### 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
• (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|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|The Law of Large Numbers and the Frequentist

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。

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