### 物理代写|统计力学代写Statistical mechanics代考|PHYSICS3542

statistics-lab™ 为您的留学生涯保驾护航 在代写统计力学Statistical mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计力学Statistical mechanics代写方面经验极为丰富，各种代写统计力学Statistical mechanics相关的作业也就用不着说。

• 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

## 物理代写|统计力学代写Statistical mechanics代考|Bayesian Updating

Suppose that we have a certain number of hypotheses $H_{1}, H_{2}, \ldots, H_{n}$ and that we have assigned probabilities $P\left(H_{i}\right)$ to each of them, probabilities that exhaust all possibilities and are mutually exclusive:
$$\sum_{i=1}^{n} P\left(H_{i}\right)=1$$
Those probabilities are called the prior probabilities.
Now, we collect new data (D) and we want to know how to change our assignments of probabilities to those various hypotheses. We will write $P\left(H_{i} \mid D\right)$ for the (updated) probability of hypothesis $H_{i}$, given $D$.

We assume that we know enough about the system to compute the probabilities of the data, for each hypothesis: $P\left(D \mid H_{i}\right), i=1, \ldots, n$. Those probabilities are called the likelihoods.
Then we simply use Bayes’ formula:
$$P\left(H_{i} \mid D\right)=\frac{P\left(D \mid H_{i}\right) P\left(H_{i}\right)}{P(D)}$$
where $P(D)=\sum_{i=1}^{n} P\left(D \mid H_{i}\right) P\left(H_{i}\right)$; this implies that the new probabilities still add up to one:
$$\sum_{i=1}^{n} P\left(H_{i} \mid D\right)=1$$
The probabilities $P\left(H_{i} \mid D\right)$ are called the posterior probabilities.
To illustrate this method, consider the well known and apparently paradoxical example of “false positives” in medical testing: assume that the prevalence of a disease in the general population is $0.5 \%$ and assume that a test for that disease gives a $99 \%$ true positive rate ( $99 \%$ of the people who have the disease are tested positive)and with a false positive rate of $2 \%$ ( $2 \%$ of the people who do not have the disease are tested positive). If a random person tests positive, what is the probability that this person has the disease?

## 物理代写|统计力学代写Statistical mechanics代考|Objections to the “Subjective” Approach

Let us now consider frequent objections to the “subjective” approach.

1. Subjectivism. Some people think that a Bayesian view of probabilities presupposes some form of subjectivism, meant as a doctrine in philosophy or philosophy of science that regards what we call knowledge as basically produced by “subjects” independently of any connection to the “outside world”. But there is no logical link here: a subjectivist about probabilities may very well claim that there are objective facts in the world, that the laws governing it are also objective, and consider probabilities as being a tool used in situations where our knowledge of those facts and those laws is incomplete. In fact, one could argue that, if there is any connection between Bayesianism and philosophical subjectivism, it goes in the opposite direction; a Bayesian should naturally think that one and only one among the ‘possible’ states is actually realized, but that there is a difference between what really happens in the world and what we know about it. On the contrary, the philosophical subjectivist position often starts by confusing the world and our knowledge of it (for example, much of loose talk about everything being ‘information’ often ignores the fact that ‘information’ is ultimately information about something which itself is not information).

Moreover, there is nothing arbitrary or subjective in the assignment of rational “subjective” probabilities. What is subjective here is simply the fact that there are no true or real probabilities “out there” in the world. But the choice of probabilities obeys rules (maximizing Shannon’s entropy and doing Bayesian updating) that do not depend on any individual’s whims, although it does depend on his or her information.

1. (Ir)relevance to physics. One may think that the Bayesian approach is useful in games of chance or in various practical problems of forecasting (as in insurance) but not for physics. Our answer in Sect. $2.5$ will be based on the law of large numbers (discussed in Sect. 2.3).
2. Ambiguities in the assignment of probabilities. It is often difficult to assign unambiguously a (subjective) probability to an event. It is easy, of course, for coin tossing or similar experiments where there are finitely many possible outcomes, which, moreover, are related by symmetry. In general, one may use maximum entropy principles, but then, one may encounter various problems: how to choose the right set of variables, how to assign an a priori distribution on those, corresponding to maximal ignorance, and how to incorporate the “knowledge that we have”.

A paradigmatic example of such problems is “Bertrand’s paradox”, invented by the $19^{t h}$ century French mathematician Joseph Bertrand [27].

Consider a circle and a set of straws that are thrown “at random” onto that circle. Assuming that the straw crosses that circle, its two points of intersection with the circle will define a chord. What is the probability that this chord is longer than the side of an equilateral triangle inscribed in that circle?

This was considered by Bertrand as an example of an ill-posed problem, because one obtains opposite answers depending on how one defines “at random”. Here are several possibilities, where, in each case, ‘random’ means that we choose a uniform distribution, but on different variables:

1. One could draw a radius of the circle perpendicular to one of the sides of the equilateral triangle (the intersection of that radius with the side of the equilateral triangle will be the midpoint of that radius). Now choose “at random” a point on that radius and draw the chord having that point as its midpoint. Then, the chord is longer than a side of the triangle if the chosen point is nearer the center of the circle than the point where the side of the triangle intersects the radius, see Fig. 2.1. Since that intersection is the midpoint of the radius, the probability that this chord is longer than the side of an equilateral triangle inscribed in that circle is $\frac{1}{2}$.
2. One could choose at random the angle (comprised between 0 and 180 degrees) between the chord and the tangent of the circle at one of its intersections. The chord will be longer than a side of the triangle if that angle is greater than 60 degrees and less than 120 degrees, see Fig. 2.2. So the probability is $\frac{1}{3}$.

∑一世=1n磷(H一世)=1

∑一世=1n磷(H一世∣D)=1

## 物理代写|统计力学代写Statistical mechanics代考|Objections to the “Subjective” Approach

1. 主观主义。有些人认为，贝叶斯的概率观预设了某种形式的主观主义，即哲学或科学哲学中的一种学说，认为我们所谓的知识基本上是由“主体”产生的，与“外部世界”的任何联系无关。但这里没有逻辑联系：关于概率的主观主义者很可能声称世界上存在客观事实，支配它的规律也是客观的，并将概率视为在我们了解这些事实的情况下使用的工具这些法律是不完整的。事实上，有人可能会争辩说，如果贝叶斯主义和哲学主观主义之间有任何联系，那就是相反的方向；贝叶斯主义者自然应该认为“可能”状态中只有一个是实际实现的，但是世界上真正发生的事情和我们所知道的事情之间是有区别的。相反，哲学主观主义的立场往往从混淆世界和我们对它的知识开始（例如，关于一切都是“信息”的松散谈论往往忽略了这样一个事实，即“信息”最终是关于某物本身不是的信息信息）。

1. (Ir) 与物理学的相关性。人们可能认为贝叶斯方法在机会游戏或各种实际预测问题（如保险）中很有用，但对物理学却没有。我们在教派中的答案。2.5将基于大数定律（在第 2.3 节中讨论）。
2. 概率分配中的歧义。通常很难为事件明确分配（主观）概率。当然，对于抛硬币或类似的实验来说，这很容易，因为这些实验有很多可能的结果，而且这些结果是通过对称性相关的。一般来说，人们可能会使用最大熵原理，但随后可能会遇到各种问题：如何选择正确的变量集，如何在这些变量上分配先验分布，对应于最大无知，以及如何结合“知识我们有”。

Bertrand 认为这是一个不适定问题的例子，因为一个人根据如何定义“随机”而获得相反的答案。这里有几种可能性，在每种情况下，“随机”意味着我们选择均匀分布，但在不同的变量上：

1. 可以绘制一个垂直于等边三角形边之一的圆的半径（该半径与等边三角形边的交点将是该半径的中点）。现在在该半径上“随机”选择一个点，并绘制以该点为中点的弦。然后，如果所选点比三角形边与半径相交的点更靠近圆心，则弦长于三角形的边，见图 2.1。由于该交点是半径的中点，因此该弦长于该圆中内接等边三角形的边的概率为12.
2. 可以随机选择弦和圆的切线之间的角度（包括在 0 和 180 度之间）在它的一个交点处。如果角度大于 60 度且小于 120 度，则弦将比三角形的边长，见图 2.2。所以概率是13.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。