Efficient Bayesian inference remains a computational challenge in hierarchical models. Simulation-based approaches such as Markov Chain Monte Carlo methods are still popular but have a large computational cost. When dealing with the large class of Latent Gaussian Models, the INLA methodology embedded in the R-INLA software provides accurate Bayesian inference by computing deterministic mixture representation to approximate the joint posterior, from which marginals are computed. The INLA approach has from the beginning been targeting to approximate univariate posteriors. In this paper we lay out the development foundation of the tools for also providing joint approximations for subsets of the latent field. These approximations inherit Gaussian copula structure and additionally provide corrections for skewness. The same idea is carried forward also to sampling from the mixture representation, which we now can adjust for skewness.
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Variational inference (VI) is a technique to approximate difficult to compute posteriors by optimization. In contrast to MCMC, VI scales to many observations. In the case of complex posteriors, however, state-of-the-art VI approaches often yield unsatisfactory posterior approximations. This paper presents Bernstein flow variational inference (BF-VI), a robust and easy-to-use method, flexible enough to approximate complex multivariate posteriors. BF-VI combines ideas from normalizing flows and Bernstein polynomial-based transformation models. In benchmark experiments, we compare BF-VI solutions with exact posteriors, MCMC solutions, and state-of-the-art VI methods including normalizing flow based VI. We show for low-dimensional models that BF-VI accurately approximates the true posterior; in higher-dimensional models, BF-VI outperforms other VI methods. Further, we develop with BF-VI a Bayesian model for the semi-structured Melanoma challenge data, combining a CNN model part for image data with an interpretable model part for tabular data, and demonstrate for the first time how the use of VI in semi-structured models.
Off-Policy Evaluation (OPE) serves as one of the cornerstones in Reinforcement Learning (RL). Fitted Q Evaluation (FQE) with various function approximators, especially deep neural networks, has gained practical success. While statistical analysis has proved FQE to be minimax-optimal with tabular, linear and several nonparametric function families, its practical performance with more general function approximator is less theoretically understood. We focus on FQE with general differentiable function approximators, making our theory applicable to neural function approximations. We approach this problem using the Z-estimation theory and establish the following results: The FQE estimation error is asymptotically normal with explicit variance determined jointly by the tangent space of the function class at the ground truth, the reward structure, and the distribution shift due to off-policy learning; The finite-sample FQE error bound is dominated by the same variance term, and it can also be bounded by function class-dependent divergence, which measures how the off-policy distribution shift intertwines with the function approximator. In addition, we study bootstrapping FQE estimators for error distribution inference and estimating confidence intervals, accompanied by a Cramer-Rao lower bound that matches our upper bounds. The Z-estimation analysis provides a generalizable theoretical framework for studying off-policy estimation in RL and provides sharp statistical theory for FQE with differentiable function approximators.
Simulator-based models are models for which the likelihood is intractable but simulation of synthetic data is possible. They are often used to describe complex real-world phenomena, and as such can often be misspecified in practice. Unfortunately, existing Bayesian approaches for simulators are known to perform poorly in those cases. In this paper, we propose a novel algorithm based on the posterior bootstrap and maximum mean discrepancy estimators. This leads to a highly-parallelisable Bayesian inference algorithm with strong robustness properties. This is demonstrated through an in-depth theoretical study which includes generalisation bounds and proofs of frequentist consistency and robustness of our posterior. The approach is then assessed on a range of examples including a g-and-k distribution and a toggle-switch model.
Neural density estimators have proven remarkably powerful in performing efficient simulation-based Bayesian inference in various research domains. In particular, the BayesFlow framework uses a two-step approach to enable amortized parameter estimation in settings where the likelihood function is implicitly defined by a simulation program. But how faithful is such inference when simulations are poor representations of reality? In this paper, we conceptualize the types of model misspecification arising in simulation-based inference and systematically investigate the performance of the BayesFlow framework under these misspecifications. We propose an augmented optimization objective which imposes a probabilistic structure on the latent data space and utilize maximum mean discrepancy (MMD) to detect potentially catastrophic misspecifications during inference undermining the validity of the obtained results. We verify our detection criterion on a number of artificial and realistic misspecifications, ranging from toy conjugate models to complex models of decision making and disease outbreak dynamics applied to real data. Further, we show that posterior inference errors increase as a function of the distance between the true data-generating distribution and the typical set of simulations in the latent summary space. Thus, we demonstrate the dual utility of MMD as a method for detecting model misspecification and as a proxy for verifying the faithfulness of amortized Bayesian inference.
Considering the worst-case scenario, junction tree algorithm remains the most general solution for exact MAP inference with polynomial run-time guarantees. Unfortunately, its main tractability assumption requires the treewidth of a corresponding MRF to be bounded strongly limiting the range of admissible applications. In fact, many practical problems in the area of structured prediction require modelling of global dependencies by either directly introducing global factors or enforcing global constraints on the prediction variables. That, however, always results in a fully-connected graph making exact inference by means of this algorithm intractable. Previous work [1]-[4] focusing on the problem of loss-augmented inference has demonstrated how efficient inference can be performed on models with specific global factors representing non-decomposable loss functions within the training regime of SSVMs. In this paper, we extend the framework for an efficient exact inference proposed in in [3] by allowing much finer interactions between the energy of the core model and the sufficient statistics of the global terms with no additional computation costs. We demonstrate the usefulness of our method in several use cases, including one that cannot be handled by any of the previous approaches. Finally, we propose a new graph transformation technique via node cloning which ensures a polynomial run-time for solving our target problem independently of the form of a corresponding clique tree. This is important for the efficiency of the main algorithm and greatly improves upon the theoretical guarantees of the previous works.
In this paper, we propose Latent Relation Language Models (LRLMs), a class of language models that parameterizes the joint distribution over the words in a document and the entities that occur therein via knowledge graph relations. This model has a number of attractive properties: it not only improves language modeling performance, but is also able to annotate the posterior probability of entity spans for a given text through relations. Experiments demonstrate empirical improvements over both a word-based baseline language model and a previous approach that incorporates knowledge graph information. Qualitative analysis further demonstrates the proposed model's ability to learn to predict appropriate relations in context.
This work focuses on combining nonparametric topic models with Auto-Encoding Variational Bayes (AEVB). Specifically, we first propose iTM-VAE, where the topics are treated as trainable parameters and the document-specific topic proportions are obtained by a stick-breaking construction. The inference of iTM-VAE is modeled by neural networks such that it can be computed in a simple feed-forward manner. We also describe how to introduce a hyper-prior into iTM-VAE so as to model the uncertainty of the prior parameter. Actually, the hyper-prior technique is quite general and we show that it can be applied to other AEVB based models to alleviate the {\it collapse-to-prior} problem elegantly. Moreover, we also propose HiTM-VAE, where the document-specific topic distributions are generated in a hierarchical manner. HiTM-VAE is even more flexible and can generate topic distributions with better variability. Experimental results on 20News and Reuters RCV1-V2 datasets show that the proposed models outperform the state-of-the-art baselines significantly. The advantages of the hyper-prior technique and the hierarchical model construction are also confirmed by experiments.
Topic models have been widely explored as probabilistic generative models of documents. Traditional inference methods have sought closed-form derivations for updating the models, however as the expressiveness of these models grows, so does the difficulty of performing fast and accurate inference over their parameters. This paper presents alternative neural approaches to topic modelling by providing parameterisable distributions over topics which permit training by backpropagation in the framework of neural variational inference. In addition, with the help of a stick-breaking construction, we propose a recurrent network that is able to discover a notionally unbounded number of topics, analogous to Bayesian non-parametric topic models. Experimental results on the MXM Song Lyrics, 20NewsGroups and Reuters News datasets demonstrate the effectiveness and efficiency of these neural topic models.
Recurrent models for sequences have been recently successful at many tasks, especially for language modeling and machine translation. Nevertheless, it remains challenging to extract good representations from these models. For instance, even though language has a clear hierarchical structure going from characters through words to sentences, it is not apparent in current language models. We propose to improve the representation in sequence models by augmenting current approaches with an autoencoder that is forced to compress the sequence through an intermediate discrete latent space. In order to propagate gradients though this discrete representation we introduce an improved semantic hashing technique. We show that this technique performs well on a newly proposed quantitative efficiency measure. We also analyze latent codes produced by the model showing how they correspond to words and phrases. Finally, we present an application of the autoencoder-augmented model to generating diverse translations.
Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.
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