We consider the problem of variational Bayesian inference in a latent variable model where a (possibly complex) observed stochastic process is governed by the solution of a latent stochastic differential equation (SDE). Motivated by the challenges that arise when trying to learn an (almost arbitrary) latent neural SDE from data, such as efficient gradient computation, we take a step back and study a specific subclass instead. In our case, the SDE evolves on a homogeneous latent space and is induced by stochastic dynamics of the corresponding (matrix) Lie group. In learning problems, SDEs on the unit n-sphere are arguably the most relevant incarnation of this setup. Notably, for variational inference, the sphere not only facilitates using a truly uninformative prior, but we also obtain a particularly simple and intuitive expression for the Kullback-Leibler divergence between the approximate posterior and prior process in the evidence lower bound. Experiments demonstrate that a latent SDE of the proposed type can be learned efficiently by means of an existing one-step geometric Euler-Maruyama scheme. Despite restricting ourselves to a less rich class of SDEs, we achieve competitive or even state-of-the-art results on various time series interpolation/classification problems.
Despite the promise of Neural Posterior Estimation (NPE) methods in astronomy, the adaptation of NPE into the routine inference workflow has been slow. We identify three critical issues: the need for custom featurizer networks tailored to the observed data, the inference inexactness, and the under-specification of physical forward models. To address the first two issues, we introduce a new framework and open-source software nbi (Neural Bayesian Inference), which supports both amortized and sequential NPE. First, nbi provides built-in "featurizer" networks with demonstrated efficacy on sequential data, such as light curve and spectra, thus obviating the need for this customization on the user end. Second, we introduce a modified algorithm SNPE-IS, which facilities asymptotically exact inference by using the surrogate posterior under NPE only as a proposal distribution for importance sampling. These features allow nbi to be applied off-the-shelf to astronomical inference problems involving light curves and spectra. We discuss how nbi may serve as an effective alternative to existing methods such as Nested Sampling. Our package is at https://github.com/kmzzhang/nbi.
The ice shelves buttressing the Antarctic ice sheet determine the rate of ice-discharge into the surrounding oceans. The geometry of ice shelves, and hence their buttressing strength, is determined by ice flow as well as by the local surface accumulation and basal melt rates, governed by atmospheric and oceanic conditions. Contemporary methods resolve one of these rates, but typically not both. Moreover, there is little information of how they changed in time. We present a new method to simultaneously infer the surface accumulation and basal melt rates averaged over decadal and centennial timescales. We infer the spatial dependence of these rates along flow line transects using internal stratigraphy observed by radars, using a kinematic forward model of internal stratigraphy. We solve the inverse problem using simulation-based inference (SBI). SBI performs Bayesian inference by training neural networks on simulations of the forward model to approximate the posterior distribution, allowing us to also quantify uncertainties over the inferred parameters. We demonstrate the validity of our method on a synthetic example, and apply it to Ekstr\"om Ice Shelf, Antarctica, for which newly acquired radar measurements are available. We obtain posterior distributions of surface accumulation and basal melt averaging over 42, 84, 146, and 188 years before 2022. Our results suggest stable atmospheric and oceanographic conditions over this period in this catchment of Antarctica. Use of observed internal stratigraphy can separate the effects of surface accumulation and basal melt, allowing them to be interpreted in a historical context of the last centuries and beyond.
This paper proposes an interpretation of RLAIF as Bayesian inference by introducing distilled Self-Critique (dSC), which refines the outputs of a LLM through a Gibbs sampler that is later distilled into a fine-tuned model. Only requiring synthetic data, dSC is exercised in experiments regarding safety, sentiment, and privacy control, showing it can be a viable and cheap alternative to align LLMs. Code released at \url{https://github.com/vicgalle/distilled-self-critique}.
We consider the problem of learning causal Directed Acyclic Graphs (DAGs) using combinations of observational and interventional experimental data. Current methods tailored to this setting assume that interventions either destroy parent-child relations of the intervened (target) nodes or only alter such relations without modifying the parent sets, even when the intervention targets are unknown. We relax this assumption by proposing a Bayesian method for causal discovery from general interventions, which allow for modifications of the parent sets of the unknown targets. Even in this framework, DAGs and general interventions may be identifiable only up to some equivalence classes. We provide graphical characterizations of such interventional Markov equivalence and devise compatible priors for Bayesian inference that guarantee score equivalence of indistinguishable structures. We then develop a Markov Chain Monte Carlo (MCMC) scheme to approximate the posterior distribution over DAGs, intervention targets and induced parent sets. Finally, we evaluate the proposed methodology on both simulated and real protein expression data.
Stein thinning is a promising algorithm proposed by (Riabiz et al., 2022) for post-processing outputs of Markov chain Monte Carlo (MCMC). The main principle is to greedily minimize the kernelized Stein discrepancy (KSD), which only requires the gradient of the log-target distribution, and is thus well-suited for Bayesian inference. The main advantages of Stein thinning are the automatic remove of the burn-in period, the correction of the bias introduced by recent MCMC algorithms, and the asymptotic properties of convergence towards the target distribution. Nevertheless, Stein thinning suffers from several empirical pathologies, which may result in poor approximations, as observed in the literature. In this article, we conduct a theoretical analysis of these pathologies, to clearly identify the mechanisms at stake, and suggest improved strategies. Then, we introduce the regularized Stein thinning algorithm to alleviate the identified pathologies. Finally, theoretical guarantees and extensive experiments show the high efficiency of the proposed algorithm. An implementation of regularized Stein thinning as the kernax library in python and JAX is available at https://gitlab.com/drti/kernax.
Variational inference (VI) is a method to approximate the computationally intractable posterior distributions that arise in Bayesian statistics. Typically, VI fits a simple parametric distribution to the target posterior by minimizing an appropriate objective such as the evidence lower bound (ELBO). In this work, we present a new approach to VI based on the principle of score matching, that if two distributions are equal then their score functions (i.e., gradients of the log density) are equal at every point on their support. With this, we develop score matching VI, an iterative algorithm that seeks to match the scores between the variational approximation and the exact posterior. At each iteration, score matching VI solves an inner optimization, one that minimally adjusts the current variational estimate to match the scores at a newly sampled value of the latent variables. We show that when the variational family is a Gaussian, this inner optimization enjoys a closed form solution, which we call Gaussian score matching VI (GSM-VI). GSM-VI is also a ``black box'' variational algorithm in that it only requires a differentiable joint distribution, and as such it can be applied to a wide class of models. We compare GSM-VI to black box variational inference (BBVI), which has similar requirements but instead optimizes the ELBO. We study how GSM-VI behaves as a function of the problem dimensionality, the condition number of the target covariance matrix (when the target is Gaussian), and the degree of mismatch between the approximating and exact posterior distribution. We also study GSM-VI on a collection of real-world Bayesian inference problems from the posteriorDB database of datasets and models. In all of our studies we find that GSM-VI is faster than BBVI, but without sacrificing accuracy. It requires 10-100x fewer gradient evaluations to obtain a comparable quality of approximation.
We present an exact Bayesian inference method for discrete statistical models, which can find exact solutions to a large class of discrete inference problems, even with infinite support and continuous priors. To express such models, we introduce a probabilistic programming language that supports discrete and continuous sampling, discrete observations, affine functions, (stochastic) branching, and conditioning on discrete events. Our key tool is probability generating functions: they provide a compact closed-form representation of distributions that are definable by programs, thus enabling the exact computation of posterior probabilities, expectation, variance, and higher moments. Our inference method is provably correct and fully automated in a tool called Genfer, which uses automatic differentiation (specifically, Taylor polynomials), but does not require computer algebra. Our experiments show that Genfer is often faster than the existing exact inference tools PSI, Dice, and Prodigy. On a range of real-world inference problems that none of these exact tools can solve, Genfer's performance is competitive with approximate Monte Carlo methods, while avoiding approximation errors.
Most applications of machine learning to classification assume a closed set of balanced classes. This is at odds with the real world, where class occurrence statistics often follow a long-tailed power-law distribution and it is unlikely that all classes are seen in a single sample. Nonparametric Bayesian models naturally capture this phenomenon, but have significant practical barriers to widespread adoption, namely implementation complexity and computational inefficiency. To address this, we present a method for extracting the inductive bias from a nonparametric Bayesian model and transferring it to an artificial neural network. By simulating data with a nonparametric Bayesian prior, we can metalearn a sequence model that performs inference over an unlimited set of classes. After training, this "neural circuit" has distilled the corresponding inductive bias and can successfully perform sequential inference over an open set of classes. Our experimental results show that the metalearned neural circuit achieves comparable or better performance than particle filter-based methods for inference in these models while being faster and simpler to use than methods that explicitly incorporate Bayesian nonparametric inference.
Sequential neural posterior estimation (SNPE) techniques have been recently proposed for dealing with simulation-based models with intractable likelihoods. Unlike approximate Bayesian computation, SNPE techniques learn the posterior from sequential simulation using neural network-based conditional density estimators by minimizing a specific loss function. The SNPE method proposed by Lueckmann et al. (2017) used a calibration kernel to boost the sample weights around the observed data, resulting in a concentrated loss function. However, the use of calibration kernels may increase the variances of both the empirical loss and its gradient, making the training inefficient. To improve the stability of SNPE, this paper proposes to use an adaptive calibration kernel and several variance reduction techniques. The proposed method greatly speeds up the process of training and provides a better approximation of the posterior than the original SNPE method and some existing competitors as confirmed by numerical experiments. We also managed to demonstrate the superiority of the proposed method for a high-dimensional model with a real-world dataset.
The Exponential Mechanism (ExpM), designed for private optimization, has been historically sidelined from use on continuous sample spaces, as it requires sampling from a generally intractable density, and, to a lesser extent, bounding the sensitivity of the objective function. Any differential privacy (DP) mechanism can be instantiated as ExpM, and ExpM poses an elegant solution for private machine learning (ML) that bypasses inherent inefficiencies of DPSGD. This paper seeks to operationalize ExpM for private optimization and ML by using an auxiliary Normalizing Flow (NF), an expressive deep network for density learning, to approximately sample from ExpM density. The method, ExpM+NF is an alternative to SGD methods for model training. We prove a sensitivity bound for the $\ell^2$ loss permitting ExpM use with any sampling method. To test feasibility, we present results on MIMIC-III health data comparing (non-private) SGD, DPSGD, and ExpM+NF training methods' accuracy and training time. We find that a model sampled from ExpM+NF is nearly as accurate as non-private SGD, more accurate than DPSGD, and ExpM+NF trains faster than Opacus' DPSGD implementation. Unable to provide a privacy proof for the NF approximation, we present empirical results to investigate privacy including the LiRA membership inference attack of Carlini et al. and the recent privacy auditing lower bound method of Steinke et al. Our findings suggest ExpM+NF provides more privacy than non-private SGD, but not as much as DPSGD, although many attacks are impotent against any model. Ancillary benefits of this work include pushing the SOTA of privacy and accuracy on MIMIC-III healthcare data, exhibiting the use of ExpM+NF for Bayesian inference, showing the limitations of empirical privacy auditing in practice, and providing several privacy theorems applicable to distribution learning.
We present USLR, a computational framework for longitudinal registration of brain MRI scans to estimate nonlinear image trajectories that are smooth across time, unbiased to any timepoint, and robust to imaging artefacts. It operates on the Lie algebra parameterisation of spatial transforms (which is compatible with rigid transforms and stationary velocity fields for nonlinear deformation) and takes advantage of log-domain properties to solve the problem using Bayesian inference. USRL estimates rigid and nonlinear registrations that: (i) bring all timepoints to an unbiased subject-specific space; and (i) compute a smooth trajectory across the imaging time-series. We capitalise on learning-based registration algorithms and closed-form expressions for fast inference. A use-case Alzheimer's disease study is used to showcase the benefits of the pipeline in multiple fronts, such as time-consistent image segmentation to reduce intra-subject variability, subject-specific prediction or population analysis using tensor-based morphometry. We demonstrate that such approach improves upon cross-sectional methods in identifying group differences, which can be helpful in detecting more subtle atrophy levels or in reducing sample sizes in clinical trials. The code is publicly available in https://github.com/acasamitjana/uslr
Hyperdimensional computing (HDC) is an increasingly popular computing paradigm with immense potential for future intelligent applications. Although the main ideas already took form in the 1990s, HDC recently gained significant attention, especially in the field of machine learning and data science. Next to efficiency, interoperability and explainability, HDC offers attractive properties for generalization as it can be seen as an attempt to combine connectionist ideas from neural networks with symbolic aspects. In recent work, we introduced the hyperdimensional transform, revealing deep theoretical foundations for representing functions and distributions as high-dimensional holographic vectors. Here, we present the power of the hyperdimensional transform to a broad data science audience. We use the hyperdimensional transform as a theoretical basis and provide insight into state-of-the-art HDC approaches for machine learning. We show how existing algorithms can be modified and how this transform can lead to a novel, well-founded toolbox. Next to the standard regression and classification tasks of machine learning, our discussion includes various aspects of statistical modelling, such as representation, learning and deconvolving distributions, sampling, Bayesian inference, and uncertainty estimation.
Transformer neural networks can exhibit a surprising capacity for in-context learning (ICL) despite not being explicitly trained for it. Prior work has provided a deeper understanding of how ICL emerges in transformers, e.g. through the lens of mechanistic interpretability, Bayesian inference, or by examining the distributional properties of training data. However, in each of these cases, ICL is treated largely as a persistent phenomenon; namely, once ICL emerges, it is assumed to persist asymptotically. Here, we show that the emergence of ICL during transformer training is, in fact, often transient. We train transformers on synthetic data designed so that both ICL and in-weights learning (IWL) strategies can lead to correct predictions. We find that ICL first emerges, then disappears and gives way to IWL, all while the training loss decreases, indicating an asymptotic preference for IWL. The transient nature of ICL is observed in transformers across a range of model sizes and datasets, raising the question of how much to "overtrain" transformers when seeking compact, cheaper-to-run models. We find that L2 regularization may offer a path to more persistent ICL that removes the need for early stopping based on ICL-style validation tasks. Finally, we present initial evidence that ICL transience may be caused by competition between ICL and IWL circuits.
There is a considerable body of work on data cleaning which employs various principles to rectify erroneous data and transform a dirty dataset into a cleaner one. One of prevalent approaches is probabilistic methods, including Bayesian methods. However, existing probabilistic methods often assume a simplistic distribution (e.g., Gaussian distribution), which is frequently underfitted in practice, or they necessitate experts to provide a complex prior distribution (e.g., via a programming language). This requirement is both labor-intensive and costly, rendering these methods less suitable for real-world applications. In this paper, we propose BClean, a Bayesian Cleaning system that features automatic Bayesian network construction and user interaction. We recast the data cleaning problem as a Bayesian inference that fully exploits the relationships between attributes in the observed dataset and any prior information provided by users. To this end, we present an automatic Bayesian network construction method that extends a structure learning-based functional dependency discovery method with similarity functions to capture the relationships between attributes. Furthermore, our system allows users to modify the generated Bayesian network in order to specify prior information or correct inaccuracies identified by the automatic generation process. We also design an effective scoring model (called the compensative scoring model) necessary for the Bayesian inference. To enhance the efficiency of data cleaning, we propose several approximation strategies for the Bayesian inference, including graph partitioning, domain pruning, and pre-detection. By evaluating on both real-world and synthetic datasets, we demonstrate that BClean is capable of achieving an F-measure of up to 0.9 in data cleaning, outperforming existing Bayesian methods by 2% and other data cleaning methods by 15%.
We present a systematic study of conditional triangular transport maps in function spaces from the perspective of optimal transportation and with a view towards amortized Bayesian inference. More specifically, we develop a theory of constrained optimal transport problems that describe block-triangular Monge maps that characterize conditional measures along with their Kantorovich relaxations. This generalizes the theory of optimal triangular transport to separable infinite-dimensional function spaces with general cost functions. We further tailor our results to the case of Bayesian inference problems and obtain regularity estimates on the conditioning maps from the prior to the posterior. Finally, we present numerical experiments that demonstrate the computational applicability of our theoretical results for amortized and likelihood-free inference of functional parameters.
Laplace's method approximates a target density with a Gaussian distribution at its mode. It is computationally efficient and asymptotically exact for Bayesian inference due to the Bernstein-von Mises theorem, but for complex targets and finite-data posteriors it is often too crude an approximation. A recent generalization of the Laplace Approximation transforms the Gaussian approximation according to a chosen Riemannian geometry providing a richer approximation family, while still retaining computational efficiency. However, as shown here, its properties depend heavily on the chosen metric, indeed the metric adopted in previous work results in approximations that are overly narrow as well as being biased even at the limit of infinite data. We correct this shortcoming by developing the approximation family further, deriving two alternative variants that are exact at the limit of infinite data, extending the theoretical analysis of the method, and demonstrating practical improvements in a range of experiments.
To enable closed form conditioning, a common assumption in Gaussian process (GP) regression is independent and identically distributed Gaussian observation noise. This strong and simplistic assumption is often violated in practice, which leads to unreliable inferences and uncertainty quantification. Unfortunately, existing methods for robustifying GPs break closed-form conditioning, which makes them less attractive to practitioners and significantly more computationally expensive. In this paper, we demonstrate how to perform provably robust and conjugate Gaussian process (RCGP) regression at virtually no additional cost using generalised Bayesian inference. RCGP is particularly versatile as it enables exact conjugate closed form updates in all settings where standard GPs admit them. To demonstrate its strong empirical performance, we deploy RCGP for problems ranging from Bayesian optimisation to sparse variational Gaussian processes.
Bayesian inference with deep generative prior has received considerable interest for solving imaging inverse problems in many scientific and engineering fields. The selection of the prior distribution is learned from, and therefore an important representation learning of, available prior measurements. The SA-Roundtrip, a novel deep generative prior, is introduced to enable controlled sampling generation and identify the data's intrinsic dimension. This prior incorporates a self-attention structure within a bidirectional generative adversarial network. Subsequently, Bayesian inference is applied to the posterior distribution in the low-dimensional latent space using the Hamiltonian Monte Carlo with preconditioned Crank-Nicolson (HMC-pCN) algorithm, which is proven to be ergodic under specific conditions. Experiments conducted on computed tomography (CT) reconstruction with the MNIST and TomoPhantom datasets reveal that the proposed method outperforms state-of-the-art comparisons, consistently yielding a robust and superior point estimator along with precise uncertainty quantification.
We present two neural network approaches that approximate the solutions of static and dynamic $\unicode{x1D450}\unicode{x1D45C}\unicode{x1D45B}\unicode{x1D451}\unicode{x1D456}\unicode{x1D461}\unicode{x1D456}\unicode{x1D45C}\unicode{x1D45B}\unicode{x1D44E}\unicode{x1D459}\unicode{x0020}\unicode{x1D45C}\unicode{x1D45D}\unicode{x1D461}\unicode{x1D456}\unicode{x1D45A}\unicode{x1D44E}\unicode{x1D459}\unicode{x0020}\unicode{x1D461}\unicode{x1D45F}\unicode{x1D44E}\unicode{x1D45B}\unicode{x1D460}\unicode{x1D45D}\unicode{x1D45C}\unicode{x1D45F}\unicode{x1D461}$ (COT) problems. Both approaches enable conditional sampling and conditional density estimation, which are core tasks in Bayesian inference$\unicode{x2013}$particularly in the simulation-based ($\unicode{x201C}$likelihood-free$\unicode{x201D}$) setting. Our methods represent the target conditional distribution as a transformation of a tractable reference distribution. Obtaining such a transformation, chosen here to be an approximation of the COT map, is computationally challenging even in moderate dimensions. To improve scalability, our numerical algorithms use neural networks to parameterize candidate maps and further exploit the structure of the COT problem. Our static approach approximates the map as the gradient of a partially input-convex neural network. It uses a novel numerical implementation to increase computational efficiency compared to state-of-the-art alternatives. Our dynamic approach approximates the conditional optimal transport via the flow map of a regularized neural ODE; compared to the static approach, it is slower to train but offers more modeling choices and can lead to faster sampling. We demonstrate both algorithms numerically, comparing them with competing state-of-the-art approaches, using benchmark datasets and simulation-based Bayesian inverse problems.