This paper is intended to appear as a chapter for the Handbook of Markov Chain Monte Carlo. The goal of this chapter is to unify various problems at the intersection of Markov chain Monte Carlo (MCMC) and machine learning$\unicode{x2014}$which includes black-box variational inference, adaptive MCMC, normalizing flow construction and transport-assisted MCMC, surrogate-likelihood MCMC, coreset construction for MCMC with big data, Markov chain gradient descent, Markovian score climbing, and more$\unicode{x2014}$within one common framework. By doing so, the theory and methods developed for each may be translated and generalized.

Inspired by Bayesian approaches to brain function in neuroscience, we give a simple theory of probabilistic inference for a unified account of reasoning and learning. We simply model how data cause symbolic knowledge in terms of its satisfiability in formal logic. The underlying idea is that reasoning is a process of deriving symbolic knowledge from data via abstraction, i.e., selective ignorance. The logical consequence relation is discussed for its proof-based theoretical correctness. The MNIST dataset is discussed for its experiment-based empirical correctness.

This study tackles the challenges of adversarial corruption in model-based reinforcement learning (RL), where the transition dynamics can be corrupted by an adversary. Existing studies on corruption-robust RL mostly focus on the setting of model-free RL, where robust least-square regression is often employed for value function estimation. However, these techniques cannot be directly applied to model-based RL. In this paper, we focus on model-based RL and take the maximum likelihood estimation (MLE) approach to learn transition model. Our work encompasses both online and offline settings. In the online setting, we introduce an algorithm called corruption-robust optimistic MLE (CR-OMLE), which leverages total-variation (TV)-based information ratios as uncertainty weights for MLE. We prove that CR-OMLE achieves a regret of $\tilde{\mathcal{O}}(\sqrt{T} + C)$, where $C$ denotes the cumulative corruption level after $T$ episodes. We also prove a lower bound to show that the additive dependence on $C$ is optimal. We extend our weighting technique to the offline setting, and propose an algorithm named corruption-robust pessimistic MLE (CR-PMLE). Under a uniform coverage condition, CR-PMLE exhibits suboptimality worsened by $\mathcal{O}(C/n)$, nearly matching the lower bound. To the best of our knowledge, this is the first work on corruption-robust model-based RL algorithms with provable guarantees.

Abstract-- This work develops a distributed optimization algorithm for multi-robot 3-D semantic mapping using streaming range and visual observations and single-hop communication. Our approach relies on gradient-based optimization of the observation log-likelihood of each robot subject to a map consensus constraint to build a common multi-class map of the environment. This formulation leads to closed-form updates which resemble Bayes rule with one-hop prior averaging. To reduce the amount of information exchanged among the robots, we utilize an octree data structure that compresses the multiclass map distribution using adaptive-resolution. Figure 1: Semantically annotated point cloud obtained from a range and category observation z Dense semantically-rich environment representations, such as multi-class volumetric maps [1], can be built online by distribution by each robot. Through a simulated experiment, mobile robots thanks to the availability of on-board GPUaccelerated we show that our algorithm leads to globally consistent segmentation models.

For example, they have the implicit assumption consequence relation, an empirical consequence that the method used to extract symbolic knowledge relation, maximal consistent sets, maximal possible from data cannot be applied to the method used to perform sets and maximum likelihood estimation. The logical reasoning over the symbolic knowledge, and vice theory gives new insights into reasoning towards versa.

Prompt tuning, in which prompts are optimized to adapt large-scale pre-trained language models to downstream tasks instead of fine-tuning the full model parameters, has been shown to be particularly effective when the prompts are trained in a multi-task transfer learning setting. These methods generally involve individually training prompts for each source task and then aggregating them to provide the initialization of the prompt for the target task. However, this approach critically ignores the fact that some of the source tasks could be negatively or positively interfering with each other. We argue that when we extract knowledge from source tasks via training source prompts, we need to consider this correlation among source tasks for better transfer to target tasks. To this end, we propose a Bayesian approach where we work with the posterior distribution of prompts across source tasks. We obtain representative source prompts corresponding to the samples from the posterior utilizing Stein Variational Gradient Descent, which are then aggregated to constitute the initial target prompt. We show extensive experimental results on the standard benchmark NLP tasks, where our Bayesian multi-task transfer learning approach outperforms the state-of-the-art methods in many settings. Furthermore, our approach requires no auxiliary models other than the prompt itself, achieving a high degree of parameter efficiency.

We present novel bounds for estimating discrete probability distributions under the $\ell_\infty$ norm. These are nearly optimal in various precise senses, including a kind of instance-optimality. Our data-dependent convergence guarantees for the maximum likelihood estimator significantly improve upon the currently known results. A variety of techniques are utilized and innovated upon, including Chernoff-type inequalities and empirical Bernstein bounds. We illustrate our results in synthetic and real-world experiments. Finally, we apply our proposed framework to a basic selective inference problem, where we estimate the most frequent probabilities in a sample.

Motivated by modern applications such as computerized adaptive testing, sequential rank aggregation, and heterogeneous data source selection, we study the problem of active sequential estimation, which involves adaptively selecting experiments for sequentially collected data. The goal is to design experiment selection rules for more accurate model estimation. Greedy information-based experiment selection methods, optimizing the information gain for one-step ahead, have been employed in practice thanks to their computational convenience, flexibility to context or task changes, and broad applicability. However, statistical analysis is restricted to one-dimensional cases due to the problem's combinatorial nature and the seemingly limited capacity of greedy algorithms, leaving the multidimensional problem open. In this study, we close the gap for multidimensional problems. In particular, we propose adopting a class of greedy experiment selection methods and provide statistical analysis for the maximum likelihood estimator following these selection rules. This class encompasses both existing methods and introduces new methods with improved numerical efficiency. We prove that these methods produce consistent and asymptotically normal estimators. Additionally, within a decision theory framework, we establish that the proposed methods achieve asymptotic optimality when the risk measure aligns with the selection rule. We also conduct extensive numerical studies on both simulated and real data to illustrate the efficacy of the proposed methods. From a technical perspective, we devise new analytical tools to address theoretical challenges. These analytical tools are of independent theoretical interest and may be reused in related problems involving stochastic approximation and sequential designs.

In this paper, we are concerned with estimating the joint probability of random variables $X$ and $Y$, given $N$ independent observation blocks $(\boldsymbol{x}^i,\boldsymbol{y}^i)$, $i=1,\ldots,N$, each of $M$ samples $(\boldsymbol{x}^i,\boldsymbol{y}^i) = \bigl((x^i_j, y^i_{\sigma^i(j)}) \bigr)_{j=1}^M$, where $\sigma^i$ denotes an unknown permutation of i.i.d. sampled pairs $(x^i_j,y_j^i)$, $j=1,\ldots,M$. This means that the internal ordering of the $M$ samples within an observation block is not known. We derive a maximum-likelihood inference functional, propose a computationally tractable approximation and analyze their properties. In particular, we prove a $\Gamma$-convergence result showing that we can recover the true density from empirical approximations as the number $N$ of blocks goes to infinity. Using entropic optimal transport kernels, we model a class of hypothesis spaces of density functions over which the inference functional can be minimized. This hypothesis class is particularly suited for approximate inference of transfer operators from data. We solve the resulting discrete minimization problem by a modification of the EMML algorithm to take addional transition probability constraints into account and prove the convergence of this algorithm. Proof-of-concept examples demonstrate the potential of our method.

Causal graph discovery is a significant problem with applications across various disciplines. However, with observational data alone, the underlying causal graph can only be recovered up to its Markov equivalence class, and further assumptions or interventions are necessary to narrow down the true graph. This work addresses the causal discovery problem under the setting of stochastic interventions with the natural goal of minimizing the number of interventions performed. We propose the following stochastic intervention model which subsumes existing adaptive noiseless interventions in the literature while capturing scenarios such as fat-hand interventions and CRISPR gene knockouts: any intervention attempt results in an actual intervention on a random subset of vertices, drawn from a distribution dependent on attempted action. Under this model, we study the two fundamental problems in causal discovery of verification and search and provide approximation algorithms with polylogarithmic competitive ratios and provide some preliminary experimental results.