De Finetti's theorem states that, if observations $(x_1, x_2, x_3, \cdots)$ are infinitely exchangeable, then their joint probability $p(x_1, x_2, \cdots, x_N)$ has a representation as a mixture:
$$p(x_1, x_2, \cdots, x_N) = \int(\Pi_{i=1}^N p(x_i | \theta)) dP(\theta)$$
for some random variable $\theta$ (the definition is from a lecture by Michael Jordan).
If this is true and if our data is sampled iid (which implies exchangeability), then why aren't we using this theorem for things like linear regression and logistic regression?