17 problems

The nonlinear regression model is approximated by a correlated Gaussian sequence experiment whose mean is a scalar multiple of $\mu_\ast$, and the empirical Bayes procedure uses on…

Consider the SPDE approach with coefficient regularity parameter $\beta$ and the associated posterior contraction rate for the unknown function under the conditions described above…

Consider the sequence model and Sobolev truth $f_0\in\mathcal{S}^{\beta}(F)$ from Theorem 1, with the truncated Horseshoe prior and scaling $\tau=1/n$. The posterior is denoted by…

Let the truth and the prior both be $alpha$-smooth for some large value $alpha\bin (0,+infty]$. Suppose that the parent sets are norming sets with cardinality … where…

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…

Consider the quantile martingale posterior with hyperparameters chosen in a more reasonable way than the settings used for the stated results. Conjecture on the posterior contracti…

General-input conjecture. Up to constants, the same lower bounds as in the paper can be derived for this model. The paper establishes the lower bounds for a more restricted input a…

Let $\mathcal{G}$ be the class of generalized additive models considered in the paper, and let $n^{-(2+d)/(4+4d)}$ be the lower-bound rate established for arbitrary mean-zero Gauss…

The paper considers posterior contraction rates in statistical models whose parameter spaces may be equipped with Wasserstein structure, and distinguishes parameter spaces with a l…

Gaussian-prior conjecture. The use of a Gaussian prior will lead to a degradation of the convergence rate unless the norm of the true coefficients $\beta^*$ is appropriately bounde…

Horseshoe nonactive-contraction conjecture. Because this horseshoe choice satisfies the lower-bound condition on $\tau$ but not the upper-bound condition, the horseshoe prior shoul…

Consider a general regression model with design matrix $X$, using a polynomially decaying shrinkage prior with polynomial-order parameter $\alpha$. Nearly orthogonal design conject…

Let $V$ and $A$ be distributions on $[0,bfinfty)$, and consider the Gaussian scale mixture prior distribution … Let the corresponding Bayes estimator be based on this prior. Scale-…

In the sparse normal means model, consider heavy-tailed prior distributions such as the horseshoe, the normal-exponential-gamma, and the generalized double Pareto priors. Bhattacha…

Consider global-local shrinkage priors with a global shrinkage parameter and heavy-tailed local prior components, including the horseshoe, generalized double Pareto, and normal-exp…

A hierarchical Bayesian procedure uses a Gaussian prior whose scale is randomly rescaled, and the initial Gaussian prior has a reproducing kernel Hilbert space (RKHS) consisting of…

Suboptimal contraction-rate conjecture. Bayesian posteriors may suffer from suboptimal contraction rates in density estimation problems under the $L^r$-loss when $r>2$.