62 problems

Let $f_0$ be an $b1$-Hölder continuous density, with $b1\in(0,1]$, and consider the Bayes histogram estimator and its convergence rate from Theorem. Sharpness conjecture. The rate…

Let $f_0$ denote the true uniform density in the cosine-based model, and consider oscillatory densities in the model's tail that weakly target $f_0$. A prior is called reasonable i…

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…

Variance-offset conjecture. Variance inflation induced by sequential updates offsets the usual variance underestimation associated with KL-based variational approximations.

Asymptotic quasi-Bayesian learning conjecture. Results analogous to these posterior-concentration and coverage conditions can be established in the quasi-Bayesian context of the ar…

Fractional-posterior uncertainty-quantification conjecture. Choosing $alpha<1$ might automatically accommodate the inflation needed for posterior credible regions to be asymptotica…

Let $\mathcal{S}$ be the central subspace and let the hierarchical prior assign mass to subspaces of varying dimension, including dimensions greater than the intrinsic dimension…

Let $d_n=d$ and $d^*_n=d^*$ be fixed, and let $\mathcal{S}_n=\mathcal{S}$ denote the central subspace. Consider the hierarchical Bayes procedure described in the paper, whose poste…

Let the parameter $\theta$ of the triangular distribution have parameter space $(0,1)$, and let the Jeffreys prior be the prior proportional to the positive square root of the dete…

Let $Y\sim {\rm N}_d(\theta,N^{-1}I_d)$, with the mean vector split into disjoint blocks $\theta^{(1)},\dots,\theta^{(B)}$ of sizes $d_1,\dots,d_B$, where $d_1+\dots+d_B=d$. Define…

Expected posterior concentration conjecture. Under reasonable regularity assumptions, this expected posterior concentration condition implies condition (v). The specific content of…

The Bayesian hierarchy involves the missingness probability $p$, the observed sample means $\bar X_1,\ldots,\bar X_{N_n}$, the number of observations $N_n$, a random distribution…

The preceding discussion concerns a single-parameter family of densities and its multivariate exponential-family generalization. In settings where posterior asymptotics such as the…

Let $V(n,\pi)$ be the value function defined in, where $n$ denotes the time index, $\pi$ is the relevant state variable, $\mu$ is a prior distribution, and $\nu$ is the parameter a…

Let global-local shrinkage priors be priors with a global shrinkage parameter and local shrinkage parameters, and consider their use for precision matrix estimation under $0-1$ los…

Uniform-approximation generalization conjecture. The posterior consistency result can be generalized as long as the basis functions have the uniform approximation property.

Let $\theta$ be a piecewise constant sequence, with $k$ denoting the model complexity or number of segments, and let the concentration rate describe the rate at which a Bayesian po…

Consider networks generated from a diagonally dominant stochastic block model. The c-SBM is the conjugate stochastic block model discussed in the source, while consistent estimatio…

Let $\theta_0\in\Theta$ be the true parameter, let $y\in\Omega$ be sampled according to $\mu_{\theta_0}$, and let $\pi_n(\cdot,y)$ denote the posterior probability measure on…

DP mixture rate conjecture. Unlike the Bayesian procedure proposed in the paper, the posterior of the DP mixture model cannot obtain an improved convergence rate for estimating the…

Posterior divergence conjecture. Posterior divergence also holds for DPMMs.

Consider independent observations from the gamma distribution with shape parameter $\theta>0$ and scale $\beta=1$, and let $T=\overline{\ln(X)}$ be sufficient. Define the fiducial…

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…