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.

Causal discovery from observational data holds great promise, but existing methods rely on strong assumptions about the underlying causal structure, often requiring full observability of all relevant variables. We tackle these challenges by leveraging the score function $\nabla \log p(X)$ of observed variables for causal discovery and propose the following contributions. First, we generalize the existing results of identifiability with the score to additive noise models with minimal requirements on the causal mechanisms. Second, we establish conditions for inferring causal relations from the score even in the presence of hidden variables; this result is two-faced: we demonstrate the score's potential as an alternative to conditional independence tests to infer the equivalence class of causal graphs with hidden variables, and we provide the necessary conditions for identifying direct causes in latent variable models. Building on these insights, we propose a flexible algorithm for causal discovery across linear, nonlinear, and latent variable models, which we empirically validate.

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 ].

We propose a variational inference approach to sample from the posterior distribution for solving inverse problems. From a pre-trained diffusion model, our approach trains a conditional flow model to minimize the divergence between the proposal variational distribution and the posterior distribution implicitly defined through the diffusion model. Once trained, the flow model is capable of sampling from the posterior distribution with a single NFE, amortized with respect to the measurement. The proposed method paves a new path for distilling a diffusion prior for efficient posterior sampling. We show that our method is applicable to standard signals in Euclidean space, as well as signals on manifold.

Boolean circuits in d-DNNF form enable tractable probabilistic inference. However, as a key insight of this work, we show that commonly used d-DNNF compilation approaches introduce irrelevant subcircuits. We call these subcircuits Tseitin artifacts, as they are introduced due to the Tseitin transformation step -- a well-established procedure to transform any circuit into the CNF format required by several d-DNNF knowledge compilers. We discuss how to detect and remove both Tseitin variables and Tseitin artifacts, leading to more succinct circuits. We empirically observe an average size reduction of 77.5% when removing both Tseitin variables and artifacts. The additional pruning of Tseitin artifacts reduces the size by 22.2% on average. This significantly improves downstream tasks that benefit from a more succinct circuit, e.g., probabilistic inference tasks.

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.

Humans and many animals show remarkably adaptive behavior and can respond differently to the same input depending on their internal goals. The brain not only represents the intermediate abstractions needed to perform a computation but also actively maintains a representation of the computation itself (task abstraction). Such separation of the computation and its abstraction is associated with faster learning, flexible decision-making, and broad generalization capacity. We investigate if such benefits might extend to neural networks trained with task abstractions. For such benefits to emerge, one needs a task inference mechanism that possesses two crucial abilities: First, the ability to infer abstract task representations when no longer explicitly provided (task inference), and second, manipulate task representations to adapt to novel problems (task recomposition). To tackle this, we cast task inference as an optimization problem from a variational inference perspective and ground our approach in an expectation-maximization framework. We show that gradients backpropagated through a neural network to a task representation layer are an efficient heuristic to infer current task demands, a process we refer to as gradient-based inference (GBI). Further iterative optimization of the task representation layer allows for recomposing abstractions to adapt to novel situations. Using a toy example, a novel image classifier, and a language model, we demonstrate that GBI provides higher learning efficiency and generalization to novel tasks and limits forgetting. Moreover, we show that GBI has unique advantages such as preserving information for uncertainty estimation and detecting out-of-distribution samples.

We propose a globally optimal Bayesian network structure discovery algorithm based on a progressively leveled scoring approach. Bayesian network structure discovery is a fundamental yet NP-hard problem in the field of probabilistic graphical models, and as the number of variables increases, memory usage grows exponentially. The simple and effective method proposed by Silander and Myllym\"aki has been widely applied in this field, as it incrementally calculates local scores to achieve global optimality. However, existing methods that utilize disk storage, while capable of handling networks with a larger number of variables, introduce issues such as latency, fragmentation, and additional overhead associated with disk I/O operations. To avoid these problems, we explore how to further enhance computational efficiency and reduce peak memory usage using only memory. We introduce an efficient hierarchical computation method that requires only a single traversal of all local structures, retaining only the data and information necessary for the current computation, thereby improving efficiency and significantly reducing memory requirements. Experimental results indicate that our method, when using only memory, not only reduces peak memory usage but also improves computational efficiency compared to existing methods, demonstrating good scalability for handling larger networks and exhibiting stable experimental results. Ultimately, we successfully achieved the processing of a Bayesian network with 28 variables using only memory.

Recent studies seek to provide Graph Neural Network (GNN) interpretability via multiple unsupervised learning models. Due to the scarcity of datasets, current methods easily suffer from learning bias. To solve this problem, we embed a Large Language Model (LLM) as knowledge into the GNN explanation network to avoid the learning bias problem. We inject LLM as a Bayesian Inference (BI) module to mitigate learning bias. The efficacy of the BI module has been proven both theoretically and experimentally. We conduct experiments on both synthetic and real-world datasets. The innovation of our work lies in two parts: 1. We provide a novel view of the possibility of an LLM functioning as a Bayesian inference to improve the performance of existing algorithms; 2. We are the first to discuss the learning bias issues in the GNN explanation problem.

Colorectal polyps are generally benign alterations that, if not identified promptly and managed successfully, can progress to cancer and cause affectations on the colon mucosa, known as adenocarcinoma. Today advances in Deep Learning have demonstrated the ability to achieve significant performance in image classification and detection in medical diagnosis applications. Nevertheless, these models are prone to overfitting, and making decisions based only on point estimations may provide incorrect predictions. Thus, to obtain a more informed decision, we must consider point estimations along with their reliable uncertainty quantification. In this paper, we built different Bayesian neural network approaches based on the flexibility of posterior distribution to develop semantic segmentation of colorectal polyp images. We found that these models not only provide state-of-the-art performance on the segmentation of this medical dataset but also, yield accurate uncertainty estimates. We applied multiplicative normalized flows(MNF) and reparameterization trick on the UNET, FPN, and LINKNET architectures tested with multiple backbones in deterministic and Bayesian versions. We report that the FPN + EfficientnetB7 architecture with MNF is the most promising option given its IOU of 0.94 and Expected Calibration Error (ECE) of 0.004, combined with its superiority in identifying difficult-to-detect colorectal polyps, which is effective in clinical areas where early detection prevents the development of colon cancer.