Recent advances in artificial intelligence have seen Large Language Models (LLMs) demonstrate notable proficiency in causal discovery tasks. This study explores the factors influencing the performance of LLMs in causal discovery tasks. Utilizing open-source LLMs, we examine how the frequency of causal relations within their pre-training corpora affects their ability to accurately respond to causal discovery queries. Our findings reveal that a higher frequency of causal mentions correlates with better model performance, suggesting that extensive exposure to causal information during training enhances the models' causal discovery capabilities. Additionally, we investigate the impact of context on the validity of causal relations. Our results indicate that LLMs might exhibit divergent predictions for identical causal relations when presented in different contexts. This paper provides the first comprehensive analysis of how different factors contribute to LLM performance in causal discovery tasks.

We consider the problem of Bayesian regression with trustworthy uncertainty quantification. We define that the uncertainty quantification is trustworthy if the ground truth can be captured by intervals dependent on the predictive distributions with a pre-specified probability. Furthermore, we propose, Trust-Bayes, a novel optimization framework for Bayesian meta learning which is cognizant of trustworthy uncertainty quantification without explicit assumptions on the prior model/distribution of the functions. We characterize the lower bounds of the probabilities of the ground truth being captured by the specified intervals and analyze the sample complexity with respect to the feasible probability for trustworthy uncertainty quantification. Monte Carlo simulation of a case study using Gaussian process regression is conducted for verification and comparison with the Meta-prior algorithm.

This paper describes a discrete state-space model -- and accompanying methods -- for generative modelling. This model generalises partially observed Markov decision processes to include paths as latent variables, rendering it suitable for active inference and learning in a dynamic setting. Specifically, we consider deep or hierarchical forms using the renormalisation group. The ensuing renormalising generative models (RGM) can be regarded as discrete homologues of deep convolutional neural networks or continuous state-space models in generalised coordinates of motion. By construction, these scale-invariant models can be used to learn compositionality over space and time, furnishing models of paths or orbits; i.e., events of increasing temporal depth and itinerancy. This technical note illustrates the automatic discovery, learning and deployment of RGMs using a series of applications. We start with image classification and then consider the compression and generation of movies and music. Finally, we apply the same variational principles to the learning of Atari-like games.

Determining the number of clusters is a fundamental issue in data clustering. Several algorithms have been proposed, including centroid-based algorithms using the Euclidean distance and model-based algorithms using a mixture of probability distributions. Among these, greedy algorithms for searching the number of clusters by repeatedly splitting or merging clusters have advantages in terms of computation time for problems with large sample sizes. However, studies comparing these methods in systematic evaluation experiments still need to be included. This study examines centroid- and model-based cluster search algorithms in various cases that Gaussian mixture models (GMMs) can generate. The cases are generated by combining five factors: dimensionality, sample size, the number of clusters, cluster overlap, and covariance type. The results show that some cluster-splitting criteria based on Euclidean distance make unreasonable decisions when clusters overlap. The results also show that model-based algorithms are insensitive to covariance type and cluster overlap compared to the centroid-based method if the sample size is sufficient. Our cluster search implementation codes are available at https://github.com/lipryou/searchClustK

Models for categorical sequences typically assume exchangeable or first-order dependent sequence elements. These are common assumptions, for example, in models of computer malware traces and protein sequences. Although such simplifying assumptions lead to computational tractability, these models fail to capture long-range, complex dependence structures that may be harnessed for greater predictive power. To this end, a Bayesian modelling framework is proposed to parsimoniously capture rich dependence structures in categorical sequences, with memory efficiency suitable for real-time processing of data streams. Parsimonious Bayesian context trees are introduced as a form of variable-order Markov model with conjugate prior distributions. The novel framework requires fewer parameters than fixed-order Markov models by dropping redundant dependencies and clustering sequential contexts. Approximate inference on the context tree structure is performed via a computationally efficient model-based agglomerative clustering procedure. The proposed framework is tested on synthetic and real-world data examples, and it outperforms existing sequence models when fitted to real protein sequences and honeypot computer terminal sessions.

Since its introduction to the public, ChatGPT has had an unprecedented impact. While some experts praised AI advancements and highlighted their potential risks, others have been critical about the accuracy and usefulness of Large Language Models (LLMs). In this paper, we are interested in the ability of LLMs to identify causal relationships. We focus on the well-established GPT-4 (Turbo) and evaluate its performance under the most restrictive conditions, by isolating its ability to infer causal relationships based solely on the variable labels without being given any context, demonstrating the minimum level of effectiveness one can expect when it is provided with label-only information. We show that questionnaire participants judge the GPT-4 graphs as the most accurate in the evaluated categories, closely followed by knowledge graphs constructed by domain experts, with causal Machine Learning (ML) far behind. We use these results to highlight the important limitation of causal ML, which often produces causal graphs that violate common sense, affecting trust in them. However, we show that pairing GPT-4 with causal ML overcomes this limitation, resulting in graphical structures learnt from real data that align more closely with those identified by domain experts, compared to structures learnt by causal ML alone. Overall, our findings suggest that despite GPT-4 not being explicitly designed to reason causally, it can still be a valuable tool for causal representation, as it improves the causal discovery process of causal ML algorithms that are designed to do just that.

In this work, we present a sampling algorithm for single hidden layer neural networks. This algorithm is built upon a recursive series of Bayesian posteriors using a method we call Greedy Bayes. Sampling of the Bayesian posterior for neuron weight vectors $w$ of dimension $d$ is challenging because of its multimodality. Our algorithm to tackle this problem is based on a coupling of the posterior density for $w$ with an auxiliary random variable $\xi$. The resulting reverse conditional $w|\xi$ of neuron weights given auxiliary random variable is shown to be log concave. In the construction of the posterior distributions we provide some freedom in the choice of the prior. In particular, for Gaussian priors on $w$ with suitably small variance, the resulting marginal density of the auxiliary variable $\xi$ is proven to be strictly log concave for all dimensions $d$. For a uniform prior on the unit $\ell_1$ ball, evidence is given that the density of $\xi$ is again strictly log concave for sufficiently large $d$. The score of the marginal density of the auxiliary random variable $\xi$ is determined by an expectation over $w|\xi$ and thus can be computed by various rapidly mixing Markov Chain Monte Carlo methods. Moreover, the computation of the score of $\xi$ permits methods of sampling $\xi$ by a stochastic diffusion (Langevin dynamics) with drift function built from this score. With such dynamics, information-theoretic methods pioneered by Bakry and Emery show that accurate sampling of $\xi$ is obtained rapidly when its density is indeed strictly log-concave. After which, one more draw from $w|\xi$, produces neuron weights $w$ whose marginal distribution is from the desired posterior.

The Expectation Maximization (EM) algorithm has been a cent ral part of the statistician's toolbox since being formalised by [ 22 ] as an effective general computational solution to the marginal maximum likelihood problem. At that time the algor ithm had been proposed previously in numerous special contexts, including that of empirical Bayes [ 27 ]. Empirical Bayes methods have received considerable attention in the m odern machine learning literature, where they are widely used to specify hyper-paramete rs in high-dimensional models. In recent years there has been a great deal of interest within the Bayesian statistics and machine learning communities in the construction of gradie nt flows, especially Wasserstein gradient flows, which underlie Langevin Monte Carlo algorit hms. Some recent work has focussed on the intersection of empirical Bayes type method s and gradient flow-based algorithms. Our aim is to demonstrate here that some of the tools, particularly those emerging from optimal transport and Wasserstein geometry, which hav e been developed in the context of these modern computational methods provide a natura l approach to the analysis of the EM algorithm itself--and many of its approximations. S uch analysis is quite direct, requires limited further technical work and yields state-o f-the-art conclusions under conditions which are, if anything, weaker than those ordinaril y employed in the quantitative analysis of EM algorithms. 1 In this paper we utilize the connection between EM and a coord inate-wise minimization algorithm applied to the free energy functional identified b y [ 43 ] to provide non-asymptotic error bounds for EM algorithms under an extended form of the l og-Sobolev inequality. To do this, we extend an argument commonly used to understand Eu clidean coordinate descent algorithms by comparison with gradient descent via the desc ent lemma [ 9, 8, 10 ], together with recently developed results for using and understandin g gradients on the product of Euclidean and Wasserstein spaces [ 13 ].

It is designed to help students and researchers to quickly familiarize themselves with the area and to provide a foundation for the development of university courses on the mathematics of deep learning. Our main goal in the composition of this book was to present various rigorous, but easy to grasp, results that help to build an understanding of fundamental mathematical concepts in deep learning. To achieve this, we prioritize simplicity over generality. As a mathematical introduction to deep learning, this book does not aim to give an exhaustive survey of the entire (and rapidly growing) field, and some important research directions are missing. In particular, we have favored mathematical results over empirical research, even though an accurate account of the theory of deep learning requires both.

Sequential Monte Carlo Squared (SMC$^2$) is a Bayesian method which can infer the states and parameters of non-linear, non-Gaussian state-space models. The standard random-walk proposal in SMC$^2$ faces challenges, particularly with high-dimensional parameter spaces. This study outlines a novel approach by harnessing first-order gradients derived from a Common Random Numbers - Particle Filter (CRN-PF) using PyTorch. The resulting gradients can be leveraged within a Langevin proposal without accept/reject. Including Langevin dynamics within the proposal can result in a higher effective sample size and more accurate parameter estimates when compared with the random-walk. The resulting algorithm is parallelized on distributed memory using Message Passing Interface (MPI) and runs in $\mathcal{O}(\log_2N)$ time complexity. Utilizing 64 computational cores we obtain a 51x speed-up when compared to a single core. A GitHub link is given which provides access to the code.