物理代写|统计力学代写Statistical mechanics代考|PHYS3020

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  • 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|PHYS3020

物理代写|统计力学代写Statistical mechanics代考|The Law of Large Numbers and the Frequentist

A frequentist might want to use the law of large numbers, specially in its strong form, (2.3.9) or (2.3.13), in order to define the concept of probability. That might answer the objection that, if one defines that concept as a frequency of results in the repetition of a large number of the “same” experiment, the notion of “large” is imprecise. But, if we take the limit $N \rightarrow \infty$, then “large” becomes precise.

There are two obvious objections to that answer: first of all, it is true that many idealizations are made in physics by taking appropriate limits, of infinite time intervals or infinite spatial extension, ${ }^{14}$ but it is difficult to see how a concept can be defined only through such a limit. Since the limit is never reached in nature and if the concept of probability makes no sense for finite sequences of experiments, then it cannot be used in the natural sciences.

The next objection is that statements like $(2.3 .9)$ or $(2.3 .13)$ are probabilistic statements even if they refer to events having probability one. But it is circular to define a concept by using a formula that involves that very concept.

The mathematician and defender of the frequentist interpretation, Richard von Mises, avoids referring to the law of large numbers and defines probabilities as limits of frequencies of particular attributes (like falling heads for a coin or landing on a 5 for a die) within what he calls a “collective.” [324]
A collective is defined as an unlimited sequence of observations so that:

  1. The limits of frequencies of particular attributes within the collective exist.
  2. These limits are not affected by place selection, which means that the same limit would be obtained if we choose a subsequence of the original sequence of the collective according to some rule, for example the subsequence of events indexed by even numbers or by prime numbers or by numbers that are squares of integers. Of course, the rule must be specified independently of the results of the sequence of observations.

This is a way of guaranteeing that the collective is “random”. Consider the sequence $0,1,0,1,0,1,0,1, \ldots$ The limits of frequencies of 0 ‘s and 1 ‘s is obviously $\frac{1}{2}$, but it would not be so if we chose the subsequence of events indexed by even numbers or by odd numbers. And that is a way of characterizing the sequence as nonrandom. ${ }^{15}$

物理代写|统计力学代写Statistical mechanics代考|Explanations and Probabilistic Explanations

One way to connect probabilities to the physical world is via the so-called Cournot’s principle which says that, if the probability of an event $A$ is very small, given some set of conditions $C$, then one can be practically certain that the event $A$ will not occur on a single realization of those conditions. ${ }^{20}$

Of course, the event and its probability have to be specified before doing the experiment where that event could occur. Otherwise, if one tosses one thousand

coins, we will obtain a definite sequence of heads and tails and that sequence does occur, although its a priori probability is very small: $\frac{1}{2^{\text {Doxn }}}$.

Besides, the probability assigned to $A$ must be properly chosen: if one were to assign probabilities $\left(\frac{1}{3}, \frac{2}{3}\right)$ to heads and tails and toss a thousand coins, the event $A$ defined by (2.3.2) would have a very small probability (exercise: estimate that probability), although it has a probability close to 1 if one assigns the usual probabilities $\left(\frac{1}{2}, \frac{1}{2}\right)$ to heads and tails.

Another way to state Cournot’s principle is that atypical events never occur. ${ }^{21}$ However, in reality, atypical events do occur: a series of coin tossing could give significant deviations from the $\left(\frac{1}{2}, \frac{1}{2}\right)$ frequencies. But that would mean that one has to revise one’s probabilities (and this is the basis of Bayesian updating: adjust your probabilities in light of the data).

But Cournot’s principle helps us to understand what is a probabilistic explanation of “random” physical phenomena.

A first form of scientific explanation is given by laws. If state A produces state $B$, according to deterministic laws, then the occurence of $B$ can be explained by the occurrence of $A$ and the existence of those laws. ${ }^{22}$ If $A$ is prepare in the laboratory, this kind of explanation is rather satisfactory, since the initial state A is produced by us.

But if B is some natural phenomena, like today’s weather and A is some meteorological condition yesterday, then $\mathrm{A}$ itself has to be explained, and that leads potentially to an “infinite” regress, going back in principle to the beginning of the universe. In practice, nobody goes back that far, and A is simply taken as “given”, which means that our explanations are in practice limited when we go backwards in time.

It is worth noting that there is something “anthropomorphic” even in this type of explanation: for example if $\mathrm{A}$ is something very special, one will try to explain A as being caused by anterior events that are not so special. Otherwise our explanation of B in terms of A will look unsatisfactory. Both the situations A and B and the laws are perfectly objective but the notion of explanation is “subjective” in the sense that it depends on what we, humans, regard as a valid explanation.

Consider now a situation where probabilities are involved, starting with the simplest example, coin tossing, and trying to use that example to build up our intuition about what constitutes a valid explanation.

物理代写|统计力学代写Statistical mechanics代考|Final Remarks

The opposition between the frequentist and Bayesian approaches to probability theory can be viewed, at least in some versions of that opposition, as part of a larger opposition between a certain version of empiricism and a certain version of rationalism. By this we mean that Bayesianism relies on the notion of rational (inductive) inference, which by definition, goes beyond mere analysis of data. The link to rationalism is that it trusts human reason of being able to make rational judgments that are not limited to “observations”. By contrast, frequentism is related to a form of skepticism with respect to the reliability of such judgments, in part because their answers can be ambiguous, as exemplified by Bertrand’s paradoxes.

Therefore, the frequentist will say, let’s limit the theory of probability to frequencies or to “data” that can be observed and forget about those uncertain reasonings. And that reaction has definitely an empiricist flavor. We have already explained our objections to that approach in Sect. 2.4. We will simply add here the remark that this move away from rationalism and towards some form of empiricism occurred simultaneously in different fields in the beginning of the twentieth century and was a somewhat understandable reaction to the “crises in the sciences” caused by the replacement of classical mechanics, that had been the bedrock of science for centuries, both by the theories of relativity and by quantum mechanics.
Here are some examples, besides frequentism, of such moves:

Logical positivism in the philosophy of science: in the Vienna Circle, there was a strong emphasis on observations or sense-data as being the only sort of things one can meaningfully speak about or “verify”. On the other hand, the logical positivists were (rightly) reacting to the metaphysical traditions in philosophy and they were also interested in inductive logic.

Formalism in the philosophy of mathematics: while realists in the philosophy of mathematics (often called Platonists) think that mathematics studies something real (numbers or sets), formalists argued that mathematicians are just deducing theorems from axioms according to given rules, but nothing more and that the axioms and the rules do not attempt to capture some “hidden” reality.

Behaviorism in the philosophy of mind: instead of postulating some “hidden” mental structures, behaviorists insisted that on should only study the links between stimuli and reactions.

The Copenhagen interpretation of quantum mechanics: the goal of science, for Bohr, Heisenberg and their followers, as opposed to the “realists” like Einstein and Schrödinger, is not to discover the properties of some microscopic reality but only to predict “results of measurements.”

Of course, for each of these positions, there are pros and cons and various nuances of these positions and we do not intend to discuss them in detail. But what is common to them is a sort of modesty with respect to science and knowledge: let’s focus on what we know for certain: empirical frequencies, sense-data, formal manipulations, inputoutput reactions, or “measurements”. But, in doing so, one abandons the explanatory character of science.

In general, modesty is praiseworthy; the problem here is that, if it goes too far, it tends to make science devoid of meaning and therefore, of interest.

物理代写|统计力学代写Statistical mechanics代考|PHYS3020


物理代写|统计力学代写Statistical mechanics代考|The Law of Large Numbers and the Frequentist



下一个反对意见是这样的陈述(2.3.9)或者(2.3.13)是概率陈述,即使它们指的是概率为 1 的事件。但是通过使用包含该概念的公式来定义一个概念是循环的。

频率论解释的数学家和捍卫者理查德·冯·米塞斯(Richard von Mises)避免提及大数定律,并将概率定义为特定属性的频率限制(例如硬币掉头或骰子落在 5 上)在他的范围内。称为“集体”。[324]

  1. 集体中特定属性的频率限制是存在的。
  2. 这些限制不受地点选择的影响,这意味着如果我们根据某些规则选择集合的原始序列的子序列,例如由偶数或素数索引的事件的子序列或由整数平方的数字。当然,必须独立于观察序列的结果来指定规则。

这是保证集体是“随机的”的一种方式。考虑序列0,1,0,1,0,1,0,1,…0 和 1 的频率限制显然是12,但如果我们选择由偶数或奇数索引的事件的子序列,情况就不会如此。这是将序列表征为非随机的一种方式。15

物理代写|统计力学代写Statistical mechanics代考|Explanations and Probabilistic Explanations



硬币,我们将获得一个确定的正面和反面序列,并且该序列确实发生,尽管它的先验概率非常小:12多克斯 .

此外,分配给的概率一个必须正确选择:如果要分配概率(13,23)正面反面抛一千枚硬币,事件一个由 (2.3.2) 定义的概率非常小(练习:估计该概率),尽管如果分配通常的概率,它的概率接近 1(12,12)到头和尾。



第一种形式的科学解释是由法律给出的。如果状态 A 产生状态乙,根据确定性定律,那么乙可以解释为发生一个以及这些法律的存在。22如果一个是在实验室准备的,这种解释还是比较满意的,因为初始状态A是我们制作的。

但是如果 B 是一些自然现象,比如今天的天气,A 是昨天的一些气象条件,那么一个必须对其自身进行解释,这可能会导致“无限”倒退,原则上可以追溯到宇宙的开始。在实践中,没有人能回溯那么远,A 被简单地视为“给定的”,这意味着当我们在时间上倒退时,我们的解释在实践中是有限的。

值得注意的是,即使在这种类型的解释中,也有一些“拟人化”的东西:例如,如果一个是非常特别的东西,人们会尝试将 A 解释为由不那么特别的先前事件引起。否则,我们根据 A 对 B 的解释将看起来不能令人满意。情况 A 和 B 以及法律都是完全客观的,但解释的概念是“主观的”,因为它取决于我们人类所认为的有效解释。


物理代写|统计力学代写Statistical mechanics代考|Final Remarks


因此,频率论者会说,让我们将概率理论限制在可以观察到的频率或“数据”上,而忘掉那些不确定的推理。这种反应绝对具有经验主义的味道。我们已经在 Sect 中解释了我们对这种方法的反对意见。2.4. 我们将在这里简单地补充一句,从理性主义转向某种形式的经验主义在 20 世纪初同时发生在不同的领域,这是对由替代理论引起的“科学危机”的某种可以理解的反应。经典力学,几个世纪以来一直是科学的基石,无论是相对论还是量子力学。




量子力学的哥本哈根解释:对于玻尔、海森堡和他们的追随者来说,与爱因斯坦和薛定谔这样的“现实主义者”相反,科学的目标不是发现某些微观现实的性质,而只是预测“测量结果” 。”



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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。