We present a new method for linear and nonlinear, lagged and contemporaneous constraint-based causal discovery from observational time series in the presence of latent confounders. We show that existing causal discovery methods such as FCI and variants suffer from low recall in the autocorrelated time series case and identify low effect size of conditional independence tests as the main reason. Information-theoretical arguments show that effect size can often be increased if causal parents are included in the conditioning sets. To identify parents early on, we suggest an iterative procedure that utilizes novel orientation rules to determine ancestral relationships already during the edge removal phase. We prove that the method is order-independent, and sound and complete in the oracle case. Extensive simulation studies for different numbers of variables, time lags, sample sizes, and further cases demonstrate that our method indeed achieves much higher recall than existing methods while keeping false positives at the desired level. This performance gain grows with stronger autocorrelation. Our method also covers causal discovery for non-time series data as a special case. We provide Python code for all methods involved in the simulation studies.

We first consider the sources of uncertainty in NNs, and briefly review Bayesian Neural Networks (BNN), the group of Bayesian approaches to tackle uncertainties in NNs. After presenting mathematical formulation of MC dropout, we proceed to suggesting potential benefits and associated costs for using MC dropout in typical NN models, with the results from our experiments.

We consider an online learning problem where the learner interacts with a Markov decision process in a sequence of episodes, where the reward function is allowed to change between episodes in an adversarial manner and the learner only gets to observe the rewards associated with its actions. We allow the state space to be arbitrarily large, but we assume that all action-value functions can be represented as linear functions in terms of a known low-dimensional feature map, and that the learner has access to a simulator of the environment that allows generating trajectories from the true MDP dynamics. Our main contribution is developing a computationally efficient algorithm that we call MDP-LinExp3, and prove that its regret is bounded by $\widetilde{\mathcal{O}}\big(H^2 T^{2/3} (dK)^{1/3}\big)$, where $T$ is the number of episodes, $H$ is the number of steps in each episode, $K$ is the number of actions, and $d$ is the dimension of the feature map. We also show that the regret can be improved to $\widetilde{\mathcal{O}}\big(H^2 \sqrt{TdK}\big)$ under much stronger assumptions on the MDP dynamics. To our knowledge, MDP-LinExp3 is the first provably efficient algorithm for this problem setting.

Information-theoretic quantities like entropy and mutual information have found numerous uses in machine learning. It is well known that there is a strong connection between these entropic quantities and submodularity since entropy over a set of random variables is submodular. In this paper, we study combinatorial information measures that generalize independence, (conditional) entropy, (conditional) mutual information, and total correlation defined over sets of (not necessarily random) variables. These measures strictly generalize the corresponding entropic measures since they are all parameterized via submodular functions that themselves strictly generalize entropy. Critically, we show that, unlike entropic mutual information in general, the submodular mutual information is actually submodular in one argument, holding the other fixed, for a large class of submodular functions whose third-order partial derivatives satisfy a non-negativity property. This turns out to include a number of practically useful cases such as the facility location and set-cover functions. We study specific instantiations of the submodular information measures on these, as well as the probabilistic coverage, graph-cut, and saturated coverage functions, and see that they all have mathematically intuitive and practically useful expressions. Regarding applications, we connect the maximization of submodular (conditional) mutual information to problems such as mutual-information-based, query-based, and privacy-preserving summarization -- and we connect optimizing the multi-set submodular mutual information to clustering and robust partitioning.

Resolving the exploration-exploitation trade-off remains a fundamental problem in the design and implementation of reinforcement learning (RL) algorithms. In this paper, we focus on model-free RL using the epsilon-greedy exploration policy, which despite its simplicity, remains one of the most frequently used forms of exploration. However, a key limitation of this policy is the specification of $\varepsilon$. In this paper, we provide a novel Bayesian perspective of $\varepsilon$ as a measure of the uniformity of the Q-value function. We introduce a closed-form Bayesian model update based on Bayesian model combination (BMC), based on this new perspective, which allows us to adapt $\varepsilon$ using experiences from the environment in constant time with monotone convergence guarantees. We demonstrate that our proposed algorithm, $\varepsilon$-\texttt{BMC}, efficiently balances exploration and exploitation on different problems, performing comparably or outperforming the best tuned fixed annealing schedules and an alternative data-dependent $\varepsilon$ adaptation scheme proposed in the literature.

Non-uniform sampling arises when an experimenter does not have full control over the sampling characteristics of the process under investigation. Moreover, it is introduced intentionally in algorithms such as Bayesian optimization and compressive sensing. We argue that Stochastic Differential Equations (SDEs) are especially well-suited for characterizing second order moments of such time series. We introduce new initial estimates for the numerical optimization of the likelihood, based on incremental estimation and initialization from autoregressive models. Furthermore, we introduce model truncation as a purely data-driven method to reduce the order of the estimated model based on the SDE likelihood. We show the increased accuracy achieved with the new estimator in simulation experiments, covering all challenging circumstances that may be encountered in characterizing a non-uniformly sampled time series. Finally, we apply the new estimator to experimental rainfall variability data.

Bayesian Networks (BNs) are graphical probabilistic models that have proven popular in medical applications. While numerous medical BNs have been published, most are presented fait accompli without explanation of how the network structure was developed or justification of why it represents the correct structure for the given medical application. This means that the process of building medical BNs from experts is typically ad hoc and offers little opportunity for methodological improvement. This paper proposes generally applicable and reusable medical reasoning patterns to aid those developing medical BNs. The proposed method complements and extends the idiom-based approach introduced by Neil, Fenton, and Nielsen in 2000. We propose instances of their generic idioms that are specific to medical BNs. We refer to the proposed medical reasoning patterns as medical idioms. In addition, we extend the use of idioms to represent interventional and counterfactual reasoning. We believe that the proposed medical idioms are logical reasoning patterns that can be combined, reused and applied generically to help develop medical BNs. All proposed medical idioms have been illustrated using medical examples on coronary artery disease. The method has also been applied to other ongoing BNs being developed with medical experts. Finally, we show that applying the proposed medical idioms to published BN models results in models with a clearer structure.

Stationary stochastic processes (SPs) are a key component of many probabilistic models, such as those for off-the-grid spatio-temporal data. They enable the statistical symmetry of underlying physical phenomena to be leveraged, thereby aiding generalization. Prediction in such models can be viewed as a translation equivariant map from observed data sets to predictive SPs, emphasizing the intimate relationship between stationarity and equivariance. Building on this, we propose the Convolutional Neural Process (ConvNP), which endows Neural Processes (NPs) with translation equivariance and extends convolutional conditional NPs to allow for dependencies in the predictive distribution. The latter enables ConvNPs to be deployed in settings which require coherent samples, such as Thompson sampling or conditional image completion. Moreover, we propose a new maximum-likelihood objective to replace the standard ELBO objective in NPs, which conceptually simplifies the framework and empirically improves performance. We demonstrate the strong performance and generalization capabilities of ConvNPs on 1D regression, image completion, and various tasks with real-world spatio-temporal data.

Structured stochastic processes evolving in continuous time present a widely adopted framework to model phenomena occurring in nature and engineering. However, such models are often chosen to satisfy the Markov property to maintain tractability. One of the more popular of such memoryless models are Continuous Time Bayesian Networks (CTBNs). In this work, we lift its restriction to exponential survival times to arbitrary distributions. Current extensions achieve this via auxiliary states, which hinder tractability. To avoid that, we introduce a set of node-wise clocks to construct a collection of graph-coupled semi-Markov chains. We provide algorithms for parameter and structure inference, which make use of local dependencies and conduct experiments on synthetic data and a data-set generated through a benchmark tool for gene regulatory networks. In doing so, we point out advantages compared to current CTBN extensions.

In the conventional Takagi-Sugeno-Kang (TSK)-type fuzzy models, constant or linear functions are usually utilized as the consequent parts of the fuzzy rules, but they cannot effectively describe the behavior within local regions defined by the antecedent parts. In this article, a theoretical and practical design methodology is developed to address this problem. First, the information granulation (Fuzzy C-Means) method is applied to capture the structure in the data and split the input space into subspaces, as well as form the antecedent parts. Second, the quadratic polynomials (QPs) are employed as the consequent parts. Compared with constant and linear functions, QPs can describe the input-output behavior within the local regions (subspaces) by refining the relationship between input and output variables. However, although QP can improve the approximation ability of the model, it could lead to the deterioration of the prediction ability of the model (e.g., overfitting). To handle this issue, we introduce an exponential weight approach inspired by the weight function theory encountered in harmonic analysis. More specifically, we adopt the exponential functions as the targeted penalty terms, which are equipped with l2 regularization (l2) (i.e., exponential weighted l2, ewl_2) to match the proposed reinforced second-order fuzzy rule-based model (RSFRM) properly. The advantage of el 2 compared to ordinary l2 lies in separately identifying and penalizing different types of polynomial terms in the coefficient estimation, and its results not only alleviate the overfitting and prevent the deterioration of generalization ability but also effectively release the prediction potential of the model.