42 problems

Consider the belief propagation equations for the matrix factorization model and their fixed points. Bayes-optimal inference refers to inference under the true generative distribut…

Conjectured improvement. The condition could be improved to

Let $\pmb{\mu}_{\mathrm{pr}}$ be the prior mean and let $\mathbf{L}_{\mathrm{pr}}$ be a prior covariance factor appearing in the finite-dimensional linear Gaussian inverse problem,…

The A-OTF method uses training data consisting of input-output samples for learning an approximation to a transport map, and its training objective is the optimization problem in e…

Let $a_{n[i]}^t$ denote the accumulated gradient associated with coordinate $i$ at iteration $t$, and let $y\in(0,1]$ be the exponent governing the prior's dependence on…

High-SNR Bayesian advantage conjecture. As in the case of discrete priors, the non-rotationally invariant nature of the prior should be exploitable to construct a Bayesian estimato…

Consider the noiseless ($\Delta=0$) one-hidden-layer neural network with quadratic activation and a target function matching this architecture. Let randomly initialized gradient de…

Metastable systems exhibit behaviour that appears stable on some time scale but becomes unstable on longer time scales. In this setting, an inference scheme is metaconsistent when…

Consider a high-dimensional Bayesian inference problem for which approximate message passing (AMP) has a state evolution and a Bayes-optimal solution given by the posterior mean es…

Prior-dependent mixture conjecture. The approximating distribution of the posterior is conjectured to be a mixture that depends on the prior distribution.

PINN posterior rate conjecture. We conjecture that the PINN posterior distribution actually converges faster than the nonparametric minimax rate.

Let $\alpha$ denote the power to which the likelihood is raised in the posterior, with standard posteriors corresponding to $\alpha=1$. Standard-posterior conjecture. The theoretic…

All-orders posterior-linearity conjecture. The predictive posterior of the deep nonlinear network coincides with the posterior of a deep linear network on the transformed data…

Let $\eta^\mathbf y_\beta$ denote the finite-dimensional non-Gaussian measure arising in the decomposition of the conditioned Gaussian measure, with observation data $\mathbf y$ an…

Nonlinearity conjecture. All non-linear hypotheses about $\Theta$ have at least a mild case of false confidence.

Universality conjecture. The optimality of the low-degree polynomial estimators holds beyond strictly orthogonally invariant priors; in particular, this universality phenomenon inc…

Bayes–Unexpectedness correspondence conjecture. Bayes’ rule is a specific instantiation of a more general template captured in ST by Unexpectedness. This conjecture proposes a theo…

Optimality conjecture. The analytical formulas for rotation-invariant estimators are optimal in the large-dimension limit, in the sense that they minimize the average mean-square e…

A skewed posterior approximation may replace the Gaussian density in the symmetric component with a density obtained by extending the Laplace method to fourth order. Fourth-order s…

Consider the paper's Bayesian-optimal inference setting, equivalently statistical-mechanical models on the Nishimori line, in the asymptotic limit $N\to\infty$, and let the paper's…

Let $\rho(x)\,dx$ be the prescribed limiting noise-eigenvalue distribution, and consider the simpler noise ensemble in which the $N$ eigenvalues are drawn independently and identic…

Let BAMP denote the novel approximate message passing algorithm using the pre-processed data matrix $\mathbf{J}(\mathbf{Y})$ and its multi-stage state-evolution recursion, and let…

Consider a high-dimensional Bayesian estimation problem in which approximate message passing (AMP) has a state-evolution characterization and achieves Bayes-optimal performance for…

Let $P_{{\rm mis}}(d\bx\mid \bY)$ be the mismatched posterior, let $\bX$ be the planted signal, and let $M(\lambda,\lambda_*)$ be the function defined by … Here…

Let $\mathbf{QBS}$ be the category of quasi-Borel spaces, and let $\mathbb{P}:\mathbf{QBS}\to\mathbf{QBS}$ denote the probabilistic construction. For objects $A$ and $B$, a morphis…