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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|随机过程代写stochastic process代考|Computational Bayesian decision analysis

We now briefly address computational issues in relation with Bayesian decision analysis problems. In principle, this involves two operations: (1) integration to obtain expected utilities of alternatives and (2) optimization to determine the alternative with maximum expected utility. To fix ideas, we shall assume that we aim at solving problem (2.4), that is finding the alternative of maximum posterior expected utility. If the posterior distribution is independent of the action chosen, then we may drop the denominator $\int f(\mathbf{x} \mid \boldsymbol{\theta}) f(\boldsymbol{\theta}) \mathrm{d} \boldsymbol{\theta}$, solving the possibly simpler problem
$$\max _a \int u(a, \boldsymbol{\theta}) f(\mathbf{x} \mid \boldsymbol{\theta}) f(\boldsymbol{\theta}) \mathrm{d} \boldsymbol{\theta} .$$
Also recall that for standard statistical decision theoretical problems, the solution of the optimization problem is well known. For example, in an estimation problem with absolute value loss, the optimal estimate will be the posterior median. We shall refer here to problems with general utility functions. We first describe two simulationbased methods and then present a key optimization principle in sequential problems, Bellman’s dynamic programming principle, which will be relevant when dealing with stochastic processes.

The first approach we describe is called sample path optimization in the simulation literature and was introduced in statistical decision theory in Shao (1989). To be most effective, it requires that the posterior does not depend on the action chosen. In such cases, we may use the following strategy:

1. Select a sample $\boldsymbol{\theta}^1, \ldots, \boldsymbol{\theta}^N \sim p(\boldsymbol{\theta} \mid \mathbf{x})$.
2. Solve the optimization problem
$$\max {a \in \mathcal{A}} \frac{1}{N} \sum{i=1}^N u\left(a, \boldsymbol{\theta}^i\right)$$
yielding $a_N^$. If the maximum expected utility alternative $a^$ is unique, we may prove that $a_N^* \rightarrow a^$, almost surely. Note that the auxiliary problem used to find $a_N^$ is a standard mathematical programming problem, see Nemhauser et al. (1990) for ample information.
Suppose now that the posterior actually depends on the chosen action. Assume that the posterior is $f(\boldsymbol{\theta} \mid \mathbf{x}, a)>0$, for each pair $(a, \boldsymbol{\theta})$. If the utility function is positive and integrable, we may define an artificial distribution on the augmented product space $\mathcal{A} \times \boldsymbol{\Theta}$ with density $h(a, \boldsymbol{\theta})$ proportional to the product of the utility function and the posterior probability density
3. $$4. h(a, \boldsymbol{\theta}) \propto u(a, \boldsymbol{\theta}) \times p(\boldsymbol{\theta} \mid \mathbf{x}, a) 5.$$

## 统计代写|随机过程代写stochastic process代考|Discrete time Markov processes with continuous state space

As noted in Chapter 1, Markov processes can be defined with both discrete and continuous state spaces. We have seen that for a Markov chain with discrete state space, the condition for the chain to have an equilibrium distribution is that the chain is aperiodic and that all states are positive recurrent. Although the condition of positive recurrence cannot be sensibly applied to chains with continuous state space, a similar condition known as Harris recurrence applies to chains with continuous state space, which essentially means that the chain can get close to any point in the future. It is known that Harris recurrent, aperiodic chains also possess an equilibrium distribution, so that if the conditional probability distribution of the chain is $P\left(X_n \mid X_{n-1}\right)$, then the equilibrium density $\pi$ satisfies
$$\pi(x)=\int P(x \mid y) \pi(y) \mathrm{d} y .$$
As with Markov chains with discrete state space, a sufficient condition for a process to possess an equilibrium distribution is to be reversible.

Example 3.2: Simple examples of continuous space Markov chain models are the autoregressive (AR) models. The first-order AR process was outlined in Example 1.1. Higher order dependence can also be incorporated. $\operatorname{An} \operatorname{AR}(k)$ model is defined by
$$X_n=\phi_0+\sum_{i=1}^k \phi_i X_{n-i}+\epsilon_n .$$

The condition for this process to be (weakly) stationary is the well-known unit roots condition that all roots of the polynomial
$$\phi_0 z^k-\sum_{i=1}^k \phi_i z^{k-i}$$
must lie within the unit circle, that is, each root $z_i$ must satisfy $\left|z_i\right|<1$.
Inference for AR processes and other continuous state space processes is briefly reviewed in Section 3.4.3.

# 随机过程代考

## 统计代写|随机过程代写随机过程代考|计算贝叶斯决策分析

$$\max _a \int u(a, \boldsymbol{\theta}) f(\mathbf{x} \mid \boldsymbol{\theta}) f(\boldsymbol{\theta}) \mathrm{d} \boldsymbol{\theta} .$$

1. 选择一个样本$\boldsymbol{\theta}^1, \ldots, \boldsymbol{\theta}^N \sim p(\boldsymbol{\theta} \mid \mathbf{x})$ .
2. 解决优化问题
$$\max {a \in \mathcal{A}} \frac{1}{N} \sum{i=1}^N u\left(a, \boldsymbol{\theta}^i\right)$$
yield $a_N^$。如果最大期望效用替代$a^$是唯一的，我们可以几乎肯定地证明$a_N^* \rightarrow a^$。请注意，用于查找$a_N^$的辅助问题是一个标准的数学规划问题，参见Nemhauser等人(1990)获得大量信息。现在假设后验实际上取决于所选的动作。假设后验为$f(\boldsymbol{\theta} \mid \mathbf{x}, a)>0$，对于每一对$(a, \boldsymbol{\theta})$。如果效用函数是正的且可积的，我们可以在增宽积空间$\mathcal{A} \times \boldsymbol{\Theta}$上定义一个人工分布，其密度$h(a, \boldsymbol{\theta})$与效用函数和后验概率密度的乘积成正比
3. $$4. h(a, \boldsymbol{\theta}) \propto u(a, \boldsymbol{\theta}) \times p(\boldsymbol{\theta} \mid \mathbf{x}, a) 5.$$

## 统计代写|随机过程代写随机过程代考|连续状态空间的离散时间Markov过程

$$\pi(x)=\int P(x \mid y) \pi(y) \mathrm{d} y .$$

$$X_n=\phi_0+\sum_{i=1}^k \phi_i X_{n-i}+\epsilon_n .$$ 定义

$$\phi_0 z^k-\sum_{i=1}^k \phi_i z^{k-i}$$

AR过程和其他连续状态空间过程的推理在第3.4.3节中简要回顾

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师