I present a very simple example in which a full-support prior over distribution functions has an inconsistent posterior, which I believe has instructive value. The example is $Y_i \stackrel{\text{iid}}{\sim} P_0$ under the Bayesian hierarchical model $[Y_i \mid P] \stackrel{\text{iid}}{\sim} P$ and $P \sim \int \mbox{DP}(\alpha, H) \, \pi_\alpha(\alpha) \ d\alpha$ when $\pi_\alpha(\cdot)$ has exponential (or heavier) tails, where $\mbox{DP}(\alpha, H)$ denotes a Dirichlet process with concentration parameter $\alpha$ and mean $H(\cdot)$; on the other hand, consistency is obtained under a light-tailed prior. This unifies and generalizes a consistency result of Freedman and Diaconis (1983) with an inconsistency result described by Ferguson et al. (1992).
Consider the $\mathbb Z_2$ synchronization problem \[ \boldsymbol{Y} = \frac{\lambda}{\sqrt n} \theta\theta^{\top} + \boldsymbol{Z}, \] where $\theta$ is uniform on $\{-1,1\}^n$ and $\boldsymbol{Z}$ is an independent Gaussian Wigner matrix with off-diagonal variance one. We give a polynomial-time posterior sampling alg...
The proofs exploit cancellation at stationarity and smoothing using techniques from Malliavin calculus, outperforming previous upper bounds based on standard couplings and shows surprisingly that in high dimensions, deterministic discretizations can outperform their random counterparts.
Fan Chen, Sinho Chewi, Jian-Feng Lu et al.· 0 citations
We study online convex optimization with one unbiased stochastic subgradient per round and an unknown finite conditional $p$th noise moment, $1<p\le2$. For every fixed interval $I$ of length $n$ and comparator path with $\Lambda_I=1+P_I/D$, one learner achieves \[ E[Regret_I(u)]\le\min(GDn, C[GD\sqrt{n(\Lambda_I+\log^2...
We study finite-sample valid tests of the one-sided mean hypothesis $H_0:\mu\leq 1$ against $H_1:\mu>1$ for nonnegative random variables. To do so, we develop a leave-one-out dual certificate framework, where certain pointwise inequalities imply p-value validity under the conditional mean null $\mathbb{E}[X_i\mid\mathb...
Statistical precision and scientific relevance operate on different scales. In regular problems, sampling uncertainty contracts at rate $n^{-1/2}$, whereas the magnitude below which an effect is scientifically negligible may be fixed or may vary with information. Let $\Delta_n$ denote a relevance threshold and $I_0$ Fi...
Let \(\{Z_n\}_{n\geq0}\) be a supercritical branching random walk with deterministic rooted \(b\)-ary tree, \(b\geq2\), and symmetric displacements satisfying \(\lim_{x\to+\infty}x^{-\alpha}\log\Pp(X>x)=-\lambda \) with \(\alpha,\lambda>0\). Set $\overline Z_n(\cdot):=Z_n(\cdot)/Z_n(\mathbb{R}),~n\geq0.$ For a finite u...
Shu-Xiong Zhang, Ya-Ping Zhu· 0 citations
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