Existing methods for graph out-of-distribution (OOD) generalization primarily rely on empirical studies on synthetic datasets. Such approaches tend to overemphasize the causal relationships between invariant sub-graphs and labels, thereby neglecting the non-negligible role of environment in real-world scenarios. In contrast to previous studies that impose rigid independence assumptions on environments and invariant sub-graphs, this paper presents the theorems of environment-label dependency and mutable rationale invariance, where the former characterizes the usefulness of environments in determining graph labels while the latter refers to the mutable importance of graph rationales. Based on analytic investigations, a novel variational inference based method named ``Probability Dependency on Environments and Rationales for OOD Graphs on Real-world Data'' (DEROG) is introduced. To alleviate the adverse effect of unknown prior knowledge on environments and rationales, DEROG utilizes generalized Bayesian inference. Further, DEROG employs an EM-based algorithm for optimization. Finally, extensive experiments on real-world datasets under different distribution shifts are conducted to show the superiority of DEROG. Our code is publicly available at https://anonymous.4open.science/r/DEROG-536B.
A new variational inference method, SPH-ParVI, based on smoothed particle hydrodynamics (SPH), is proposed for sampling partially known densities (e.g. up to a constant) or sampling using gradients. SPH-ParVI simulates the flow of a fluid under external effects driven by the target density; transient or steady state of the fluid approximates the target density. The continuum fluid is modelled as an interacting particle system (IPS) via SPH, where each particle carries smoothed properties, interacts and evolves as per the Navier-Stokes equations. This mesh-free, Lagrangian simulation method offers fast, flexible, scalable and deterministic sampling and inference for a class of probabilistic models such as those encountered in Bayesian inference and generative modelling.
We present MELLE, a novel continuous-valued token based language modeling approach for text-to-speech synthesis (TTS). MELLE autoregressively generates continuous mel-spectrogram frames directly from text condition, bypassing the need for vector quantization, which is typically designed for audio compression and sacrifices fidelity compared to continuous representations. Specifically, (i) instead of cross-entropy loss, we apply regression loss with a proposed spectrogram flux loss function to model the probability distribution of the continuous-valued tokens; (ii) we have incorporated variational inference into MELLE to facilitate sampling mechanisms, thereby enhancing the output diversity and model robustness. Experiments demonstrate that, compared to the two-stage codec language model VALL-E and its variants, the single-stage MELLE mitigates robustness issues by avoiding the inherent flaws of sampling vector-quantized codes, achieves superior performance across multiple metrics, and, most importantly, offers a more streamlined paradigm. The demos of our work are provided at https://aka.ms/melle.
We develop a novel measurement framework based on a Gaussian process coregionalization model to address a long-lasting debate in psychometrics: whether psychological features like personality share a common structure across the population, vary uniquely for individuals, or some combination. We propose the idiographic personality Gaussian process (IPGP) framework, an intermediate model that accommodates both shared trait structure across a population and "idiographic" deviations for individuals. IPGP leverages the Gaussian process coregionalization model to handle the grouped nature of battery responses, but adjusted to non-Gaussian ordinal data. We further exploit stochastic variational inference for efficient latent factor estimation required for idiographic modeling at scale. Using synthetic and real data, we show that IPGP improves both prediction of actual responses and estimation of individualized factor structures relative to existing benchmarks. In a third study, we show that IPGP also identifies unique clusters of personality taxonomies in real-world data, displaying great potential in advancing individualized approaches to psychological diagnosis and treatment.
Bayesian causal discovery offers the power to quantify epistemic uncertainties among a broad range of structurally diverse causal theories potentially explaining the data, represented in forms of directed acyclic graphs (DAGs). However, existing methods struggle with efficient DAG sampling due to the complex acyclicity constraint. In this study, we propose a scalable Bayesian approach to effectively learn the posterior distribution over causal graphs given observational data thanks to the ability to generate DAGs without explicitly enforcing acyclicity. Specifically, we introduce a novel differentiable DAG sampling method that can generate a valid acyclic causal graph by mapping an unconstrained distribution of implicit topological orders to a distribution over DAGs. Given this efficient DAG sampling scheme, we are able to model the posterior distribution over causal graphs using a simple variational distribution over a continuous domain, which can be learned via the variational inference framework. Extensive empirical experiments on both simulated and real datasets demonstrate the superior performance of the proposed model compared to several state-of-the-art baselines.
Best-of-N (BoN) is a popular and effective algorithm for aligning language models to human preferences. The algorithm works as follows: at inference time, N samples are drawn from the language model, and the sample with the highest reward, as judged by a reward model, is returned as the output. Despite its effectiveness, BoN is computationally expensive; it reduces sampling throughput by a factor of N. To make BoN more efficient at inference time, one strategy is to fine-tune the language model to mimic what BoN does during inference. To achieve this, we derive the distribution induced by the BoN algorithm. We then propose to fine-tune the language model to minimize backward KL divergence to the BoN distribution. Our approach is analogous to mean-field variational inference and, thus, we term it variational BoN (vBoN). To the extent this fine-tuning is successful and we end up with a good approximation, we have reduced the inference cost by a factor of N. Our experiments on a controlled generation task suggest that while variational BoN is not as effective as BoN in aligning language models, it is close to BoN performance as vBoN appears more often on the Pareto frontier of reward and KL divergence compared to models trained with KL-constrained RL objective.
We propose a randomized physics-informed neural network (PINN) or rPINN method for uncertainty quantification in inverse partial differential equation (PDE) problems with noisy data. This method is used to quantify uncertainty in the inverse PDE PINN solutions. Recently, the Bayesian PINN (BPINN) method was proposed, where the posterior distribution of the PINN parameters was formulated using the Bayes' theorem and sampled using approximate inference methods such as the Hamiltonian Monte Carlo (HMC) and variational inference (VI) methods. In this work, we demonstrate that HMC fails to converge for non-linear inverse PDE problems. As an alternative to HMC, we sample the distribution by solving the stochastic optimization problem obtained by randomizing the PINN loss function. The effectiveness of the rPINN method is tested for linear and non-linear Poisson equations, and the diffusion equation with a high-dimensional space-dependent diffusion coefficient. The rPINN method provides informative distributions for all considered problems. For the linear Poisson equation, HMC and rPINN produce similar distributions, but rPINN is on average 27 times faster than HMC. For the non-linear Poison and diffusion equations, the HMC method fails to converge because a single HMC chain cannot sample multiple modes of the posterior distribution of the PINN parameters in a reasonable amount of time.
Knowledge graph (KG) reasoning is a task that aims to predict unknown facts based on known factual samples. Reasoning methods can be divided into two categories: rule-based methods and KG-embedding based methods. The former possesses precise reasoning capabilities but finds it challenging to reason efficiently over large-scale knowledge graphs. While gaining the ability to reason over large-scale knowledge graphs, the latter sacrifices reasoning accuracy. This paper aims to design a reasoning framework called Neural Probabilistic Logic Learning(NPLL) that achieves accurate reasoning on knowledge graphs. Our approach introduces a scoring module that effectively enhances the expressive power of embedding networks, striking a balance between model simplicity and reasoning capabilities. We improve the interpretability of the model by incorporating a Markov Logic Network based on variational inference. We empirically evaluate our approach on several benchmark datasets, and the experimental results validate that our method substantially enhances the accuracy and quality of the reasoning results.
Group Equivariant CNNs (G-CNNs) have shown promising efficacy in various tasks, owing to their ability to capture hierarchical features in an equivariant manner. However, their equivariance is fixed to the symmetry of the whole group, limiting adaptability to diverse partial symmetries in real-world datasets, such as limited rotation symmetry of handwritten digit images and limited color-shift symmetry of flower images. Recent efforts address this limitation, one example being Partial G-CNN which restricts the output group space of convolution layers to break full equivariance. However, such an approach still fails to adjust equivariance levels across data. In this paper, we propose a novel approach, Variational Partial G-CNN (VP G-CNN), to capture varying levels of partial equivariance specific to each data instance. VP G-CNN redesigns the distribution of the output group elements to be conditioned on input data, leveraging variational inference to avoid overfitting. This enables the model to adjust its equivariance levels according to the needs of individual data points. Additionally, we address training instability inherent in discrete group equivariance models by redesigning the reparametrizable distribution. We demonstrate the effectiveness of VP G-CNN on both toy and real-world datasets, including MNIST67-180, CIFAR10, ColorMNIST, and Flowers102. Our results show robust performance, even in uncertainty metrics.
Semi-implicit variational inference (SIVI) enriches the expressiveness of variational families by utilizing a kernel and a mixing distribution to hierarchically define the variational distribution. Existing SIVI methods parameterize the mixing distribution using implicit distributions, leading to intractable variational densities. As a result, directly maximizing the evidence lower bound (ELBO) is not possible, so they resort to one of the following: optimizing bounds on the ELBO, employing costly inner-loop Markov chain Monte Carlo runs, or solving minimax objectives. In this paper, we propose a novel method for SIVI called Particle Variational Inference (PVI) which employs empirical measures to approximate the optimal mixing distributions characterized as the minimizer of a free energy functional. PVI arises naturally as a particle approximation of a Euclidean--Wasserstein gradient flow and, unlike prior works, it directly optimizes the ELBO whilst making no parametric assumption about the mixing distribution. Our empirical results demonstrate that PVI performs favourably compared to other SIVI methods across various tasks. Moreover, we provide a theoretical analysis of the behaviour of the gradient flow of a related free energy functional: establishing the existence and uniqueness of solutions as well as propagation of chaos results.
Most scientific machine learning (SciML) applications of neural networks involve hundreds to thousands of parameters, and hence, uncertainty quantification for such models is plagued by the curse of dimensionality. Using physical applications, we show that $L_0$ sparsification prior to Stein variational gradient descent ($L_0$+SVGD) is a more robust and efficient means of uncertainty quantification, in terms of computational cost and performance than the direct application of SGVD or projected SGVD methods. Specifically, $L_0$+SVGD demonstrates superior resilience to noise, the ability to perform well in extrapolated regions, and a faster convergence rate to an optimal solution.
Deep Gaussian Processes (DGPs) leverage a compositional structure to model non-stationary processes. DGPs typically rely on local inducing point approximations across intermediate GP layers. Recent advances in DGP inference have shown that incorporating global Fourier features from the Reproducing Kernel Hilbert Space (RKHS) can enhance the DGPs' capability to capture complex non-stationary patterns. This paper extends the use of these features to compositional GPs involving linear transformations. In particular, we introduce Ordinary Differential Equation(ODE)--based RKHS Fourier features that allow for adaptive amplitude and phase modulation through convolution operations. This convolutional formulation relates our work to recently proposed deep latent force models, a multi-layer structure designed for modelling nonlinear dynamical systems. By embedding these adjustable RKHS Fourier features within a doubly stochastic variational inference framework, our model exhibits improved predictive performance across various regression tasks.
The Multi-Output Gaussian Process is is a popular tool for modelling data from multiple sources. A typical choice to build a covariance function for a MOGP is the Linear Model of Coregionalization (LMC) which parametrically models the covariance between outputs. The Latent Variable MOGP (LV-MOGP) generalises this idea by modelling the covariance between outputs using a kernel applied to latent variables, one per output, leading to a flexible MOGP model that allows efficient generalization to new outputs with few data points. Computational complexity in LV-MOGP grows linearly with the number of outputs, which makes it unsuitable for problems with a large number of outputs. In this paper, we propose a stochastic variational inference approach for the LV-MOGP that allows mini-batches for both inputs and outputs, making computational complexity per training iteration independent of the number of outputs.
In this paper, we propose a novel method of model-based time series clustering with mixtures of general state space models (MSSMs). Each component of MSSMs is associated with each cluster. An advantage of the proposed method is that it enables the use of time series models appropriate to the specific time series. This not only improves clustering and prediction accuracy but also enhances the interpretability of the estimated parameters. The parameters of the MSSMs are estimated using stochastic variational inference, a subtype of variational inference. The proposed method estimates the latent variables of an arbitrary state space model by using neural networks with a normalizing flow as a variational estimator. The number of clusters can be estimated using the Bayesian information criterion. In addition, to prevent MSSMs from converging to the local optimum, we propose several optimization tricks, including an additional penalty term called entropy annealing. To our best knowledge, the proposed method is the first computationally feasible one for time series clustering based on general (possibly nonlinear, non-Gaussian) state space models. Experiments on simulated datasets show that the proposed method is effective for clustering, parameter estimation, and estimating the number of clusters.
Bayesian inference has predominantly relied on the Markov chain Monte Carlo (MCMC) algorithm for many years. However, MCMC is computationally laborious, especially for complex phylogenetic models of time trees. This bottleneck has led to the search for alternatives, such as variational Bayes, which can scale better to large datasets. In this paper, we introduce torchtree, a framework written in Python that allows developers to easily implement rich phylogenetic models and algorithms using a fixed tree topology. One can either use automatic differentiation, or leverage torchtree's plug-in system to compute gradients analytically for model components for which automatic differentiation is slow. We demonstrate that the torchtree variational inference framework performs similarly to BEAST in terms of speed and approximation accuracy. Furthermore, we explore the use of the forward KL divergence as an optimizing criterion for variational inference, which can handle discontinuous and non-differentiable models. Our experiments show that inference using the forward KL divergence tends to be faster per iteration compared to the evidence lower bound (ELBO) criterion, although the ELBO-based inference may converge faster in some cases. Overall, torchtree provides a flexible and efficient framework for phylogenetic model development and inference using PyTorch.
Graph learning models have been widely deployed in collaborative filtering (CF) based recommendation systems. Due to the issue of data sparsity, the graph structure of the original input lacks potential positive preference edges, which significantly reduces the performance of recommendations. In this paper, we study how to enhance the graph structure for CF more effectively, thereby optimizing the representation of graph nodes. Previous works introduced matrix completion techniques into CF, proposing the use of either stochastic completion methods or superficial structure completion to address this issue. However, most of these approaches employ random numerical filling that lack control over noise perturbations and limit the in-depth exploration of higher-order interaction features of nodes, resulting in biased graph representations. In this paper, we propose an Amplify Graph Learning framework based on Sparsity Completion (called AGL-SC). First, we utilize graph neural network to mine direct interaction features between user and item nodes, which are used as the inputs of the encoder. Second, we design a factorization-based method to mine higher-order interaction features. These features serve as perturbation factors in the latent space of the hidden layer to facilitate generative enhancement. Finally, by employing the variational inference, the above multi-order features are integrated to implement the completion and enhancement of missing graph structures. We conducted benchmark and strategy experiments on four real-world datasets related to recommendation tasks. The experimental results demonstrate that AGL-SC significantly outperforms the state-of-the-art methods.
Filtering - the task of estimating the conditional distribution for states of a dynamical system given partial and noisy observations - is important in many areas of science and engineering, including weather and climate prediction. However, the filtering distribution is generally intractable to obtain for high-dimensional, nonlinear systems. Filters used in practice, such as the ensemble Kalman filter (EnKF), provide biased probabilistic estimates for nonlinear systems and have numerous tuning parameters. Here, we present a framework for learning a parameterized analysis map - the transformation that takes samples from a forecast distribution, and combines with an observation, to update the approximate filtering distribution - using variational inference. In principle this can lead to a better approximation of the filtering distribution, and hence smaller bias. We show that this methodology can be used to learn the gain matrix, in an affine analysis map, for filtering linear and nonlinear dynamical systems; we also study the learning of inflation and localization parameters for an EnKF. The framework developed here can also be used to learn new filtering algorithms with more general forms for the analysis map.
In this work, we demonstrate the Empirical Bayes approach to learning a Dynamic Bayesian Network. By starting with several point estimates of structure and weights, we can use a data-driven prior to subsequently obtain a model to quantify uncertainty. This approach uses a recent development of Generalized Variational Inference, and indicates the potential of sampling the uncertainty of a mixture of DAG structures as well as a parameter posterior.
Multiple types of inference are available for probabilistic graphical models, e.g., marginal, maximum-a-posteriori, and even marginal maximum-a-posteriori. Which one do researchers mean when they talk about "planning as inference"? There is no consistency in the literature, different types are used, and their ability to do planning is further entangled with specific approximations or additional constraints. In this work we use the variational framework to show that, just like all commonly used types of inference correspond to different weightings of the entropy terms in the variational problem, planning corresponds exactly to a different set of weights. This means that all the tricks of variational inference are readily applicable to planning. We develop an analogue of loopy belief propagation that allows us to perform approximate planning in factored-state Markov decisions processes without incurring intractability due to the exponentially large state space. The variational perspective shows that the previous types of inference for planning are only adequate in environments with low stochasticity, and allows us to characterize each type by its own merits, disentangling the type of inference from the additional approximations that its practical use requires. We validate these results empirically on synthetic MDPs and tasks posed in the International Planning Competition.
When studying ecosystems, hierarchical trees are often used to organize entities based on proximity criteria, such as the taxonomy in microbiology, social classes in geography, or product types in retail businesses, offering valuable insights into entity relationships. Despite their significance, current count-data models do not leverage this structured information. In particular, the widely used Poisson log-normal (PLN) model, known for its ability to model interactions between entities from count data, lacks the possibility to incorporate such hierarchical tree structures, limiting its applicability in domains characterized by such complexities. To address this matter, we introduce the PLN-Tree model as an extension of the PLN model, specifically designed for modeling hierarchical count data. By integrating structured variational inference techniques, we propose an adapted training procedure and establish identifiability results, enhancing both theoretical foundations and practical interpretability. Experiments on synthetic datasets and human gut microbiome data highlight generative improvements when using PLN-Tree, demonstrating the practical interest of knowledge graphs like the taxonomy in microbiome modeling. Additionally, we present a proof-of-concept implication of the identifiability results by illustrating the practical benefits of using identifiable features for classification tasks, showcasing the versatility of the framework.