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# 物理代写|统计力学代写Statistical Mechanics代考|PHY405 “Subjective” Versus “Objective” Probabilities

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## 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 $\mathrm{E}$ under condition $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ equals p, one means that, if one reproduces the ‘same’ conditions $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ ‘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.

## 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.

## 物理代写|统计力学代写STATISTICAL MECHANICS 代考|THE INDIFFERENCE PRINCIPLE

$$P(i)=\frac{1}{N}, \forall i=1, \ldots, N \text {. }$$

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