This principle of minimal updating is enforced via two axioms, which we paraphrase from [15]: (i) Subset independence: Probabilities conditioned on one domain should not be affected by information about a different, nonoverlapping domain. Consequently, nonoverlapping domains contribute additively to the entropy. (ii) Subsystem independence: If we assume two variables are unrelated, and if we receive information about each variable separately, then we should not infer any relationship between the variables on the basis of this information. In other words, if we measure the marginal distributions $\{\rho (\mathit{v}),\rho (\mathit{w})\}$, and if the prior does not encode any relationship between $\mathit{v}$ and $\mathit{w}$, then the posterior should be the product of the marginals: $\rho (\mathit{v},\mathit{w})=\rho (\mathit{v})\rho (\mathit{w})$.

One way to fit the 4D or 6D covariance matrix to all measured projections is to leverage the Cholesky decomposition $\mathbf{\Sigma }=\mathit{L}{\mathit{L}}^T$, where $\mathit{L}$ is a lower-diagonal matrix with positive diagonal elements, which ensures the covariance matrix is positive semidefinite. Alternatively, the diagonal or block-diagonal elements of the covariance matrix are typically available from faster measurements of the 2D phase space distributions $\{\rho (x,x^{\prime }),\rho (y,y^{\prime }),\rho (z,z^{\prime })\}$. Cross-plane correlations ignored in this scenario are typically much weaker than within-plane correlations, such as quadrupole focusing. If 2D measurements are unavailable, one may estimate the beam size by sampling from a uniform MENT prior and refining until the beam size can be estimated. (The prior is nonzero only in the regions allowed by the measurements).

For a fixed number of measurements $K=N(N-1)/2$ of a distribution without linear correlations between planes, we hypothesize that the ideal set of measurements is the set of $K$ two-dimensional marginals.

In this case, we extracted the covariance matrix elements from the previous GPSR fit.

The distribution exhibits some fluctuations during training, which may indicate that the sampling chains have not fully converged to the target distribution. However, the final distribution seems fairly stable with respect to random number generator seed. Additional convergence diagnostics (beyond visual checks) should be used in future studies to resolve this issue.