Tractable probabilistic models such as cutset networks which admit exact linear time posterior marginal inference are often preferred in practice over intractable models such as Bayesian and Markov networks. This is because although tractable models, when learned from data, are slightly inferior to the intractable ones in terms of goodness-of-fit measures such as log-likelihood, they do not use approximate inference at prediction time and as a result exhibit superior predictive performance. In this paper, we consider the problem of improving a tractable model using a large number of local probability estimates, each defined over a small subset of variables that are either available from experts or via an external process. Given a model learned from fully-observed, but small amount of possibly noisy data, the key idea in our approach is to update the parameters of the model via a gradient descent procedure that seeks to minimize a convex combination of two quantities: one that enforces closeness via KL divergence to the local estimates and another that enforces closeness to the given model. We show that although the gradients are NP-hard to compute on arbitrary graphical models, they can be efficiently computed over tractable models. We show via experiments that our approach yields tractable models that are significantly superior to the ones learned from small amount of possibly noisy data, even when the local estimates are inconsistent.

Tractable probabilistic models such as cutset networks which admit exact linear time posterior marginal inference are often preferred in practice over intractable models such as Bayesian and Markov networks. This is because although tractable models, when learned from data, are slightly inferior to the intractable ones in terms of goodness-of-fit measures such as log-likelihood, they do not use approximate inference at prediction time and as a result exhibit superior predictive performance. In this paper, we consider the problem of improving a tractable model using a large number of local probability estimates, each defined over a small subset of variables that are either available from experts or via an external process. Given a model learned from fully-observed, but small amount of possibly noisy data, the key idea in our approach is to update the parameters of the model via a gradient descent procedure that seeks to minimize a convex combination of two quantities: one that enforces closeness via KL divergence to the local estimates and another that enforces closeness to the given model. We show that although the gradients are NP-hard to compute on arbitrary graphical models, they can be efficiently computed over tractable models.

Large-scale probabilistic representations, including statistical knowledge bases and graphical models, are increasingly in demand. They are built by mining massive sources of structured and unstructured data, the latter often derived from natural language processing techniques. The very nature of the enterprise makes the extracted representations probabilistic. In particular, inducing relations and facts from noisy and incomplete sources via statistical machine learning models means that the labels are either already probabilistic, or that probabilities approximate confidence. While the progress is impressive, extracted representations essentially enforce the closed-world assumption, which means that all facts in the database are accorded the corresponding probability, but all other facts have probability zero. The CWA is deeply problematic in most machine learning contexts. A principled solution is needed for representing incomplete and indeterminate knowledge in such models, imprecise probability models such as credal networks being an example. In this work, we are interested in the foundational problem of learning such open-world probabilistic models. However, since exact inference in probabilistic graphical models is intractable, the paradigm of tractable learning has emerged to learn data structures (such as arithmetic circuits) that support efficient probabilistic querying. We show here how the computational machinery underlying tractable learning has to be generalized for imprecise probabilities. Our empirical evaluations demonstrate that our regime is also effective.

In recent years, several probabilistic techniques have been applied to various debugging problems. However, most existing probabilistic debugging systems use relatively simple statistical models, and fail to generalize across multiple programs. In this work, we propose Tractable Fault Localization Models (TFLMs) that can be learned from data, and probabilistically infer the location of the bug. While most previous statistical debugging methods generalize over many executions of a single program, TFLMs are trained on a corpus of previously seen buggy programs, and learn to identify recurring patterns of bugs. Widely-used fault localization techniques such as TARANTULA evaluate the suspiciousness of each line in isolation; in contrast, a TFLM defines a joint probability distribution over buggy indicator variables for each line. Joint distributions with rich dependency structure are often computationally intractable; TFLMs avoid this by exploiting recent developments in tractable probabilistic models (specifically, Relational SPNs). Further, TFLMs can incorporate additional sources of information, including coverage-based features such as TARANTULA. We evaluate the fault localization performance of TFLMs that include TARANTULA scores as features in the probabilistic model. Our study shows that the learned TFLMs isolate bugs more effectively than previous statistical methods or using TARANTULA directly.