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.

The ability to learn effective transformations of complex data sources, sometimes called representation learning, is an essential primitive in modern machine learning, leading to remarkable achievements in language modeling, vision, and serving as a partial explanation for the success of deep learning more broadly (Bengio et al., 2013). In Reinforcement Learning (RL), several works have shown empirically that learning succinct representations of perceptual inputs can accelerate the search for decision-making policies (Pathak et al., 2017; Tang et al., 2017; Oord et al., 2018; Srinivas et al., 2020). However, representation learning for RL is far more subtle than it is for supervised learning (Du et al., 2019a; Van Roy and Dong, 2019; Lattimore and Szepesvari, 2019), and the theoretical foundations of representation learning for RL are nascent. The first question that arises in this context is: what is a good representation? Intuitively, a good representation should help us achieve greater sample efficiency on downstream tasks.

There are multiple ways to estimate a Stan model in R, but I choose to build the Stan code directly rather than using the brms or rstanarm packages. In the Stan code, we need to define the data structure, specify the parameters, specify any transformed parameters (which are just a function of the parameters), and then build the model – which includes laying out the prior distributions as well as the likelihood. In this case, the model is slightly different from what was presented in the context of a mixed effects model. The key difference is that there are prior distributions on \(\Delta\) and \(\tau\), introducing an additional level of uncertainty into the estimate. I would expect that the estimate of the overall treatment effect \(\Delta\) will have a wider 95% CI (credible interval in this context) than the 95% CI (confidence interval) for \(\delta_0\) in the mixed effects model.

The use of sophisticated machine learning models for critical decision making is faced with a challenge that these models are often applied as a "black-box". This has led to an increased interest in interpretable machine learning, where post hoc interpretation presents a useful mechanism for generating interpretations of complex learning models. In this paper, we propose a novel approach underpinned by an extended framework of Bayesian networks for generating post hoc interpretations of a black-box predictive model. The framework supports extracting a Bayesian network as an approximation of the black-box model for a specific prediction. Compared to the existing post hoc interpretation methods, the contribution of our approach is three-fold. Firstly, the extracted Bayesian network, as a probabilistic graphical model, can provide interpretations about not only what input features but also why these features contributed to a prediction. Secondly, for complex decision problems with many features, a Markov blanket can be generated from the extracted Bayesian network to provide interpretations with a focused view on those input features that directly contributed to a prediction. Thirdly, the extracted Bayesian network enables the identification of four different rules which can inform the decision-maker about the confidence level in a prediction, thus helping the decision-maker assess the reliability of predictions learned by a black-box model. We implemented the proposed approach, applied it in the context of two well-known public datasets and analysed the results, which are made available in an open-source repository.

The No Free Lunch theorems prove that under a uniform distribution over induction problems (search problems or learning problems), all induction algorithms perform equally. As I discuss in this chapter, the importance of the theorems arises by using them to analyze scenarios involving {non-uniform} distributions, and to compare different algorithms, without any assumption about the distribution over problems at all. In particular, the theorems prove that {anti}-cross-validation (choosing among a set of candidate algorithms based on which has {worst} out-of-sample behavior) performs as well as cross-validation, unless one makes an assumption -- which has never been formalized -- about how the distribution over induction problems, on the one hand, is related to the set of algorithms one is choosing among using (anti-)cross validation, on the other. In addition, they establish strong caveats concerning the significance of the many results in the literature which establish the strength of a particular algorithm without assuming a particular distribution. They also motivate a ``dictionary'' between supervised learning and improve blackbox optimization, which allows one to ``translate'' techniques from supervised learning into the domain of blackbox optimization, thereby strengthening blackbox optimization algorithms. In addition to these topics, I also briefly discuss their implications for philosophy of science.

We study the problem usually referred to as group testing in the context of COVID-19. Given n samples collected from patients, how should we select and test mixtures of samples to maximize information and minimize the number of tests? Group testing is a well-studied problem with several appealing solutions, but recent biological studies impose practical constraints for COVID-19 that are incompatible with traditional methods. Furthermore, existing methods use unnecessarily restrictive solutions, which were devised for settings with more memory and compute constraints than the problem at hand. This results in poor utility. In the new setting, we obtain strong solutions for small values of n using evolutionary strategies. We then develop a new method combining Bloom filters with belief propagation to scale to larger values of n (more than 100) with good empirical results. We also present a more accurate decoding algorithm that is tailored for specific COVID-19 settings. This work demonstrates the practical gap between dedicated algorithms and well-known generic solutions. Our efforts results in a new and practical multiplex method yielding strong empirical performance without mixing more than a chosen number of patients into the same probe. Finally, we briefly discuss adaptive methods, casting them into the framework of adaptive sub-modularity.