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Register a new userProbabilistic Data with Continuous Distributions
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Added by
Martin Grohe
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Statistical models of real world data typically involve continuous probability distributions such as normal, Laplace, or exponential distributions. Such distributions are supported by many probabilistic modelling formalisms, including probabilistic database systems. Yet, the traditional theoretical framework of probabilistic databases focusses entirely on finite probabilistic databases. Only recently, we set out to develop the mathematical theory of infinite probabilistic databases. The present paper is an exposition of two recent papers which are cornerstones of this theory. In (Grohe, Lindner; ICDT 2020) we propose a very general framework for probabilistic databases, possibly involving continuous probability distributions, and show that queries have a well-defined semantics in this framework. In (Grohe, Kaminski, Katoen, Lindner; PODS 2020) we extend the declarative probabilistic programming language Generative Datalog, proposed by (Barany et al.~2017) for discrete probability distributions, to continuous probability distributions and show that such programs yield generative models of continuous probabilistic databases.
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Arguing for the need to combine declarative and probabilistic programming, Barany et al. (TODS 2017) recently introduced a probabilistic extension of Datalog as a purely declarative probabilistic programming language. We revisit this language and propose a more principled approach towards defining its semantics based on stochastic kernels and Markov processes - standard notions from probability theory. This allows us to extend the semantics to continuous probability distributions, thereby settling an open problem posed by Barany et al. We show that our semantics is fairly robust, allowing both parallel execution and arbitrary chase orders when evaluating a program. We cast our semantics in the framework of infinite probabilistic databases (Grohe and Lindner, ICDT 2020), and show that the semantics remains meaningful even when the input of a probabilistic Datalog program is an arbitrary probabilistic database.
We study termination of higher-order probabilistic functional programs with recursion, stochastic conditioning and sampling from continuous distributions. Reasoning about the termination probability of programs with continuous distributions is hard, because the enumeration of terminating executions cannot provide any non-trivial bounds. We present a new operational semantics based on traces of intervals, which is sound and complete with respect to the standard sampling-based semantics, in which (countable) enumeration can provide arbitrarily tight lower bounds. Consequently we obtain the first proof that deciding almost-sure termination (AST) for programs with continuous distributions is $Pi^0_2$-complete. We also provide a compositional representation of our semantics in terms of an intersection type system. In the second part, we present a method of proving AST for non-affine programs, i.e., recursive programs that can, during the evaluation of the recursive body, make multiple recursive calls (of a first-order function) from distinct call sites. Unlike in a deterministic language, the number of recursion call sites has direct consequences on the termination probability. Our framework supports a proof system that can verify AST for programs that are well beyond the scope of existing methods. We have constructed prototype implementations of our method of computing lower bounds of termination probability, and AST verification.
Interactive data visualization and exploration (DVE) applications are often network-bottlenecked due to bursty request patterns, large response sizes, and heterogeneous deployments over a range of networks and devices. This makes it difficult to ensure consistently low response times (< 100ms). Khameleon is a framework for DVE applications that uses a novel combination of prefetching and response tuning to dynamically trade-off response quality for low latency. Khameleon exploits DVEs approximation tolerance: immediate lower-quality responses are preferable to waiting for complete results. To this end, Khameleon progressively encodes responses, and runs a server-side scheduler that proactively streams portions of responses using available bandwidth to maximize users perceived interactivity. The scheduler involves a complex optimization based on available resources, predicted user interactions, and response quality levels; yet, decisions must also be real-time. To overcome this, Khameleon uses a fast greedy approximation which closely mimics the optimal approach. Using image exploration and visualization applications with real user interaction traces, we show that across a wide range of network and client resource conditions, Khameleon outperforms classic prefetching approaches that benefit from perfect prediction models: response latencies with Khameleon are never higher, and typically between 2 to 3 orders of magnitude lower while response quality remains within 50%-80%.
Canonical Correlation Analysis (CCA) is a classic technique for multi-view data analysis. To overcome the deficiency of linear correlation in practical multi-view learning tasks, various CCA variants were proposed to capture nonlinear dependency. However, it is non-trivial to have an in-principle understanding of these variants due to their inherent restrictive assumption on the data and latent code distributions. Although some works have studied probabilistic interpretation for CCA, these models still require the explicit form of the distributions to achieve a tractable solution for the inference. In this work, we study probabilistic interpretation for CCA based on implicit distributions. We present Conditional Mutual Information (CMI) as a new criterion for CCA to consider both linear and nonlinear dependency for arbitrarily distributed data. To eliminate direct estimation for CMI, in which explicit form of the distributions is still required, we derive an objective which can provide an estimation for CMI with efficient inference methods. To facilitate Bayesian inference of multi-view analysis, we propose Adversarial CCA (ACCA), which achieves consistent encoding for multi-view data with the consistent constraint imposed on the marginalization of the implicit posteriors. Such a model would achieve superiority in the alignment of the multi-view data with implicit distributions. It is interesting to note that most of the existing CCA variants can be connected with our proposed CCA model by assigning specific form for the posterior and likelihood distributions. Extensive experiments on nonlinear correlation analysis and cross-view generation on benchmark and real-world datasets demonstrate the superiority of our model.
Probabilistic databases play a crucial role in the management and understanding of uncertain data. However, incorporating probabilities into the semantics of incomplete databases has posed many challenges, forcing systems to sacrifice modeling power, scalability, or restrict the class of relational algebra formula under which they are closed. We propose an alternative approach where the underlying relational database always represents a single world, and an external factor graph encodes a distribution over possible worlds; Markov chain Monte Carlo (MCMC) inference is then used to recover this uncertainty to a desired level of fidelity. Our approach allows the efficient evaluation of arbitrary queries over probabilistic databases with arbitrary dependencies expressed by graphical models with structure that changes during inference. MCMC sampling provides efficiency by hypothesizing {em modifications} to possible worlds rather than generating entire worlds from scratch. Queries are then run over the portions of the world that change, avoiding the onerous cost of running full queries over each sampled world. A significant innovation of this work is the connection between MCMC sampling and materialized view maintenance techniques: we find empirically that using view maintenance techniques is several orders of magnitude faster than naively querying each sampled world. We also demonstrate our systems ability to answer relational queries with aggregation, and demonstrate additional scalability through the use of parallelization.
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