We study grammar induction with mildly context-sensitive grammars for unsupervised discontinuous parsing. Using the probabilistic linear context-free rewriting system (LCFRS) formalism, our approach fixes the rule structure in advance and focuses on parameter learning with maximum likelihood. To reduce the computational complexity of both parsing and parameter estimation, we restrict the grammar formalism to LCFRS-2 (i.e., binary LCFRS with fan-out two) and further discard rules that require O(n^6) time to parse, reducing inference to O(n^5). We find that using a large number of nonterminals is beneficial and thus make use of tensor decomposition-based rank-space dynamic programming with an embedding-based parameterization of rule probabilities to scale up the number of nonterminals. Experiments on German and Dutch show that our approach is able to induce linguistically meaningful trees with continuous and discontinuous structures

Within the classical Condorcet error model for collective binary decisions, we establish equivalence results between elections and markets, showing that the alternative that would be selected by weighed majority voting (under specific weighting schemes) corresponds to the alternative with highest price in the equilibrium of the market (under specific assumptions on the market type). This makes it possible to implement specific weighted majority elections, which are known to have superior truth-tracking performance, through information markets and, crucially, without needing to elicit voters' competences.

Data transformations are essential for broad applicability of parametric regression models. However, for Bayesian analysis, joint inference of the transformation and model parameters typically involves restrictive parametric transformations or nonparametric representations that are computationally inefficient and cumbersome for implementation and theoretical analysis, which limits their usability in practice. This paper introduces a simple, general, and efficient strategy for joint posterior inference of an unknown transformation and all regression model parameters. The proposed approach directly targets the posterior distribution of the transformation by linking it with the marginal distributions of the independent and dependent variables, and then deploys a Bayesian nonparametric model via the Bayesian bootstrap. Crucially, this approach delivers (1) joint posterior consistency under general conditions, including multiple model misspecifications, and (2) efficient Monte Carlo (not Markov chain Monte Carlo) inference for the transformation and all parameters for important special cases. These tools apply across a variety of data domains, including real-valued, integer-valued, compactly-supported, and positive data. Simulation studies and an empirical application demonstrate the effectiveness and efficiency of this strategy for semiparametric Bayesian analysis with linear models, quantile regression, and Gaussian processes.

We resolve difficulties in training and sampling from a discrete generative model by learning a smoothed energy function, sampling from the smoothed data manifold with Langevin Markov chain Monte Carlo (MCMC), and projecting back to the true data manifold with one-step denoising. Our Discrete Walk-Jump Sampling formalism combines the maximum likelihood training of an energy-based model and improved sample quality of a score-based model, while simplifying training and sampling by requiring only a single noise level. We evaluate the robustness of our approach on generative modeling of antibody proteins and introduce the distributional conformity score to benchmark protein generative models. By optimizing and sampling from our models for the proposed distributional conformity score, 97-100% of generated samples are successfully expressed and purified and 35% of functional designs show equal or improved binding affinity compared to known functional antibodies on the first attempt in a single round of laboratory experiments. We also report the first demonstration of long-run fast-mixing MCMC chains where diverse antibody protein classes are visited in a single MCMC chain.

High-dimensional deep neural network representations of images and concepts can be aligned to predict human annotations of diverse stimuli. However, such alignment requires the costly collection of behavioral responses, such that, in practice, the deep-feature spaces are only ever sparsely sampled. Here, we propose an active learning approach to adaptively sampling experimental stimuli to efficiently learn a Bayesian matrix factorization model with deep side information. We observe a significant efficiency gain over a passive baseline. Furthermore, with a sequential batched sampling strategy, the algorithm is applicable not only to small datasets collected from traditional laboratory experiments but also to settings where large-scale crowdsourced data collection is needed to accurately align the high-dimensional deep feature representations derived from pre-trained networks.

Large language models generate fluent texts and can follow natural language instructions to solve a wide range of tasks without task-specific training. Nevertheless, it is notoriously difficult to control their generation to satisfy the various constraints required by different applications. In this work, we present InstructCTG, a controlled text generation framework that incorporates different constraints by conditioning on natural language descriptions and demonstrations of the constraints. In particular, we first extract the underlying constraints of natural texts through a combination of off-the-shelf NLP tools and simple heuristics. We then verbalize the constraints into natural language instructions to form weakly supervised training data. By prepending natural language descriptions of the constraints and a few demonstrations, we fine-tune a pre-trained language model to incorporate various types of constraints. Compared to existing search-based or score-based methods, InstructCTG is more flexible to different constraint types and has a much smaller impact on the generation quality and speed because it does not modify the decoding procedure. Additionally, InstructCTG allows the model to adapt to new constraints without re-training through the use of few-shot task generalization and in-context learning abilities of instruction-tuned language models.

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices \citep{lauritzen2019maximum,slawski2015estimation} and even one observation for diagonally dominant M-matrices \citep{truell2021maximum}. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted $\ell_1$-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.

The solution of probabilistic inverse problems for which the corresponding forward problem is constrained by physical principles is challenging. This is especially true if the dimension of the inferred vector is large and the prior information about it is in the form of a collection of samples. In this work, a novel deep learning based approach is developed and applied to solving these types of problems. The approach utilizes samples of the inferred vector drawn from the prior distribution and a physics-based forward model to generate training data for a conditional Wasserstein generative adversarial network (cWGAN). The cWGAN learns the probability distribution for the inferred vector conditioned on the measurement and produces samples from this distribution. The cWGAN developed in this work differs from earlier versions in that its critic is required to be 1-Lipschitz with respect to both the inferred and the measurement vectors and not just the former. This leads to a loss term with the full (and not partial) gradient penalty. It is shown that this rather simple change leads to a stronger notion of convergence for the conditional density learned by the cWGAN and a more robust and accurate sampling strategy. Through numerical examples it is shown that this change also translates to better accuracy when solving inverse problems. The numerical examples considered include illustrative problems where the true distribution and/or statistics are known, and a more complex inverse problem motivated by applications in biomechanics.

The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to note that existing methods often struggle to identify the underlying equation accurately in the presence of noise. In this study, we present a new approach to discovering PDEs by combining variational Bayes and sparse linear regression. The problem of PDE discovery has been posed as a problem to learn relevant basis from a predefined dictionary of basis functions. To accelerate the overall process, a variational Bayes-based approach for discovering partial differential equations is proposed. To ensure sparsity, we employ a spike and slab prior. We illustrate the efficacy of our strategy in several examples, including Burgers, Korteweg-de Vries, Kuramoto Sivashinsky, wave equation, and heat equation (1D as well as 2D). Our method offers a promising avenue for discovering PDEs from data and has potential applications in fields such as physics, engineering, and biology.

Positron emission tomography (PET) is an important functional medical imaging technique often used in the evaluation of certain brain disorders, whose reconstruction problem is ill-posed. The vast majority of reconstruction methods in PET imaging, both iterative and deep learning, return a single estimate without quantifying the associated uncertainty. Due to ill-posedness and noise, a single solution can be misleading or inaccurate. Thus, providing a measure of uncertainty in PET image reconstruction can help medical practitioners in making critical decisions. This paper proposes a deep learning-based method for uncertainty quantification in PET image reconstruction via posterior sampling. The method is based on training a conditional generative adversarial network whose generator approximates sampling from the posterior in Bayesian inversion. The generator is conditioned on reconstruction from a low-dose PET scan obtained by a conventional reconstruction method and a high-quality magnetic resonance image and learned to estimate a corresponding standard-dose PET scan reconstruction. We show that the proposed model generates high-quality posterior samples and yields physically-meaningful uncertainty estimates.