp-value), defined as
$$p=P{T(x, \theta) \geq T(y, \theta) \mid y} .$$
Note 1: The ppp-value is calculated under the implicit assumption that $H_{0}$ is true. Thus we could also write $p=P\left{T(x, \theta) \geq T(y, \theta) \mid y, H_{0}\right}$.
Note 2 : The discrepancy measure may or may not depend on the model parameter, $\theta$. Thus in some cases, $T(y, \theta)$ may also be written as $T(y)$.
The underlying idea behind the choice of discrepancy measure $T$ is that if the observed data $y$ is highly inconsistent with $H_{0}$ in favour of $H_{1}$ then $p$ should likely be small. This is the same idea as behind classical hypothesis testing. In fact, the classical theory may be viewed as a special case of the theory of ppp-values. The advantage of the ppp-value framework is that it is far more versatile and can be used in situations where it is not obvious how the classical theory should be applied.

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian models with multiple parameters

So far we have examined Bayesian models involving some data $y$ and a parameter $\theta$, where $\theta$ is a strictly scalar quantity. We now consider the case of Bayesian models with multiple parameters, starting with a focus on just two, say $\theta_{1}$ and $\theta_{2}$. In that case, the Bayesian model may be defined by specifying $f(y \mid \theta)$ and $f(\theta)$ in the same way as previously, but with an understanding that $\theta$ is a vector of the form $\theta=\left(\theta_{1}, \theta_{2}\right)$.
The first task now is to find the joint posterior density of $\theta_{1}$ and $\theta_{2}$, according to
$$f(\theta \mid y) \propto f(\theta) f(y \mid \theta),$$
or equivalently
$$f\left(\theta_{1}, \theta_{2} \mid y\right) \propto f\left(\theta_{1}, \theta_{2}\right) f\left(y \mid \theta_{1}, \theta_{2}\right),$$
where
$$f(\theta)=f\left(\theta_{1}, \theta_{2}\right)$$
is the joint prior density of the two parameters.
Often, this joint prior density is specified as an unconditional prior multiplied by a conditional prior, for example as
$$f\left(\theta_{1}, \theta_{2}\right)=f\left(\theta_{1}\right) f\left(\theta_{2} \mid \theta_{1}\right) .$$
Once a Bayesian model with two parameters has been defined, one task is to find the marginal posterior densities of $\theta_{1}$ and $\theta_{2}$, respectively, via the equations:
\begin{aligned} &f\left(\theta_{1} \mid y\right)=\int f\left(\theta_{1}, \theta_{2} \mid y\right) d \theta_{2} \ &f\left(\theta_{2} \mid y\right)=\int f\left(\theta_{1}, \theta_{2} \mid y\right) d \theta_{1} \end{aligned}

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

F(X∣是)=∫F(X,θ∣是)dθ =∫F(X∣是,θ)F(θ∣是)dθ

X^=和(X∣是)=∫XF(X∣是)dX.

X^=和(X∣是)=和和(X∣是,θ)∣是 =∫和(X∣是,θ)F(θ∣是)dθ

统计代写|贝叶斯分析代写Bayesian Analysis代考|Posterior predictive p-values

H0:和0

H1:和1,

(i) 定义合适的差异度量（或检验统计量），表示为吨(是,θ)，

(ii) 定义X作为数据的独立未来副本是.
(iii) 计算后验预测p-值（ppp值），定义为

p=磷吨(X,θ)≥吨(是,θ)∣是.

统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian models with multiple parameters

F(θ∣是)∝F(θ)F(是∣θ),

F(θ1,θ2∣是)∝F(θ1,θ2)F(是∣θ1,θ2),

F(θ)=F(θ1,θ2)

F(θ1,θ2)=F(θ1)F(θ2∣θ1).

F(θ1∣是)=∫F(θ1,θ2∣是)dθ2 F(θ2∣是)=∫F(θ1,θ2∣是)dθ1

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