text: Bayesian networks are a central tool in machine learning and artificial intelligence, and make use of conditional independencies to impose structure on joint distributions. However, they are generally not as expressive as deep learning models and inference is hard and slow. In contrast, deep probabilistic models such as sum-product networks (SPNs) capture joint distributions in a tractable fashion, but use little interpretable structure. Here, we extend the notion of SPNs towards conditional distributions, which combine simple conditional models into high-dimensional ones. As shown in our experiments, the resulting conditional SPNs can be naturally used to impose structure on deep probabilistic models, allow for mixed data types, while maintaining fast and efficient inference.
  @cite_9: We present conditional random fields , a framework for building probabilistic models to segment and label sequence data. Conditional random fields offer several advantages over hidden Markov models and stochastic grammars for such tasks, including the ability to relax strong independence assumptions made in those models. Conditional random fields also avoid a fundamental limitation of maximum entropy Markov models (MEMMs) and other discriminative Markov models based on directed graphical models, which can be biased towards states with few successor states. We present iterative parameter estimation algorithms for conditional random fields and compare the performance of the resulting models to HMMs and MEMMs on synthetic and natural-language data.
  @cite_1: While Gaussian processes (GPs) are the method of choice for regression tasks, they also come with practical difficulties, as inference cost scales cubic in time and quadratic in memory. In this paper, we introduce a natural and expressive way to tackle these problems, by incorporating GPs in sum-product networks (SPNs), a recently proposed tractable probabilistic model allowing exact and efficient inference. In particular, by using GPs as leaves of an SPN we obtain a novel flexible prior over functions, which implicitly represents an exponentially large mixture of local GPs. Exact and efficient posterior inference in this model can be done in a natural interplay of the inference mechanisms in GPs and SPNs. Thereby, each GP is -- similarly as in a mixture of experts approach -- responsible only for a subset of data points, which effectively reduces inference cost in a divide and conquer fashion. We show that integrating GPs into the SPN framework leads to a promising probabilistic regression model which is: (1) computational and memory efficient, (2) allows efficient and exact posterior inference, (3) is flexible enough to mix different kernel functions, and (4) naturally accounts for non-stationarities in time series. In a variate of experiments, we show that the SPN-GP model can learn input dependent parameters and hyper-parameters and is on par with or outperforms the traditional GPs as well as state of the art approximations on real-world data.
  @cite_17: We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn the hyperparameters using the marginal likelihood. We explain the practical advantages of Gaussian Process and end with conclusions and a look at the current trends in GP work.
summary: Gaussian Processes (GPs) @cite_17 and Conditional Random Fields (CRFs) @cite_9 are staples for (SOP) regression and classification. However, they have serious shortcomings when inference has to scale to high dimensional data. Moreover, they do not generally allow for exact marginalization. Deep mixtures of GPs have been introduced in @cite_1 . However, while partially alleviating GP inference scalability issues, they are limited to continuous domains and with CSPNs we directly tackle , i.e., the scenario where @math is multivariate. In a nutshell, CSPNs might be seen as an efficient way to aggregate univariate (leaf) predictor to tackle SOP in a principled probabilistic way.