We introduce graph gamma process (GGP) linear dynamical systems to model real-valued multivariate time series. For temporal pattern discovery, the latent representation under the model is used to decompose the time series into a parsimonious set of multivariate sub-sequences. In each sub-sequence, different data dimensions often share similar temporal patterns but may exhibit distinct magnitudes, and hence allowing the superposition of all sub-sequences to exhibit diverse behaviors at different data dimensions. We further generalize the proposed model by replacing the Gaussian observation layer with the negative binomial distribution to model multivariate count time series. Generated from the proposed GGP is an infinite dimensional directed sparse random graph, which is constructed by taking the logical OR operation of countably infinite binary adjacency matrices that share the same set of countably infinite nodes. Each of these adjacency matrices is associated with a weight to indicate its activation strength, and places a finite number of edges between a finite subset of nodes belonging to the same node community. We use the generated random graph, whose number of nonzero-degree nodes is finite, to define both the sparsity pattern and dimension of the latent state transition matrix of a (generalized) linear dynamical system. The activation strength of each node community relative to the overall activation strength is used to extract a multivariate sub-sequence, revealing the data pattern captured by the corresponding community. On both synthetic and real-world time series, the proposed nonparametric Bayesian dynamic models, which are initialized at random, consistently exhibit good predictive performance in comparison to a variety of baseline models, revealing interpretable latent state transition patterns and decomposing the time series into distinctly behaved sub-sequences.

Graph-based semi-supervised regression (SSR) is the problem of estimating the value of a function on a weighted graph from its values (labels) on a small subset of the vertices. This paper is concerned with the consistency of SSR in the context of classification, in the setting where the labels have small noise and the underlying graph weighting is consistent with well-clustered nodes. We present a Bayesian formulation of SSR in which the weighted graph defines a Gaussian prior, using a graph Laplacian, and the labeled data defines a likelihood. We analyze the rate of contraction of the posterior measure around the ground truth in terms of parameters that quantify the small label error and inherent clustering in the graph. We obtain bounds on the rates of contraction and illustrate their sharpness through numerical experiments. The analysis also gives insight into the choice of hyperparameters that enter the definition of the prior.

The empirical practice of using factor models to adjust for shared, unobserved confounders, $\mathbf{Z}$, in observational settings with multiple treatments, $\mathbf{A}$, is widespread in fields including genetics, networks, medicine, and politics. Wang and Blei (2019, WB) formalizes these procedures and develops the "deconfounder," a causal inference method using factor models of $\mathbf{A}$ to estimate "substitute confounders," $\hat{\mathbf{Z}}$, then estimating treatment effects by regressing the outcome, $\mathbf{Y}$, on part of $\mathbf{A}$ while adjusting for $\hat{\mathbf{Z}}$. WB claim the deconfounder is unbiased when there are no single-cause confounders and $\hat{\mathbf{Z}}$ is "pinpointed." We clarify pinpointing requires each confounder to affect infinitely many treatments. We prove under these assumptions, a na\"ive semiparametric regression of $\mathbf{Y}$ on $\mathbf{A}$ is asymptotically unbiased. Deconfounder variants nesting this regression are therefore also asymptotically unbiased, but variants using $\hat{\mathbf{Z}}$ and subsets of causes require further untestable assumptions. We replicate every deconfounder analysis with available data and find it fails to consistently outperform na\"ive regression. In practice, the deconfounder produces implausible estimates in WB's case study to movie earnings: estimates suggest comic author Stan Lee's cameo appearances causally contributed \$15.5 billion, most of Marvel movie revenue. We conclude neither approach is a viable substitute for careful research design in real-world applications.

Control schemes that learn using measurement data collected online are increasingly promising for the control of complex and uncertain systems. However, in most approaches of this kind, learning is viewed as a side effect that passively improves control performance, e.g., by updating a model of the system dynamics. Determining how improvements in control performance due to learning can be actively exploited in the control synthesis is still an open research question. In this paper, we present AntLer, a design algorithm for learning-based control laws that anticipates learning, i.e., that takes the impact of future learning in uncertain dynamic settings explicitly into account. AntLer expresses system uncertainty using a non-parametric probabilistic model. Given a cost function that measures control performance, AntLer chooses the control parameters such that the expected cost of the closed-loop system is minimized approximately. We show that AntLer approximates an optimal solution arbitrarily accurately with probability one. Furthermore, we apply AntLer to a nonlinear system, which yields better results compared to the case where learning is not anticipated.

Sequential decision making, commonly formalized as Markov Decision Process (MDP) optimization, is a key challenge in artificial intelligence. Two key approaches to this problem are reinforcement learning (RL) and planning. This paper presents a survey of the integration of both fields, better known as model-based reinforcement learning. Model-based RL has two main steps. First, we systematically cover approaches to dynamics model learning, including challenges like dealing with stochasticity, uncertainty, partial observability, and temporal abstraction. Second, we present a systematic categorization of planning-learning integration, including aspects like: where to start planning, what budgets to allocate to planning and real data collection, how to plan, and how to integrate planning in the learning and acting loop. After these two key sections, we also discuss the potential benefits of model-based RL, like enhanced data efficiency, targeted exploration, and improved stability. Along the survey, we also draw connections to several related RL fields, like hierarchical RL and transfer, and other research disciplines, like behavioural psychology. Altogether, the survey presents a broad conceptual overview of planning-learning combinations for MDP optimization.

In the causal learning setting, we wish to learn cause-and-effect relationships between variables such that we can correctly infer the effect of an intervention. While the difference between a cyclic structure and an acyclic structure may be just a single edge, cyclic causal structures have qualitatively different behavior under intervention: cycles cause feedback loops when the downstream effect of an intervention propagates back to the source variable. We present three theoretical observations about probability distributions with self-referential factorizations, i.e. distributions that could be graphically represented with a cycle. First, we prove that self-referential distributions in two variables are, in fact, independent. Second, we prove that self-referential distributions in N variables have zero mutual information. Lastly, we prove that self-referential distributions that factorize in a cycle, also factorize as though the cycle were reversed. These results suggest that cyclic causal dependence may exist even where observational data suggest independence among variables. Methods based on estimating mutual information, or heuristics based on independent causal mechanisms, are likely to fail to learn cyclic casual structures. We encourage future work in causal learning that carefully considers cycles.

We develop several new algorithms for learning Markov Decision Processes in an infinite-horizon average-reward setting with linear function approximation. Using the optimism principle and assuming that the MDP has a linear structure, we first propose a computationally inefficient algorithm with optimal $\widetilde{O}(\sqrt{T})$ regret and another computationally efficient variant with $\widetilde{O}(T^{3/4})$ regret, where $T$ is the number of interactions. Next, taking inspiration from adversarial linear bandits, we develop yet another efficient algorithm with $\widetilde{O}(\sqrt{T})$ regret under a different set of assumptions, improving the best existing result by Hao et al. (2020) with $\widetilde{O}(T^{2/3})$ regret. Moreover, we draw a connection between this algorithm and the Natural Policy Gradient algorithm proposed by Kakade (2002), and show that our analysis improves the sample complexity bound recently given by Agarwal et al. (2020).

Gaussian ODE filtering is a probabilistic numerical method to solve ordinary differential equations (ODEs). It computes a Bayesian posterior over the solution from evaluations of the vector field defining the ODE. Its most popular version, which employs an integrated Brownian motion prior, uses Taylor expansions of the mean to extrapolate forward and has the same convergence rates as classical numerical methods. As the solution of many important ODEs are periodic functions (oscillators), we raise the question whether Fourier expansions can also be brought to bear within the framework of Gaussian ODE filtering. To this end, we construct a Fourier state space model for ODEs and a `hybrid' model that combines a Taylor (Brownian motion) and Fourier state space model. We show by experiments how the hybrid model might become useful in cheaply predicting until the end of the time domain.

Scientists and engineers employ stochastic numerical simulators to model empirically observed phenomena. In contrast to purely statistical models, simulators express scientific principles that provide powerful inductive biases, improve generalization to new data or scenarios and allow for fewer, more interpretable and domain-relevant parameters. Despite these advantages, tuning a simulator's parameters so that its outputs match data is challenging. Simulation-based inference (SBI) seeks to identify parameter sets that a) are compatible with prior knowledge and b) match empirical observations. Importantly, SBI does not seek to recover a single 'best' data-compatible parameter set, but rather to identify all high probability regions of parameter space that explain observed data, and thereby to quantify parameter uncertainty. In Bayesian terminology, SBI aims to retrieve the posterior distribution over the parameters of interest. In contrast to conventional Bayesian inference, SBI is also applicable when one can run model simulations, but no formula or algorithm exists for evaluating the probability of data given parameters, i.e. the likelihood. We present $\texttt{sbi}$, a PyTorch-based package that implements SBI algorithms based on neural networks. $\texttt{sbi}$ facilitates inference on black-box simulators for practising scientists and engineers by providing a unified interface to state-of-the-art algorithms together with documentation and tutorials.

We present a new method for estimating the stochastic intensity of a doubly stochastic Poisson process. Statistical and theoretical analyses of traffic traces show that these processes are appropriate models of high intensity traffic arriving at an array of service systems. The statistical estimation of the underlying latent stochastic intensity process driving the traffic model involves a rather complicated nonlinear filtering problem. We develop a novel simulation method, using deep neural networks to approximate the path measures induced by the stochastic intensity process, for solving this nonlinear filtering problem. Our simulation studies demonstrate that the method is quite accurate on both in-sample estimation and on an out-of-sample performance prediction task for an infinite server queue.