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

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我们提供的统计力学Statistical mechanics及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|PHYSICS3542

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].

物理代写|统计力学代写Statistical mechanics代考|Bertrand’s Paradox

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 $\frac{1}{3}$.
物理代写|统计力学代写Statistical mechanics代考|PHYSICS3542


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


现在,我们收集新数据(D),我们想知道如何改变我们对这些不同假设的概率分配。我们会写磷(H一世∣D)对于假设的(更新的)概率H一世, 给定D.

我们假设我们对系统有足够的了解来计算数据的概率,对于每个假设:磷(D∣H一世),一世=1,…,n. 这些概率称为可能性。

在哪里磷(D)=∑一世=1n磷(D∣H一世)磷(H一世); 这意味着新的概率加起来仍然为 1:

为了说明这种方法,请考虑医学测试中“假阳性”的众所周知且显然自相矛盾的例子:假设一般人群中疾病的流行率是0.5%并假设对该疾病的测试给出了99%真阳性率(99%患有这种疾病的人的检测结果呈阳性),假阳性率为2% ( 2%没有这种疾病的人的检测结果呈阳性)。如果一个随机的人检测呈阳性,这个人得病的概率是多少?

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”。


物理代写|统计力学代写Statistical mechanics代考|Bertrand’s Paradox


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.
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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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。