The Kalman Filter (KF) is a powerful mathematical tool widely used for state estimation in various domains, including Simultaneous Localization and Mapping (SLAM). This paper presents an in-depth introduction to the Kalman Filter and explores its several extensions: the Extended Kalman Filter (EKF), the Error-State Kalman Filter (ESKF), the Iterated Extended Kalman Filter (IEKF), and the Iterated Error-State Kalman Filter (IESKF). Each variant is meticulously examined, with detailed derivations of their mathematical formulations and discussions on their respective advantages and limitations. By providing a comprehensive overview of these techniques, this paper aims to offer valuable insights into their applications in SLAM and enhance the understanding of state estimation methodologies in complex environments.

Recent work has suggested using Monte Carlo methods based on piecewise deterministic Markov processes (PDMPs) to sample from target distributions of interest. PDMPs are non-reversible continuous-time processes endowed with momentum, and hence can mix better than standard reversible MCMC samplers. Furthermore, they can incorporate exact sub-sampling schemes which only require access to a single (randomly selected) data point at each iteration, yet without introducing bias to the algorithm's stationary distribution. However, the range of models for which PDMPs can be used, particularly with sub-sampling, is limited. We propose approximate simulation of PDMPs with sub-sampling for scalable sampling from posterior distributions. The approximation takes the form of an Euler approximation to the true PDMP dynamics, and involves using an estimate of the gradient of the log-posterior based on a data sub-sample. We thus call this class of algorithms stochastic-gradient PDMPs. Importantly, the trajectories of stochastic-gradient PDMPs are continuous and can leverage recent ideas for sampling from measures with continuous and atomic components. We show these methods are easy to implement, present results on their approximation error and demonstrate numerically that this class of algorithms has similar efficiency to, but is more robust than, stochastic gradient Langevin dynamics.

Discovering causal structures in data is a challenging task. Ideally, we would like to input the data into an algorithm that outputs one or more plausible causal directed acyclic graphs (DAGs) linking the variables in the data. Such data driven approaches for estimating a causal DAG are known as causal discovery (or causal search, structure learning etc.). While algorithms for causal discovery were first developed in the field of computer science more than twenty years ago [1] and have, since then, continually been generalised and refined [2], their use with biomedical or epidemiological data is still rare (other than in genetics [3]). Exceptions are, for example, two applications of causal discovery to data from cohort studies finding that modifiable risk factors in early childhood or early life have mostly indirect, if any, causal relations with later health outcomes[4, 5]. Similarly, in an analysis of healthcare data considering cardiac surgery it was found that many of the known predictors were in fact only indirect causes of postoperative length of stay [6]. Also, it has been suggested to use causal discovery to improve the quality of care for hip replacement patients by investigating the complex clinical performance of implants with data from large patient registries [7]. Typical causal analyses often aim at causal effect estimation. In contrast to the above, such analyses typically assume the causal structure, i.e. the DAG, to be given, usually derived from domain

Agent-based models (ABM) provide an excellent framework for modeling outbreaks and interventions in epidemiology by explicitly accounting for diverse individual interactions and environments. However, these models are usually stochastic and highly parametrized, requiring precise calibration for predictive performance. When considering realistic numbers of agents and properly accounting for stochasticity, this high dimensional calibration can be computationally prohibitive. This paper presents a random forest based surrogate modeling technique to accelerate the evaluation of ABMs and demonstrates its use to calibrate an epidemiological ABM named CityCOVID via Markov chain Monte Carlo (MCMC). The technique is first outlined in the context of CityCOVID's quantities of interest, namely hospitalizations and deaths, by exploring dimensionality reduction via temporal decomposition with principal component analysis (PCA) and via sensitivity analysis. The calibration problem is then presented and samples are generated to best match COVID-19 hospitalization and death numbers in Chicago from March to June in 2020. These results are compared with previous approximate Bayesian calibration (IMABC) results and their predictive performance is analyzed showing improved performance with a reduction in computation.

We consider the problem of selecting an optimal subset of information sources for a hypothesis testing/classification task where the goal is to identify the true state of the world from a finite set of hypotheses, based on finite observation samples from the sources. In order to characterize the learning performance, we propose a misclassification penalty framework, which enables nonuniform treatment of different misclassification errors. In a centralized Bayesian learning setting, we study two variants of the subset selection problem: (i) selecting a minimum cost information set to ensure that the maximum penalty of misclassifying the true hypothesis is below a desired bound and (ii) selecting an optimal information set under a limited budget to minimize the maximum penalty of misclassifying the true hypothesis. Under certain assumptions, we prove that the objective (or constraints) of these combinatorial optimization problems are weak (or approximate) submodular, and establish high-probability performance guarantees for greedy algorithms. Further, we propose an alternate metric for information set selection which is based on the total penalty of misclassification. We prove that this metric is submodular and establish near-optimal guarantees for the greedy algorithms for both the information set selection problems. Finally, we present numerical simulations to validate our theoretical results over several randomly generated instances.

In sampling tasks, it is common for target distributions to be known up to a normalising constant. However, in many situations, evaluating even the unnormalised distribution can be costly or infeasible. This issue arises in scenarios such as sampling from the Bayesian posterior for tall datasets and the 'doubly-intractable' distributions. In this paper, we begin by observing that seemingly different Markov chain Monte Carlo (MCMC) algorithms, such as the exchange algorithm, PoissonMH, and TunaMH, can be unified under a simple common procedure. We then extend this procedure into a novel framework that allows the use of auxiliary variables in both the proposal and acceptance-rejection steps. We develop the theory of the new framework, applying it to existing algorithms to simplify and extend their results. Several new algorithms emerge from this framework, with improved performance demonstrated on both synthetic and real datasets.

In recent years, transformer-based models have revolutionized deep learning, particularly in sequence modeling. To better understand this phenomenon, there is a growing interest in using Markov input processes to study transformers. However, our current understanding in this regard remains limited with many fundamental questions about how transformers learn Markov chains still unanswered. In this paper, we address this by focusing on first-order Markov chains and single-layer transformers, providing a comprehensive characterization of the learning dynamics in this context. Specifically, we prove that transformer parameters trained on next-token prediction loss can either converge to global or local minima, contingent on the initialization and the Markovian data properties, and we characterize the precise conditions under which this occurs. To the best of our knowledge, this is the first result of its kind highlighting the role of initialization. We further demonstrate that our theoretical findings are corroborated by empirical evidence. Based on these insights, we provide guidelines for the initialization of transformer parameters and demonstrate their effectiveness. Finally, we outline several open problems in this arena. Code is available at: https://github.com/Bond1995/Markov.

Determining the chronology of ancient handwritten manuscripts is essential for reconstructing the evolution of ideas. For the Dead Sea Scrolls, this is particularly important. However, there is an almost complete lack of date-bearing manuscripts evenly distributed across the timeline and written in similar scripts available for palaeographic comparison. Here, we present Enoch, a state-of-the-art AI-based date-prediction model, trained on the basis of new radiocarbon-dated samples of the scrolls. Enoch uses established handwriting-style descriptors and applies Bayesian ridge regression. The challenge of this study is that the number of radiocarbon-dated manuscripts is small, while current machine learning requires an abundance of training data. We show that by using combined angular and allographic writing style feature vectors and applying Bayesian ridge regression, Enoch could predict the radiocarbon-based dates from style, supported by leave-one-out validation, with varied MAEs of 27.9 to 30.7 years relative to the radiocarbon dating. Enoch was then used to estimate the dates of 135 unseen manuscripts, revealing that 79 per cent of the samples were considered 'realistic' upon palaeographic post-hoc evaluation. We present a new chronology of the scrolls. The radiocarbon ranges and Enoch's style-based predictions are often older than the traditionally assumed palaeographic estimates. In the range of 300-50 BCE, Enoch's date prediction provides an improved granularity. The study is in line with current developments in multimodal machine-learning techniques, and the methods can be used for date prediction in other partially-dated manuscript collections. This research shows how Enoch's quantitative, probability-based approach can be a tool for palaeographers and historians, re-dating ancient Jewish key texts and contributing to current debates on Jewish and Christian origins.

In many real-world decision problems there is partially observed, hidden or latent information that remains fixed throughout an interaction. Such decision problems can be modeled as Latent Markov Decision Processes (LMDPs), where a latent variable is selected at the beginning of an interaction and is not disclosed to the agent. In the last decade, there has been significant progress in solving LMDPs under different structural assumptions. However, for general LMDPs, there is no known learning algorithm that provably matches the existing lower bound (Kwon et al., 2021). We introduce the first sample-efficient algorithm for LMDPs without any additional structural assumptions. Our result builds off a new perspective on the role of off-policy evaluation guarantees and coverage coefficients in LMDPs, a perspective, that has been overlooked in the context of exploration in partially observed environments. Specifically, we establish a novel off-policy evaluation lemma and introduce a new coverage coefficient for LMDPs. Then, we show how these can be used to derive near-optimal guarantees of an optimistic exploration algorithm. These results, we believe, can be valuable for a wide range of interactive learning problems beyond LMDPs, and especially, for partially observed environments.

Automatic speech recognition (ASR) is a key area in computational linguistics, focusing on developing technologies that enable computers to convert spoken language into text. This field combines linguistics and machine learning. ASR models, which map speech audio to transcripts through supervised learning, require handling real and unrestricted text. Text-to-speech systems directly work with real text, while ASR systems rely on language models trained on large text corpora. High-quality transcribed data is essential for training predictive models. The research involved two main components: developing a web application and designing a web interface for speech recognition. The web application, created with JavaScript and Node.js, manages large volumes of audio files and their transcriptions, facilitating collaborative human correction of ASR transcripts. It operates in real-time using a client-server architecture. The web interface for speech recognition records 16 kHz mono audio from any device running the web app, performs voice activity detection (VAD), and sends the audio to the recognition engine. VAD detects human speech presence, aiding efficient speech processing and reducing unnecessary processing during non-speech intervals, thus saving computation and network bandwidth in VoIP applications. The final phase of the research tested a neural network for accurately aligning the speech signal to hidden Markov model (HMM) states. This included implementing a novel backpropagation method that utilizes prior statistics of node co-activations.