Mean Estimation from One-Bit Measurements

We consider the problem of estimating the mean of a symmetric log-concave distribution under the constraint that only a single bit per sample from this distribution is available to the estimator. We study the mean squared error as a function of the sample size (and hence the number of bits). We consider three settings: first, a centralized setting, where an encoder may release $n$ bits given a sample of size $n$, and for which there is no asymptotic penalty for quantization; second, an adaptive setting in which each bit is a function of the current observation and previously recorded bits, where we show that the optimal relative efficiency compared to the sample mean is precisely the efficiency of the median; lastly, we show that in a distributed setting where each bit is only a function of a local sample, no estimator can achieve optimal efficiency uniformly over the parameter space. We additionally complement our results in the adaptive setting by showing that \emph{one} round of adaptivity is sufficient to achieve optimal mean-square error

Mean Estimation from Adaptive One-bit Measurements

We consider the problem of estimating the mean of a normal distribution under the following constraint: the estimator can access only a single bit from each sample from this distribution. We study the squared error risk in this estimation as a function of the number of samples and one-bit measurements $n$. We consider an adaptive estimation setting where the single-bit sent at step $n$ is a function of both the new sample and the previous $n-1$ acquired bits. For this setting, we show that no estimator can attain asymptotic mean squared error smaller than $\pi/(2n)+O(n^{-2})$ times the variance. In other words, one-bit restriction increases the number of samples required for a prescribed accuracy of estimation by a factor of at least $\pi/2$ compared to the unrestricted case. In addition, we provide an explicit estimator that attains this asymptotic error, showing that, rather surprisingly, only $\pi/2$ times more samples are required in order to attain estimation performance equivalent to the unrestricted case

The minimax risk in testing the histogram of discrete distributions for uniformity under missing ball alternatives

We consider the problem of testing the fit of a discrete sample of items from many categories to the uniform distribution over the categories. As a class of alternative hypotheses, we consider the removal of an $\ell_p$ ball of radius $\epsilon$ around the uniform rate sequence for $p \leq 2$. We deliver a sharp characterization of the asymptotic minimax risk when $\epsilon \to 0$ as the number of samples and number of dimensions go to infinity, for testing based on the occurrences' histogram (number of absent categories, singletons, collisions, ...). For example, for $p=1$ and in the limit of a small expected number of samples $n$ compared to the number of categories $N$ (aka "sub-linear" regime), the minimax risk $R^*_\epsilon$ asymptotes to $2 \bar{\Phi}\left(n \epsilon^2/\sqrt{8N}\right)$, with $\bar{\Phi}(x)$ the normal survival function. Empirical studies over a range of problem parameters show that this estimate is accurate in finite samples, and that our test is significantly better than the chisquared test or a test that only uses collisions. Our analysis is based on the asymptotic normality of histogram ordinates, the equivalence between the minimax setting to a Bayesian one, and the reduction of a multi-dimensional optimization problem to a one-dimensional problem

Separating the Human Touch from AI-Generated Text using Higher Criticism: An Information-Theoretic Approach

We propose a method to determine whether a given article was entirely written by a generative language model versus an alternative situation in which the article includes some significant edits by a different author, possibly a human. Our process involves many perplexity tests for the origin of individual sentences or other text atoms, combining these multiple tests using Higher Criticism (HC). As a by-product, the method identifies parts suspected to be edited. The method is motivated by the convergence of the log-perplexity to the cross-entropy rate and by a statistical model for edited text saying that sentences are mostly generated by the language model, except perhaps for a few sentences that might have originated via a different mechanism. We demonstrate the effectiveness of our method using real data and analyze the factors affecting its success. This analysis raises several interesting open challenges whose resolution may improve the method's effectiveness

Higher Criticism for Discriminating Word-Frequency Tables and Testing Authorship

We adapt the Higher Criticism (HC) goodness-of-fit test to measure closeness between word-frequency tables. We apply this measure to authorship attribution challenges, where the goal is to identify the author of a document using other documents whose authorship is known. The method is simple yet performs well without handcrafting and tuning; reporting accuracy at the state of the art level in various current challenges. As an inherent side effect, the HC calculation identifies a subset of discriminating words. In practice, the identified words have low variance across documents belonging to a corpus of homogeneous authorship. We conclude that in comparing the similarity of a new document and a corpus of a single author, HC is mostly affected by words characteristic of the author and is relatively unaffected by topic structure.Comment: under review (AOAS

Unification of Rare/Weak Detection Models using Moderate Deviations Analysis and Log-Chisquared P-values

Rare/Weak models for multiple hypothesis testing assume that only a small proportion of the tested hypotheses concern non-null effects and the individual effects are only moderately large, so that they generally do not stand out individually, for example in a Bonferroni analysis. Such rare/weak models have been studied in quite a few settings, for example in some cases studies focused on underlying Gaussian means model for the hypotheses being tested; in some others, Poisson. It seems not to have been noticed before that such seemingly different models have asymptotically the following common structure: Summarizing the evidence each test provides by the negative logarithm of its P-value, previous rare/weak model settings are asymptotically equivalent to detection where most negative log P-values have a standard exponential distribution but a small fraction of the P-values might have an alternative distribution which is moderately larger; we do not know which individual tests those might be, or even if there are any such. Moreover, the alternative distribution is noncentral chisquared on one degree of freedom. We characterize the asymptotic performance of global tests combining these P-values in terms of the chisquared mixture parameters: the scaling parameters controlling heteroscedasticity, the non-centrality parameter describing the effect size whenever it exists, and the parameter controlling the rarity of the non-null effects. Specifically, in a phase space involving the last two parameters, we derive a region where all tests are asymptotically powerless. Outside of this region, the Berk-Jones and the Higher Criticism tests have maximal power. Inference techniques based on the minimal P-value, false-discovery rate controlling, and Fisher's test have sub-optimal asymptotic phase diagrams. We provide various examples for multiple testing problems of the said common structure.Comment: 32 pages, 2 figures, submitte