I am interested in a concrete case of a representation in de Finetti's Theorem for expressing the distribution of exchangeable binary sequence in terms of product measures. Namely, let
$$X\_1,\ldots,X\_n$$
be a random $0-1$ string with exactly $k$ $1$'s. How could one obtain a concrete respresentation of the measure in this particular case? This could be very standard and a reference might do. Thanks.



Concrete representation in de Finetti's Theorem
The distribution over $(X\_1, \ldots, X\_n)$ you are considering is extremally exchangeable, i.e. it is a vertex of the simplex of $n$-exchangeable distributions. It can not be extended to an $(n+1)$-exchangeable distribution $(X\_1, \ldots, X\_{n+1})$ of which it is the marginal. One consequence of De Finetti's theorem is the mixtures of iid measures are precisely those sequences which can be exchangeably extended to arbitrary length. So your original question cannot have a solution.


It has been mentioned in the comments that there are fragments of De Finetti's theorem for finite exchangeable sequences. **But these involve negative probabilities.** Geometrically, the IID distributions do not generate the space of all $n$-exchangeable distributions via convex combinations, but they still do so via affine combinations (this is where the negative coefficients come from). Finite representations are not unique, so for any given length $n$ you can find many different De Finetti representations by inverting certain matrices (given in the comments).


Maybe the most canonical one is this: In an iid sequence with bias $p$, the number $K$ of ones is $Bin(n,p)$ distributed. If we want to write your target distribution (with $P(K=n)=1$) as a continuous mixture with density $d(p)$, it is necessary and sufficient that


$$\int\_0^1 \binom{n}{k} p^k(1-p)^{n-k} d(p)\mathrm dp = \delta\_{nk}$$


that is $d(p)$ is orthogonal to the Bernstein basis polynomials. Such $d(p)$ must integrate to 1, thus is a signed probability density. The polynomial basis that is *defined* by this orthogonality relation is known as [dual Bernstein basis](http://www.ag.jku.at/pubs/tdb98.pdf) and seems to have some relevance in Computer Graphics, given the ubiquity of Bernstein polynomials. Here you have a (signed) probabilistic reading :)


