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## 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: $\frac{1}{2^{1000}}$.

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).

## 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.

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