### Philosophical Interpretation of [PERSON] Finetti's Theorem  
The concept of probability has several philosophical interpretations (for a survey, see [5]). An appealing aspect of de Finetti's Theorem is that it builds a bridge between two major conflicting interpretations: the _frequentist_ and the _subjective_ interpretations. (The latter is also known as _Bayesian_ interpretation.) Let us briefly explain these through the simple experiment of coin flipping.  
The frequentist interpretation of probability says that there exists a real number \(p\in[0,1]\), such that if we keep flipping the same coin independently, then the relative frequency of heads converges to \(p\), and this value gives us the probability of heads. In this sense, the probability is an _objective_ quantity, even when we may not know its exact value. Most researchers accept this interpretation, since it is in good agreement with experiments, and provides a common-sense, testable concept. In some cases, however, it does not work so well, such as when we deal with a one-time event that cannot be indefinitely repeated. For example, it is hard to assign a precise meaning to a statement like \"candidate X will win the election tomorrow with probability 52%\".  
In contrast, the subjective (Bayesian) interpretation denies the objective existence of probability. Rather, it says that the concept only expresses one's subjective expectation that a certain event happens. For example, there is no reason to a priori assume that if among the first 100 coin flips we observed, say, 53 heads, then similar behavior has to be expected among the next 100 flips. If we still assume that the order in which the coin flips are recorded does not matter, then what we see is just an exchangeable sequence of binary values, but possibly no convergence to a _constant_.  
Which interpretation is right? The one that [PERSON] favored (see [6]), against the majority view, was the subjective interpretation. Nevertheless, his theorem provides a nice bridge between the two interpretations, in the following way. Consider two experiments:  
(1) _Bayesian_: Just keep flipping a coin and record the results. Do not presuppose the existence of a probability to which the relative frequency of heads converges, but still assume that the order of recording does not matter. Then what we obtain is an exchangeable sequence, but no specific objective probability.  
(2) _Frequentist_: Assume that an objective probability \(p\) of heads does exist, but we do not know its value exactly, so we consider it as a random quantity, drawn from some probability distribution \(\mu\). Then the experiment will be this: draw \(p\) from the distribution \(\mu\), fix it, and then keep flipping a coin that has probability \(p\) of heads, on the basis that this probability \(p\) objectively exists.  
Now de Finetti's Theorem states that the results of the above two experiments are _indistinguishable_: an exchangeable sequence of coin flips cannot be distinguished from a mix of Bernoulli sequences. In this sense, the conflicting interpretations do not lead to conflicting experimental results, so the theorem indeed builds a bridge between the subjective and frequentist views. This is a reassuring reconciliation between the conflicting interpretations!  
We need to note, however, that the above argument is only guaranteed to work if the sequence of coin flips is infinite. As already mentioned earlier, for the finite case, Theorem 1 does not always hold. This anomaly with finite sequences may be explained by the fact that the frequentist probability, as the _limiting_ value of the relative frequency, is only meaningful if we can consider infinite sequences.