From Theory of Statistics by Mark J. Schervish (page 12):


  Although DeFinetti's representation theorem 1.49 is central to motivating parametric models, it is not actually used in their implementation.


How is the theorem central to parametric models?

De Finetti's Representation Theorem gives in a single take, within the subjectivistic interpretation of probabilities, the raison d'être of statistical models and the meaning of parameters and their prior distributions. 

Suppose that the random variables $X_1,\dots,X_n$ represent the results of successive tosses of a coin, with values $1$ and $0$ corresponding to the results "Heads" and "Tails", respectively. Analyzing, within the context of a subjectivistic interpretation of the probability calculus, the meaning of the usual frequentist model under which the $X_i$'s are independent and identically distributed, De Finetti observed that the condition of independence would imply, for example, that
$$  
  P\{X_n=x_n\mid X_1=x_1,\dots,X_{n-1}=x_{n-1}\} = P\{X_n=x_n\} \, ,
$$
and, therefore, the results of the first $n-1$ tosses would not change my uncertainty about the result of $n$-th toss. For example, if I believe $\textit{a priori}$ that this is a balanced coin, then, after getting the information that the first $999$ tosses turned out to be "Heads", I would still believe, conditionally on that information, that the probability of getting "Heads" on toss 1000 is equal to  $1/2$. Effectively, the hypothesis of independence of the $X_i$'s would imply that it is impossible to learn anything about the coin by observing the results of its tosses. 

This observation led De Finetti to the introduction of a condition weaker than independence that resolves this apparent contradiction. The key to De Finetti's solution is a kind of distributional symmetry known as exchangeability.

$\textbf{Definition.}$ For a given finite set $\{X_i\}_{i=1}^n$ of random objects, let $\mu_{X_1,\dots,X_n}$ denote their joint distribution. This finite set is exchangeable if $\mu_{X_1,\dots,X_n} = \mu_{X_{\pi(1)},\dots,X_{\pi(n)}}$, for every permutation $\pi:\{1,\dots,n\}\to\{1,\dots,n\}$. A sequence $\{X_i\}_{i=1}^\infty$ of random objects is exchangeable if each of its finite subsets are exchangeable.

Supposing only that the sequence of random variables $\{X_i\}_{i=1}^\infty$ is exchangeable, De Finetti proved a notable theorem that sheds light on the meaning of commonly used statistical models. In the particular case when the $X_i$'s take the values $0$ and $1$, De Finetti's Representation Theorem says that $\{X_i\}_{i=1}^\infty$ is exchangeable if and only if there is a random variable $\Theta:\Omega\to[0,1]$, with distribution $\mu_\Theta$, such that 
$$
  P\{X_1=x_1,\dots,X_n=x_n\} = \int_{[0,1]} \theta^s(1-\theta)^{n-s}\,d\mu_\Theta(\theta) \, ,
$$
in which $s=\sum_{i=1}^n x_i$. Moreover, we have that
$$
  \bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i \xrightarrow[n\to\infty]{} \Theta \qquad \textrm{almost surely},
$$
which is known as De Finetti's Strong Law of Large Numbers.

This Representation Theorem shows how statistical models emerge in a Bayesian context: under the hypothesis of exchangeability of the observables $\{X_i\}_{i=1}^\infty$, $\textbf{there is}$ a $\textit{parameter}$ $\Theta$ such that, given the value of $\Theta$, the observables are $\textit{conditionally}$ independent and identically distributed. Moreover, De Finetti's Strong law shows that our prior opinion about the unobservable $\Theta$, represented by the distribution $\mu_\Theta$, is the opinion about the limit of $\bar{X}_n$, before we have information about the values of the realizations of any of the $X_i$'s. The parameter $\Theta$ plays the role of a useful subsidiary construction, which allows us to obtain conditional probabilities involving only observables through relations like
$$
  P\{X_n=1\mid X_1=x_1,\dots,X_{n-1}=x_{n-1}\} = \mathrm{E}\left[\Theta\mid X_1=x_1,\dots,X_{n-1}=x_{n-1}\right] \, .
$$ 
