The success of Bayesian inference with MCMC depends critically on Markov chains rapidly reaching the posterior distribution. Despite the plentitude of inferential theory for posteriors in Bayesian non-parametrics, convergence properties of MCMC algorithms that simulate from such ideal inferential targets are not thoroughly understood. This work focuses on the Bayesian CART algorithm which forms a building block of Bayesian Additive Regression Trees (BART). We derive upper bounds on mixing times for typical posteriors under various proposal distributions. Exploiting the wavelet representation of trees, we provide sufficient conditions for Bayesian CART to mix well (polynomially) under certain hierarchical connectivity restrictions on the signal. We also derive a negative result showing that Bayesian CART (based on simple grow and prune steps) cannot reach deep isolated signals in faster than exponential mixing time. To remediate myopic tree exploration, we propose Twiggy Bayesian CART which attaches/detaches entire twigs (not just single nodes) in the proposal distribution. We show polynomial mixing of Twiggy Bayesian CART without assuming that the signal is connected on a tree. Going further, we show that informed variants achieve even faster mixing. A thorough simulation study highlights discrepancies between spike-and-slab priors and Bayesian CART under a variety of proposals.

Due to the increasing adoption of electronic health records (EHR), large scale EHRs have become another rich data source for translational clinical research. Despite its potential, deriving generalizable knowledge from EHR data remains challenging. First, EHR data are generated as part of clinical care with data elements too detailed and fragmented for research. Despite recent progress in mapping EHR data to common ontology with hierarchical structures, much development is still needed to enable automatic grouping of local EHR codes to meaningful clinical concepts at a large scale. Second, the total number of unique EHR features is large, imposing methodological challenges to derive reproducible knowledge graph, especially when interest lies in conditional dependency structure. Third, the detailed EHR data on a very large patient cohort imposes additional computational challenge to deriving a knowledge network. To overcome these challenges, we propose to infer the conditional dependency structure among EHR features via a latent graphical block model (LGBM). The LGBM has a two layer structure with the first providing semantic embedding vector (SEV) representation for the EHR features and the second overlaying a graphical block model on the latent SEVs. The block structures on the graphical model also allows us to cluster synonymous features in EHR. We propose to learn the LGBM efficiently, in both statistical and computational sense, based on the empirical point mutual information matrix. We establish the statistical rates of the proposed estimators and show the perfect recovery of the block structure. Numerical results from simulation studies and real EHR data analyses suggest that the proposed LGBM estimator performs well in finite sample.

Fairness plays a crucial role in various multi-agent systems (e.g., communication networks, financial markets, etc.). Many multi-agent dynamical interactions can be cast as Markov Decision Processes (MDPs). While existing research has focused on studying fairness in known environments, the exploration of fairness in such systems for unknown environments remains open. In this paper, we propose a Reinforcement Learning (RL) approach to achieve fairness in multi-agent finite-horizon episodic MDPs. Instead of maximizing the sum of individual agents' value functions, we introduce a fairness function that ensures equitable rewards across agents. Since the classical Bellman's equation does not hold when the sum of individual value functions is not maximized, we cannot use traditional approaches. Instead, in order to explore, we maintain a confidence bound of the unknown environment and then propose an online convex optimization based approach to obtain a policy constrained to this confidence region. We show that such an approach achieves sub-linear regret in terms of the number of episodes. Additionally, we provide a probably approximately correct (PAC) guarantee based on the obtained regret bound. We also propose an offline RL algorithm and bound the optimality gap with respect to the optimal fair solution. To mitigate computational complexity, we introduce a policy-gradient type method for the fair objective. Simulation experiments also demonstrate the efficacy of our approach.

In this paper, we consider the motion planning problem in Gaussian belief space for minimum sensing navigation. Despite the extensive use of sampling-based algorithms and their rigorous analysis in the deterministic setting, there has been little formal analysis of the quality of their solutions returned by sampling algorithms in Gaussian belief space. This paper aims to address this lack of research by examining the asymptotic behavior of the cost of solutions obtained from Gaussian belief space based sampling algorithms as the number of samples increases. To that end, we propose a sampling based motion planning algorithm termed Information Geometric PRM* (IG-PRM*) for generating feasible paths that minimize a weighted sum of the Euclidean and an information-theoretic cost and show that the cost of the solution that is returned is guaranteed to approach the global optimum in the limit of large number of samples. Finally, we consider an obstacle-free scenario and compute the optimal solution using the "move and sense" strategy in literature. We then verify that the cost returned by our proposed algorithm converges to this optimal solution as the number of samples increases.

Markov jump processes are continuous-time stochastic processes with a wide range of applications in both natural and social sciences. Despite their widespread use, inference in these models is highly non-trivial and typically proceeds via either Monte Carlo or expectation-maximization methods. In this work we introduce an alternative, variational inference algorithm for Markov jump processes which relies on neural ordinary differential equations, and is trainable via back-propagation. Our methodology learns neural, continuous-time representations of the observed data, that are used to approximate the initial distribution and time-dependent transition probability rates of the posterior Markov jump process. The time-independent rates of the prior process are in contrast trained akin to generative adversarial networks. We test our approach on synthetic data sampled from ground-truth Markov jump processes, experimental switching ion channel data and molecular dynamics simulations. Source code to reproduce our experiments is available online.

Wu, 2017; Ma, 2020), multimodal learning (Srivastava & Salakhutdinov, 2012), and collaborative filtering (Salakhutdinov We address the problem of biased gradient estimation et al., 2007). Boltzmann machines also have the in deep Boltzmann machines (DBMs). The potential as powerful generative models because it is known existing method to obtain an unbiased estimator as a universal approximator of the probability mass function uses a maximal coupling based on a Gibbs sampler, on discrete variables (Le Roux & Bengio, 2008). Among but when the state is high-dimensional, it them, deep Boltzmann machines (DBMs) (Salakhutdinov & takes a long time to converge. In this study, we Larochelle, 2010), which are multi-layered undirected models, propose to use a coupling based on the Metropolis-can capture complex structures by their deep structure Hastings (MH) and to initialize the state around while retaining the advantages of the Boltzmann machine.

In online reinforcement learning (RL), instead of employing standard structural assumptions on Markov decision processes (MDPs), using a certain coverage condition (original from offline RL) is enough to ensure sample-efficient guarantees (Xie et al. 2023). In this work, we focus on this new direction by digging more possible and general coverage conditions, and study the potential and the utility of them in efficient online RL. We identify more concepts, including the $L^p$ variant of concentrability, the density ratio realizability, and trade-off on the partial/rest coverage condition, that can be also beneficial to sample-efficient online RL, achieving improved regret bound. Furthermore, if exploratory offline data are used, under our coverage conditions, both statistically and computationally efficient guarantees can be achieved for online RL. Besides, even though the MDP structure is given, e.g., linear MDP, we elucidate that, good coverage conditions are still beneficial to obtain faster regret bound beyond $\widetilde{O}(\sqrt{T})$ and even a logarithmic order regret. These results provide a good justification for the usage of general coverage conditions in efficient online RL.

We propose a novel task, G4C, to study teacher-student natural language interactions in a goal-driven and grounded environment. Dungeons and Dragons (D&D), a role-playing game, provides an ideal setting to investigate such interactions. Here, the Dungeon Master (DM), i.e., the teacher, guides the actions of several players -- students, each with their own personas and abilities -- to achieve shared goals grounded in a fantasy world. Our approach is to decompose and model these interactions into (1) the DM's intent to guide players toward a given goal; (2) the DM's guidance utterance to the players expressing this intent; and (3) a theory-of-mind (ToM) model that anticipates the players' reaction to the guidance one turn into the future. We develop a novel reinforcement learning (RL) method for training a DM that generates guidance for players by rewarding utterances where the intent matches the ToM-anticipated player actions. Human and automated evaluations show that a DM trained to explicitly model intents and incorporate ToM of the players using RL generates better-quality guidance that is 3x more likely to fulfill the DM's intent than a vanilla natural language generation (NLG) approach.

Via operator theoretic methods, we formalize the concentration phenomenon for a given observable `$r$' of a discrete time Markov chain with `$\mu_{\pi}$' as invariant ergodic measure, possibly having support on an unbounded state space. The main contribution of this paper is circumventing tedious probabilistic methods with a study of a composition of the Markov transition operator $P$ followed by a multiplication operator defined by $e^{r}$. It turns out that even if the observable/ reward function is unbounded, but for some for some $q>2$, $\|e^{r}\|_{q \rightarrow 2} \propto \exp\big(\mu_{\pi}(r) +\frac{2q}{q-2}\big) $ and $P$ is hyperbounded with norm control $\|P\|_{2 \rightarrow q }< e^{\frac{1}{2}[\frac{1}{2}-\frac{1}{q}]}$, sharp non-asymptotic concentration bounds follow. \emph{Transport-entropy} inequality ensures the aforementioned upper bound on multiplication operator for all $q>2$. The role of \emph{reversibility} in concentration phenomenon is demystified. These results are particularly useful for the reinforcement learning and controls communities as they allow for concentration inequalities w.r.t standard unbounded obersvables/reward functions where exact knowledge of the system is not available, let alone the reversibility of stationary measure.

Probabilistic modeling of multidimensional spatiotemporal data is critical to many real-world applications. As real-world spatiotemporal data often exhibits complex dependencies that are nonstationary and nonseparable, developing effective and computationally efficient statistical models to accommodate nonstationary/nonseparable processes containing both long-range and short-scale variations becomes a challenging task, in particular for large-scale datasets with various corruption/missing structures. In this paper, we propose a new statistical framework -- Bayesian Complementary Kernelized Learning (BCKL) -- to achieve scalable probabilistic modeling for multidimensional spatiotemporal data. To effectively characterize complex dependencies, BCKL integrates two complementary approaches -- kernelized low-rank tensor factorization and short-range spatiotemporal Gaussian Processes. Specifically, we use a multi-linear low-rank factorization component to capture the global/long-range correlations in the data and introduce an additive short-scale GP based on compactly supported kernel functions to characterize the remaining local variabilities. We develop an efficient Markov chain Monte Carlo (MCMC) algorithm for model inference and evaluate the proposed BCKL framework on both synthetic and real-world spatiotemporal datasets. Our experiment results show that BCKL offers superior performance in providing accurate posterior mean and high-quality uncertainty estimates, confirming the importance of both global and local components in modeling spatiotemporal data.