Let $\Psi :=p_1|\psi_1 \rangle \langle \psi_1 | + p_2|\psi_2 \rangle \langle \psi_2|$ be a density matrix/operator representing a (possibly mixed) qubit state. A projective measurement corresponds to a pair of (equivalence classes of) orthogonal vectors in $\mathbb{CP}^1$, and choosing arbitrary representatives $\langle \phi_1|$, $\langle \phi_2|$ from such an equivalence class, for a fixed $\Psi$ the projective measurement corresponds to the Bernoulli distribution $\mathbb{P}(X=1) = p_1 |\langle \phi_1 | \psi_1 \rangle |^2 + p_2|\langle \phi_1 | \psi_2 \rangle |^2$, $\mathbb{P}(X=2) = p_1 |\langle \phi_2 | \psi_1 \rangle |^2 + p_2|\langle \phi_2 | \psi_2 \rangle |^2$. The projective measurement also corresponds to a pair of antipodal points on the sphere (using the Bloch sphere parameterization).
Question: For a given qubit state $\Psi$, can all of these Bernoulli distributions corresponding to projective measurements be written as a conditional disintegration of a distribution on the sphere?
This appears to be possible in at least one case, namely if $\Psi$ is the perfectly mixed state $\frac{1}{2}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then all of the Bernoulli distributions corresponding to projective measurements are "fair coins", i.e. $\mathbb{P}(X=1) = \frac{1}{2} = \mathbb{P}(X=2)$. Hence these distributions should form a conditional disintegration of the uniform distribution on the sphere.
Note 1: Even if such a construction works for qubits, I doubt it has any real-world relevance / "physical explainability", because it would not seem to be applicable to e.g. qutrits or other quantum systems with more than two states. Cf. this question.
Note 2: For more on conditional disintegration, cf. e.g. the references [1] [2] below, or these other questions [a] [b] on math stackexchange.
Note 3: I am familiar with contextuality, counterfactual indefiniteness, and Bell-type inequalities for stochastic hidden variables. However all of these only seem to be obstacles for writing the Bernoulli distributions corresponding to projective measurements of a qubit as marginal distributions of a common joint distribution.
As far as I understand, they don't seem to say anything that prohibits writing those distributions as conditional distributions of a common "mixture distribution". Marginalization and conditioning in general can behave very differently.
Note 4: Because the family of von-Mises Fisher distributions (restricted to $0 \le \kappa \le 1$, $p=3$) has a parameterization very similar to that of the Bloch sphere, and because (according to Wikipedia) it has been used to model the interaction of electric dipoles in an electric field, it is very tempting to imagine that a conditional disintegration of these distributions could correspond to projective measurements of a qubit. That was the motivation for a question I asked previously on the stats stackexchange. However, even if the vMF distributions are unrelated, that doesn't by itself prohibit the possibility of writing the projective measurements of a qubit as a conditional disintegration of some other distribution on the sphere.
References:
[1] Conditioning as disintegration, J. T. Chang and D. Pollard, Statistica Neerlandica (1997) Vol. 51, nr. 3, pp. 287±317
[2] sections 2.2. and 2.3 of Exact Bayesian Inference by Symbolic Disintegration, Chung-chieh Shan and Norman Ramsey, POPL’17, January 15–21, 2017, Paris, France ACM. 978-1-4503-4660-3/17/01. http://dx.doi.org/10.1145/3009837.3009852