The Bayesian Context Trees (BCT) framework is a recently introduced, general collection of statistical and algorithmic tools for modelling, analysis and inference with discrete-valued time series. The foundation of this development is built in part on some well-known information-theoretic ideas and techniques, including Rissanen's tree sources and Willems et al.'s context-tree weighting algorithm. This paper presents a collection of theoretical results that provide mathematical justifications and further insight into the BCT modelling framework and the associated practical tools. It is shown that the BCT prior predictive likelihood (the probability of a time series of observations averaged over all models and parameters) is both pointwise and minimax optimal, in agreement with the MDL principle and the BIC criterion. The posterior distribution is shown to be asymptotically consistent with probability one (over both models and parameters), and asymptotically Gaussian (over the parameters). And the posterior predictive distribution is also shown to be asymptotically consistent with probability one.
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Opinion diffusion is a crucial phenomenon in social networks, often underlying the way in which a collective of agents develops a consensus on relevant decisions. The voter model is a well-known theoretical model to study opinion spreading in social networks and structured populations. Its simplest version assumes that an updating agent will adopt the opinion of a neighboring agent chosen at random. The model allows us to study, for example, the probability that a certain opinion will fixate into a consensus opinion, as well as the expected time it takes for a consensus opinion to emerge. Standard voter models are oblivious to the opinions held by the agents involved in the opinion adoption process. We propose and study a context-dependent opinion spreading process on an arbitrary social graph, in which the probability that an agent abandons opinion $a$ in favor of opinion $b$ depends on both $a$ and $b$. We discuss the relations of the model with existing voter models and then derive theoretical results for both the fixation probability and the expected consensus time for two opinions, for both the synchronous and the asynchronous update models.
Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution. Developing efficient and accurate solution strategies that account for errors on the space, time and parameter domains simultaneously is highly challenging. Indeed, it is well known that standard polynomial-based approximations on the parameter domain can incur errors that grow in time. In this work, we focus on advection-diffusion problems with parameter-dependent wind fields. A novel adaptive solution strategy is proposed that allows users to combine stochastic collocation on the parameter domain with off-the-shelf adaptive timestepping algorithms with local error control. This is a non-intrusive strategy that builds a polynomial-based surrogate that is adapted sequentially in time. The algorithm is driven by a so-called hierarchical estimator for the parametric error and balances this against an estimate for the global timestepping error which is derived from a scaling argument.
We study multi-agent reinforcement learning in the setting of episodic Markov decision processes, where multiple agents cooperate via communication through a central server. We propose a provably efficient algorithm based on value iteration that enable asynchronous communication while ensuring the advantage of cooperation with low communication overhead. With linear function approximation, we prove that our algorithm enjoys an $\tilde{\mathcal{O}}(d^{3/2}H^2\sqrt{K})$ regret with $\tilde{\mathcal{O}}(dHM^2)$ communication complexity, where $d$ is the feature dimension, $H$ is the horizon length, $M$ is the total number of agents, and $K$ is the total number of episodes. We also provide a lower bound showing that a minimal $\Omega(dM)$ communication complexity is required to improve the performance through collaboration.
A seller is pricing identical copies of a good to a stream of unit-demand buyers. Each buyer has a value on the good as his private information. The seller only knows the empirical value distribution of the buyer population and chooses the revenue-optimal price. We consider a widely studied third-degree price discrimination model where an information intermediary with perfect knowledge of the arriving buyer's value sends a signal to the seller, hence changing the seller's posterior and inducing the seller to set a personalized posted price. Prior work of Bergemann, Brooks, and Morris (American Economic Review, 2015) has shown the existence of a signaling scheme that preserves seller revenue, while always selling the item, hence maximizing consumer surplus. In a departure from prior work, we ask whether the consumer surplus generated is fairly distributed among buyers with different values. To this end, we aim to maximize welfare functions that reward more balanced surplus allocations. Our main result is the surprising existence of a novel signaling scheme that simultaneously $8$-approximates all welfare functions that are non-negative, monotonically increasing, symmetric, and concave, compared with any other signaling scheme. Classical examples of such welfare functions include the utilitarian social welfare, the Nash welfare, and the max-min welfare. Such a guarantee cannot be given by any consumer-surplus-maximizing scheme -- which are the ones typically studied in the literature. In addition, our scheme is socially efficient, and has the fairness property that buyers with higher values enjoy higher expected surplus, which is not always the case for existing schemes.
In causal inference, sensitivity analysis is important to assess the robustness of study conclusions to key assumptions. We perform sensitivity analysis of the assumption that missing outcomes are missing completely at random. We follow a Bayesian approach, which is nonparametric for the outcome distribution and can be combined with an informative prior on the sensitivity parameter. We give insight in the posterior and provide theoretical guarantees in the form of Bernstein-von Mises theorems for estimating the mean outcome. We study different parametrisations of the model involving Dirichlet process priors on the distribution of the outcome and on the distribution of the outcome conditional on the subject being treated. We show that these parametrisations incorporate a prior on the sensitivity parameter in different ways and discuss the relative merits. We also present a simulation study, showing the performance of the methods in finite sample scenarios.
We present counterfactual situation testing (CST), a causal data mining framework for detecting discrimination in classifiers. CST aims to answer in an actionable and meaningful way the intuitive question "what would have been the model outcome had the individual, or complainant, been of a different protected status?" It extends the legally-grounded situation testing of Thanh et al. (2011) by operationalizing the notion of fairness given the difference using counterfactual reasoning. For any complainant, we find and compare similar protected and non-protected instances in the dataset used by the classifier to construct a control and test group, where a difference between the decision outcomes of the two groups implies potential individual discrimination. Unlike situation testing, which builds both groups around the complainant, we build the test group on the complainant's counterfactual generated using causal knowledge. The counterfactual is intended to reflect how the protected attribute when changed affects the seemingly neutral attributes used by the classifier, which is taken for granted in many frameworks for discrimination. Under CST, we compare similar individuals within each group but dissimilar individuals across both groups due to the possible difference between the complainant and its counterfactual. Evaluating our framework on two classification scenarios, we show that it uncovers a greater number of cases than situation testing, even when the classifier satisfies the counterfactual fairness condition of Kusner et al. (2017).
We study multi-agent reinforcement learning in the setting of episodic Markov decision processes, where multiple agents cooperate via communication through a central server. We propose a provably efficient algorithm based on value iteration that enable asynchronous communication while ensuring the advantage of cooperation with low communication overhead. With linear function approximation, we prove that our algorithm enjoys an $\tilde{\mathcal{O}}(d^{3/2}H^2\sqrt{K})$ regret with $\tilde{\mathcal{O}}(dHM^2)$ communication complexity, where $d$ is the feature dimension, $H$ is the horizon length, $M$ is the total number of agents, and $K$ is the total number of episodes. We also provide a lower bound showing that a minimal $\Omega(dM)$ communication complexity is required to improve the performance through collaboration.
This paper provides a selective review of the statistical network analysis literature focused on clustering and inference problems for stochastic blockmodels and their variants. We survey asymptotic normality results for stochastic blockmodels as a means of thematically linking classical statistical concepts to contemporary research in network data analysis. Of note, multiple different forms of asymptotically Gaussian behavior arise in stochastic blockmodels and are useful for different purposes, pertaining to estimation and testing, the characterization of cluster structure in community detection, and understanding latent space geometry. This paper concludes with a discussion of open problems and ongoing research activities addressing asymptotic normality and its implications for statistical network modeling.
We study a double robust Bayesian inference procedure on the average treatment effect (ATE) under unconfoundedness. Our robust Bayesian approach involves two adjustment steps: first, we make a correction for prior distributions of the conditional mean function; second, we introduce a recentering term on the posterior distribution of the resulting ATE. We prove asymptotic equivalence of our Bayesian estimator and double robust frequentist estimators by establishing a new semiparametric Bernstein-von Mises theorem under double robustness; i.e., the lack of smoothness of conditional mean functions can be compensated by high regularity of the propensity score and vice versa. Consequently, the resulting Bayesian point estimator internalizes the bias correction as the frequentist-type doubly robust estimator, and the Bayesian credible sets form confidence intervals with asymptotically exact coverage probability. In simulations, we find that this robust Bayesian procedure leads to significant bias reduction of point estimation and accurate coverage of confidence intervals, especially when the dimensionality of covariates is large relative to the sample size and the underlying functions become complex. We illustrate our method in an application to the National Supported Work Demonstration.
A Classification of Feedback Loops and Their Relation to Biases in Automated Decision-Making Systems
Prediction-based decision-making systems are becoming increasingly prevalent in various domains. Previous studies have demonstrated that such systems are vulnerable to runaway feedback loops, e.g., when police are repeatedly sent back to the same neighborhoods regardless of the actual rate of criminal activity, which exacerbate existing biases. In practice, the automated decisions have dynamic feedback effects on the system itself that can perpetuate over time, making it difficult for short-sighted design choices to control the system's evolution. While researchers started proposing longer-term solutions to prevent adverse outcomes (such as bias towards certain groups), these interventions largely depend on ad hoc modeling assumptions and a rigorous theoretical understanding of the feedback dynamics in ML-based decision-making systems is currently missing. In this paper, we use the language of dynamical systems theory, a branch of applied mathematics that deals with the analysis of the interconnection of systems with dynamic behaviors, to rigorously classify the different types of feedback loops in the ML-based decision-making pipeline. By reviewing existing scholarly work, we show that this classification covers many examples discussed in the algorithmic fairness community, thereby providing a unifying and principled framework to study feedback loops. By qualitative analysis, and through a simulation example of recommender systems, we show which specific types of ML biases are affected by each type of feedback loop. We find that the existence of feedback loops in the ML-based decision-making pipeline can perpetuate, reinforce, or even reduce ML biases.
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