### 物理代写|统计力学代写Statistical mechanics代考|PHYSICS 7546

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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|“Subjective” Versus “Objective” Probabilities

As we said, there are, broadly speaking, two different meanings given to the word ‘probability’ in the natural sciences. The first notion is the so-called “objective” or “frequentist” one, namely the view of probability as something like a “theoretical frequency”: if one says that the probability of the event $E$ under condition $X, Y, Z$ equals $\mathrm{p}$, one means that, if one reproduces the ‘same’ conditions $X, Y, Z$ “sufficiently often”, the event E will appear with frequency p. Of course, since the world is constantly changing, it is not clear what reproducing the ‘same’ conditions means exactly; besides, the expression “sufficiently often” is vague and this is the source of much criticism of that notion of probability. ${ }^{2}$ But, putting those objections aside for a moment, probabilistic statements are, according to the “frequentist” view, factual statements that can in principle be confirmed or refuted by observations or experiments. We will come back to the discussion of the frequentist view in Sect. $2.4$ below, but now we will turn to the other meaning of the word ‘probability’, the “subjective” or Bayesian one.

In this approach, probabilities refer to a form of reasoning and not to a factual statement. Assigning a probability to an event expresses a judgment on the likelihood of that single event, based on the information available at that moment. Note that, here, one is not interested in what happens when one reproduces many times the ‘same’ event, as in the objective approach, but in the probability of a single event. This is of course very important in practice: when I wonder whether I need to take my umbrella because it may rain, or whether the stock market will crash next week, I am not mainly interested in the frequencies with which such events occur but with what will happen here and now; of course, these frequencies may be part of the information that is used in arriving at a judgment on the probability of a single event, but, typically, they are not the only information available.

One may even ask probabilistic questions, like “what is the probability of life or of intelligent life in the universe or in our galaxy?” or “what is the probability that the value of a given physical constants lies in a given interval” that do not make sense from a frequentist point of view. ${ }^{3}$ Yet, people do try to answer these questions; those answers may not be better than educated guesses, but these examples show that our intuitive notion of probability is not restricted to theoretical frequencies.

Note that, to add to the confusion, one has to make a distinction between the “objective” or “rational” Bayesian approach and the “subjective” Bayesian approach. In the latter approach, which was championed among others by the Italian mathematician de Finetti $[93,94]$, probabilities can be assigned more or less arbitrarily, provided one follows the rules of probability, like
$$P(A \cup B)=P(A)+P(B)$$
whenever $A \cap B=\emptyset$

## 物理代写|统计力学代写Statistical mechanics代考|The Indifference Principle

This principle says: first, list a series of possibilities for a “random” event, about which we know nothing, namely that we have no reason to think that one of them is more likely to occur than another one (so that “we are equally ignorant” with respect to all those possibilities). Then, assign to each of them an equal probability. If there are $N$ possibilities, we have:
$$\begin{gathered} P(i)=\frac{1}{N} \ \forall i=1, \ldots, N \end{gathered}$$

This “principle” is just another expression of our equal ignorance. ${ }^{7}$
There are many problems with this definition and several objections have been raised against it. First of all, when are we in this situation of indifference? In games of chance where there is a symmetry between the different outcomes of the random event (tossing of a coin, throwing of a die, roulette wheels etc.) it is easy to apply the indifference principle. But for more complicated situations, it is not obvious how to proceed.

Some people object that we use our ignorance to gain some information about that random event: at first, we do not know anything about it and from that we deduce that all those events are equally probable. But, from a subjectivist view of probabilities, not knowing anything about a series of possibilities and saying that all those possibilities have equal probabilities are equivalent statements, since, in that view, a probability statement is not a statement about the world but about our state of knowledge.

In more complicated situations, where there is no symmetry between the different possibilities one uses the maximum entropy principle. Namely one assigns to each probability distribution $\mathbf{p}=\left(p_{i}\right){i=1}^{N}$ over $N$ objects its Shannon entropy, given by: $$S(\mathbf{p})=-\sum{i=1}^{N} p_{i} \log p_{i}$$
One then chooses the probability distribution that has the maximum entropy, among those that satisfy certain constraints that incorporate the information that we have about the system.

The rationale behind this principle, as for the indifference principle, is not to introduce bias in our judgments, namely information that we do not have (like people who believe in lucky numbers). And one can argue that maximizing the Shannon entropy is indeed the best way to formalize that notion. We will discuss in detail this idea and its justification in Sect. 7.2, see also Shannon [291] and Jaynes [180], [183, Sect. 11.3].

## 物理代写|统计力学代写Statistical mechanics代考|Cox’ “Axioms” and Theorem

As an aside, let us mention also that, in 1946, Cox [90], inspired by previous ideas of Keynes [190], gave a foundation to the “subjective” approach to probability based on reasonings about the plausibility of propositions (see Jaynes [183] for an extensive discussion of this approach). Instead of assigning probabilities to events, as in elementary probabilities, or to sets, as in the mathematical version (see Appendix 2.A), Cox gives a numerical value to the plausibility $\mathcal{P}(p \mid q)$ of a proposition $p$ given that another proposition $q$ is true. ${ }^{8}$

Then, Cox imposes some rules of rationality on those plausibility assignments and derive from them, for a given proposition $\mathrm{r}$, the sum rule:
$$\mathcal{P}(p \text { or } q \mid r)=\mathcal{P}(p \mid r)+\mathcal{P}(q \mid r)-\mathcal{P}(p \text { and } q \mid r),$$
and the product rule:
$\mathcal{P}(p$ and $q \mid r)=\mathcal{P}(p \mid q$ and $r) \mathcal{P}(q \mid r)=\mathcal{P}(q \mid p$ and $r) \mathcal{P}(p \mid r)$
If we replace the propositions by sets (e.g. sets of events that render the propositions true), (2.2.3), expressed in terms of probabilities of sets, means:
$$P(A \cup B)=P(A)+P(B)-P(A \cap B),$$
which reduces to (2.2.1) when $p$ and $q$ are incompatible, namely when the sets $A$ and $B$ of events for which those propositions are true are disjoint; besides (2.2.5) follows by applying (2.2.1) to the disjoint union $A \cup B=(A \backslash B) \cup(B \backslash A) \cup(A \cap B)$ and using $A=(A \backslash B) \cup(A \cap B)$ and $B=(B \backslash A) \cup(A \cap B)$.
Equation (2.2.4), expressed in terms of probabilities of sets, means:
$$P(A \cap B \mid C)=P(A \mid B \cap C) P(B \mid C)=P(B \mid A \cap C) P(A \mid C),$$
where by definition,
$$P(A \mid B)=\frac{P(A \cap B)}{P(B)}$$
is the conditional probability of event $A$ given event $B$.

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