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Register a new userComposing graphical models with neural networks for structured representations and fast inference
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Added by
Matthew Johnson
Publication date
2016
fields
Mathematical Statistics
and research's language is
English
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We propose a general modeling and inference framework that composes probabilistic graphical models with deep learning methods and combines their respective strengths. Our model family augments graphical structure in latent variables with neural network observation models. For inference, we extend variational autoencoders to use graphical model approximating distributions with recognition networks that output conjugate potentials. All components of these models are learned simultaneously with a single objective, giving a scalable algorithm that leverages stochastic variational inference, natural gradients, graphical model message passing, and the reparameterization trick. We illustrate this framework with several example models and an application to mouse behavioral phenotyping.
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Gaussian state space models have been used for decades as generative models of sequential data. They admit an intuitive probabilistic interpretation, have a simple functional form, and enjoy widespread adoption. We introduce a unified algorithm to efficiently learn a broad class of linear and non-linear state space models, including variants where the emission and transition distributions are modeled by deep neural networks. Our learning algorithm simultaneously learns a compiled inference network and the generative model, leveraging a structured variational approximation parameterized by recurrent neural networks to mimic the posterior distribution. We apply the learning algorithm to both synthetic and real-world datasets, demonstrating its scalability and versatility. We find that using the structured approximation to the posterior results in models with significantly higher held-out likelihood.
Gaussian graphical models (GGMs) are probabilistic tools of choice for analyzing conditional dependencies between variables in complex systems. Finding changepoints in the structural evolution of a GGM is therefore essential to detecting anomalies in the underlying system modeled by the GGM. In order to detect structural anomalies in a GGM, we consider the problem of estimating changes in the precision matrix of the corresponding Gaussian distribution. We take a two-step approach to solving this problem:- (i) estimating a background precision matrix using system observations from the past without any anomalies, and (ii) estimating a foreground precision matrix using a sliding temporal window during anomaly monitoring. Our primary contribution is in estimating the foreground precision using a novel contrastive inverse covariance estimation procedure. In order to accurately learn only the structural changes to the GGM, we maximize a penalized log-likelihood where the penalty is the $l_1$ norm of difference between the foreground precision being estimated and the already learned background precision. We modify the alternating direction method of multipliers (ADMM) algorithm for sparse inverse covariance estimation to perform contrastive estimation of the foreground precision matrix. Our results on simulated GGM data show significant improvement in precision and recall for detecting structural changes to the GGM, compared to a non-contrastive sliding window baseline.
The last decade witnessed the development of algorithms that completely solve the identifiability problem for causal effects in hidden variable causal models associated with directed acyclic graphs. However, much of this machinery remains underutilized in practice owing to the complexity of estimating identifying functionals yielded by these algorithms. In this paper, we provide simple graphical criteria and semiparametric estimators that bridge the gap between identification and estimation for causal effects involving a single treatment and a single outcome. First, we provide influence function based doubly robust estimators that cover a significant subset of hidden variable causal models where the effect is identifiable. We further characterize an important subset of this class for which we demonstrate how to derive the estimator with the lowest asymptotic variance, i.e., one that achieves the semiparametric efficiency bound. Finally, we provide semiparametric estimators for any single treatment causal effect parameter identified via the aforementioned algorithms. The resulting estimators resemble influence function based estimators that are sequentially reweighted, and exhibit a partial double robustness property, provided the parts of the likelihood corresponding to a set of weight models are correctly specified. Our methods are easy to implement and we demonstrate their utility through simulations.
Unsupervised structure learning in high-dimensional time series data has attracted a lot of research interests. For example, segmenting and labelling high dimensional time series can be helpful in behavior understanding and medical diagnosis. Recent advances in generative sequential modeling have suggested to combine recurrent neural networks with state space models (e.g., Hidden Markov Models). This combination can model not only the long term dependency in sequential data, but also the uncertainty included in the hidden states. Inheriting these advantages of stochastic neural sequential models, we propose a structured and stochastic sequential neural network, which models both the long-term dependencies via recurrent neural networks and the uncertainty in the segmentation and labels via discrete random variables. For accurate and efficient inference, we present a bi-directional inference network by reparamterizing the categorical segmentation and labels with the recent proposed Gumbel-Softmax approximation and resort to the Stochastic Gradient Variational Bayes. We evaluate the proposed model in a number of tasks, including speech modeling, automatic segmentation and labeling in behavior understanding, and sequential multi-objects recognition. Experimental results have demonstrated that our proposed model can achieve significant improvement over the state-of-the-art methods.
Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the Sherali-Adams hierarchy. Despite the popularity of these algorithms, it is well understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees. Unfortunately, SOS relaxations for a graph with $n$ vertices require solving an SDP with $n^{Theta(d)}$ variables where $d$ is the degree in the hierarchy. In practice, for $dge 4$, this approach does not scale beyond a few tens of variables. In this paper, we propose binary SDP relaxations for MAP inference using the SOS hierarchy with two innovations focused on computational efficiency. Firstly, in analogy to BP and its variants, we only introduce decision variables corresponding to contiguous regions in the graphical model. Secondly, we solve the resulting SDP using a non-convex Burer-Monteiro style method, and develop a sequential rounding procedure. We demonstrate that the resulting algorithm can solve problems with tens of thousands of variables within minutes, and outperforms BP and GBP on practical problems such as image denoising and Ising spin glasses. Finally, for specific graph types, we establish a sufficient condition for the tightness of the proposed partial SOS relaxation.
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