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# 数学代写|随机过程Stochastic Porcesses代考|MA53200 Bayesian decision analysis

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## 数学代写|随机过程Stochastic Porcesses代考|Bayesian decision analysis

Often, the ultimate aim of statistical research will be to support decision-making. As an example, the gambler might have to decide whether or not to play the game and what initial stake to put. An important strength of the Bayesian approach is its natural inclusion into a coherent framework for decision-making, which, in practical terms, leads to Bayesian decision analysis.

If the consequences of the decisions, or actions of a decision maker $(D M)$, depend upon the future values of observations, the general description of a decision problem is as follows. For each feasible action $a \in \mathcal{A}$, with $\mathcal{A}$ the action space, and each future result $\mathbf{y}$, we associate a consequence $c(a, \mathbf{y})$. For example, in the case of the gambler’s ruin problem, if the gambler stakes a quantity $x_0$ (the action $a$ ) and wins the game after a sequence $\mathbf{y}$ of results, the consequence is that she wins a quantity $m-x_0$. This consequence will be evaluated through its utility $u(c(a, \mathbf{y}))$, which encodes the DM’s preferences and risk attitudes. The DM should choose the action maximizing her predictive expected utility
$$\max _{a \in \mathcal{A}} \int u(c(a, \mathbf{y})) f(\mathbf{y} \mid \mathbf{x}) \mathrm{d} y,$$
where $f(\mathbf{y} \mid \mathbf{x})$ represents the DM’s predictive density for $\mathbf{y}$ given her current knowledge and data, $\mathbf{x}$, described in (2.3).

In other instances, the consequences will actually depend on the parameter $\boldsymbol{\theta}$, rather than on the observable $\mathbf{y}$. In these cases, we shall be interested in maximizing the posterior expected utility
$$\max _{a \in \mathcal{A}} \int u(c(a, \boldsymbol{\theta})) f(\boldsymbol{\theta} \mid \mathbf{x}) \mathrm{d} \boldsymbol{\theta}$$

## 数学代写|随机过程Stochastic Porcesses代考|Computational Bayesian statistics

The key operation in the practical implementation of Bayesian methods is integration. In the examples we have seen so far in this chapter, most integrations are standard and may be done analytically. This is a typical consequence of the use of conjugate prior distributions: a class of priors is conjugate to a given model, if the resulting posterior belongs to the same class of distributions. When the properties of the conjugate family of distributions are known, the use of conjugate prior distributions greatly simplifies Bayesian analysis procedures since, given observed data, the calculation of the posterior distribution reduces to simply modifying the parameters of the prior distribution. However, it is important to note that conjugate prior distributions are associated with (generalized) exponential family sampling distributions, and, therefore, that conjugate prior distributions do not always exist. For example, if we consider data generated from a Cauchy distribution, then it is well known that no conjugate prior exists.

However, more complex, nonconjugate models will generally not allow for such neat computations. Various techniques for approximating Bayesian integrals can be considered.

When the sample size is sufficiently large, central limit type theorems can sometimes be applied so that the posterior distribution is approximated by a normal distribution, when integrals may often be estimated in a straightforward way. Otherwise, in low-dimensional problems such as in Example 2.7, we can often apply numerical integration techniques like Gaussian quadrature. However, in higher dimensional problems, the number of function evaluations necessary to accurately evaluate the relevant integrals increases rapidly and such methods become inaccurate. Therefore, approaches based on simulation are typically preferred. Given their increasing importance in Bayesian statistical computation, we outline such methods.

## 数学代写|随机过程Stochastic Porcesses代考|Bayesian decision analysis

$m-x_0$. 这个结果将通过它的效用来评估 $u(c(a, \mathbf{y}))$ ，它编码了 DM 的偏好和风险态庻。DM 应该选择最大化 她的预测预期效用的行动
$$\max {a \in \mathcal{A}} \int u(c(a, \mathbf{y})) f(\mathbf{y} \mid \mathbf{x}) \mathrm{d} y$$ 在哪里 $f(\mathbf{y} \mid \mathbf{x})$ 代表 DM 的预测密度 $\mathbf{y}$ 鉴于她目前的知识和数据， $\mathbf{x}$, 在 (2.3) 中描述。 在其他情况下，后果实际上取决于参数 $\boldsymbol{\theta}$ ，而不是在可观察的 $\mathbf{y}$. 在这些情况下，我们将对最大化后验期望效用 感兴趣 $$\max {a \in \mathcal{A}} \int u(c(a, \boldsymbol{\theta})) f(\boldsymbol{\theta} \mid \mathbf{x}) \mathrm{d} \boldsymbol{\theta}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。