Demonstration
de Finetti, and where it goes negative
de Finetti writes an exchangeable sequence as a mixture of i.i.d. ones. For finite sequences the mixture is still exact, but its weights can turn negative.
Take the simplest non-trivial case: two exchangeable coins. The law is fixed by three numbers, the chances of zero, one, or two heads, so it is a point in a triangle. The i.i.d. coins \(\mathrm{Bernoulli}(\theta)^{\otimes 2}\) trace the orange parabola as \(\theta\) runs from 0 to 1.
de Finetti's theorem says an exchangeable law that extends to an infinite sequence is a positive mixture of those i.i.d. points. Positive mixtures fill the shaded region, which is exactly the laws with correlation \(\rho \ge 0\). Push the two coins to be negatively correlated and the point leaves that region. The law is still exchangeable, and by Kerns–Székely it is still an exact mixture of i.i.d. coins, but now some mixing weights are negative. That signed measure is not a prior you can sample. It is the finite, non-extendable corner.
The triangle is every exchangeable two-coin law. Orange is the i.i.d. curve. The shaded lens is de Finetti's positive mixtures (\(\rho\ge 0\)). The dot is your law: blue inside, red once \(\rho\) goes negative and the mixture must be signed.
The mixing measure over three i.i.d. nodes \(\theta=0,\tfrac12,1\) that reproduces your law. With \(\rho\ge 0\) the weights are a genuine probability (all non-negative). With \(\rho<0\) the outer weights drop below zero: a negative probability, in the sense Feynman and Wigner used, exact but un-samplable.
Exchangeability buys you a mixture-of-i.i.d. representation either way. The sign of the correlation decides whether that mixture is an honest prior or a signed one. The honest case is de Finetti; the signed case is where finite, negatively-dependent data live, and it is the regime that a companion note ties to the reliability of conformal prediction.
Using conformal prediction in your own project? Tell Claude: “Read https://conformalprediction.net/SKILL.md and create a project skill from it.” It adds a check for whether your coverage is conditionally trustworthy.