### 统计代写|应用随机过程代写Stochastic process代考|Bayesian decision analysis

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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|应用随机过程代写Stochastic process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 $y$ of results, the consequence is that she wins a quantity $m-x_{0}$. This consequence will be evaluated through its utility $u(c(a, y))$, 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 $\theta$, rather than on the observable $y$. In these cases, we shall be interested in maximizing the posterior expected utility
$$\max _{a \in \mathcal{A}} \int u(c(a, \theta)) f(\theta \mid \mathbf{x}) \mathrm{d} \theta$$
In most statistical contexts, we normally talk about losses, rather than utilities, and we aim at minimizing the posterior (or predictive) expected loss. We just need to consider that utility is the negative of the loss. Note also that all the standard statistical approaches mentioned earlier may be justified within this framework. As an example, if we are interested in point estimation through the posterior mean, we may easily see that this estimate is optimal, in terms of minimizing posterior expected loss, when we use the quadratic loss function (see, e.g., French and Ríos Insua, 2000). We would like to stress, however, that we should not always appeal to such canonical utility/loss functions, but rather try to model whatever relevant consequential aspects we may deem appropriate in the problem at hand.

## 统计代写|应用随机过程代写Stochastic process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.

The key idea is that of Monte Carlo integration, which substitutes an integral by a sample mean of a sufficiently large number, say $N$, of values simulated from the relevant posterior distribution. If $\boldsymbol{\theta}^{1}, \ldots, \theta^{N}$ is a sample from $f(\theta \mid \mathbf{x})$, then we have that for some function, $g(\theta)$, with finite posterior mean and variance, then
$$\frac{1}{N} \sum_{i=1}^{N} g\left(\boldsymbol{\theta}^{(i)}\right) \cong E[g(\boldsymbol{\theta}) \mid \mathbf{x}] .$$
This result follows from the strong law of large numbers, which provides almost sure convergence of the Monte Carlo approximation to the integral. The variance of the Monte Carlo approximation provides guidance on the precision of the estimate.

## 统计代写|应用随机过程代写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
$$\max {a \in \mathcal{A}} \frac{1}{N} \sum{i=1}^{N} u\left(a, \theta^{i}\right)$$
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.

## 统计代写|应用随机过程代写Stochastic process代考|Computational Bayesian statistics

1ñ∑一世=1ñG(θ(一世))≅和[G(θ)∣X].

## 统计代写|应用随机过程代写Stochastic process代考|Computational Bayesian decision analysis

1. 选择一个样本θ1,…,θñ∼p(θ∣X).
2. 解决优化问题
最大限度一种∈一种1ñ∑一世=1ñ在(一种,θ一世)
屈服一种ñ. 如果最大期望效用替代一种是唯一的，我们可以证明一种ñ→一种，几乎可以肯定。注意用于查找的辅助问题一种ñ∗是一个标准的数学规划问题，参见 Nemhauser 等人。(1990) 以获得充足的信息。

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## MATLAB代写

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