Herding is a deterministic algorithm used to generate data points that can be regarded as random samples satisfying input moment conditions. The algorithm is based on the complex behavior of a high-dimensional dynamical system and is inspired by the maximum entropy principle of statistical inference. In this paper, we propose an extension of the herding algorithm, called entropic herding, which generates a sequence of distributions instead of points. Entropic herding is derived as the optimization of the target function obtained from the maximum entropy principle. Using the proposed entropic herding algorithm as a framework, we discuss a closer connection between herding and the maximum entropy principle. Specifically, we interpret the original herding algorithm as a tractable version of entropic herding, the ideal output distribution of which is mathematically represented. We further discuss how the complex behavior of the herding algorithm contributes to optimization. We argue that the proposed entropic herding algorithm extends the application of herding to probabilistic modeling. In contrast to original herding, entropic herding can generate a smooth distribution such that both efficient probability density calculation and sample generation become possible. To demonstrate the viability of these arguments in this study, numerical experiments were conducted, including a comparison with other conventional methods, on both synthetic and real data.

A Bayesian Network is a directed acyclic graph (DAG) on a set of $n$ random variables (identified with the vertices); a Bayesian Network Distribution (BND) is a probability distribution on the rv's that is Markovian on the graph. A finite mixture of such models is the projection on these variables of a BND on the larger graph which has an additional "hidden" (or "latent") random variable $U$, ranging in $\{1,\ldots,k\}$, and a directed edge from $U$ to every other vertex. Models of this type are fundamental to research in Causal Inference, where $U$ models a confounding effect. One extremely special case has been of longstanding interest in the theory literature: the empty graph. Such a distribution is simply a mixture of $k$ product distributions. A longstanding problem has been, given the joint distribution of a mixture of $k$ product distributions, to identify each of the product distributions, and their mixture weights. Our results are: (1) We improve the sample complexity (and runtime) for identifying mixtures of $k$ product distributions from $\exp(O(k^2))$ to $\exp(O(k \log k))$. This is almost best possible in view of a known $\exp(\Omega(k))$ lower bound. (2) We give the first algorithm for the case of non-empty graphs. The complexity for a graph of maximum degree $\Delta$ is $\exp(O(k(\Delta^2 + \log k)))$. (The above complexities are approximate and suppress dependence on secondary parameters.)

Currently, it is hard to reap the benefits of deep learning for Bayesian methods, which allow the explicit specification of prior knowledge and accurately capture model uncertainty. We present Prior-Data Fitted Networks (PFNs). PFNs leverage large-scale machine learning techniques to approximate a large set of posteriors. The only requirement for PFNs to work is the ability to sample from a prior distribution over supervised learning tasks (or functions). Our method restates the objective of posterior approximation as a supervised classification problem with a set-valued input: it repeatedly draws a task (or function) from the prior, draws a set of data points and their labels from it, masks one of the labels and learns to make probabilistic predictions for it based on the set-valued input of the rest of the data points. Presented with a set of samples from a new supervised learning task as input, PFNs make probabilistic predictions for arbitrary other data points in a single forward propagation, having learned to approximate Bayesian inference. We demonstrate that PFNs can near-perfectly mimic Gaussian processes and also enable efficient Bayesian inference for intractable problems, with over 200-fold speedups in multiple setups compared to current methods. We obtain strong results in very diverse areas such as Gaussian process regression, Bayesian neural networks, classification for small tabular data sets, and few-shot image classification, demonstrating the generality of PFNs. Code and trained PFNs are released at https://github.com/automl/TransformersCanDoBayesianInference.

We are interested in privatizing an approximate posterior inference algorithm called Expectation Propagation (EP). EP approximates the posterior by iteratively refining approximations to the local likelihoods, and is known to provide better posterior uncertainties than those by variational inference (VI). However, using EP for large-scale datasets imposes a challenge in terms of memory requirements as it needs to maintain each of the local approximates in memory. To overcome this problem, stochastic expectation propagation (SEP) was proposed, which only considers a unique local factor that captures the average effect of each likelihood term to the posterior and refines it in a way analogous to EP. In terms of privacy, SEP is more tractable than EP because at each refining step of a factor, the remaining factors are fixed to the same value and do not depend on other datapoints as in EP, which makes the sensitivity analysis tractable. We provide a theoretical analysis of the privacy-accuracy trade-off in the posterior estimates under differentially private stochastic expectation propagation (DP-SEP). Furthermore, we demonstrate the performance of our DP-SEP algorithm evaluated on both synthetic and real-world datasets in terms of the quality of posterior estimates at different levels of guaranteed privacy.

With the development of AIoT, data-driven attack detection methods for cyber-physical systems (CPSs) have attracted lots of attention. However, existing methods usually adopt tractable distributions to approximate data distributions, which are not suitable for complex systems. Besides, the correlation of the data in different channels does not attract sufficient attention. To address these issues, we use energy-based generative models, which are less restrictive on functional forms of the data distribution. In addition, graph neural networks are used to explicitly model the correlation of the data in different channels. In the end, we propose TFDPM, a general framework for attack detection tasks in CPSs. It simultaneously extracts temporal pattern and feature pattern given the historical data. Then extract features are sent to a conditional diffusion probabilistic model. Predicted values can be obtained with the conditional generative network and attacks are detected based on the difference between predicted values and observed values. In addition, to realize real-time detection, a conditional noise scheduling network is proposed to accelerate the prediction process. Experimental results show that TFDPM outperforms existing state-of-the-art attack detection methods. The noise scheduling network increases the detection speed by three times.

The development of high-throughput sequencing technologies has led to the production of largescale molecular profiling data, allowing us to gain insights into underlying biological processes (Wid lak, 2013). One such technology is microarray sequencing, in which mRNA counts are used to describe gene expression. Such data, known as transcriptomics, are widely used in the biomedical domain and when analyzed alongside survival times have provided extraordinary opportunities for biomarker characterization and prognostic modelling (Bøvelstad et al., 2007; Lloyd et al., 2015; Lightbody et al., 2019; Lu et al., 2021). However, profiling data is often high-dimensional, which introduces several statistical challenges including: (i) variable selection, (ii) effect estimation of the features, and (iii) scalable computation. The task of variable selection is particularly important, as few genes typically have an effect on the outcome. Motivated by clinical applicability, we propose a state-of-the-art scalable (variational) Bayesian variable selection method for the proportional hazards models. In recent years, several methods have been proposed to analyze sparse high-dimensional data, with one of the most popular being the LASSO (Tibshirani, 1996). As biomedical studies are often concerned with clinical phenotypes, such as time to disease recurrence or overall survival time, these methods have been adapted to support survival analysis (Antoniadis et al., 2010; Witten and Tibshirani, 2010). For instance, the LASSO, ridge and elastic-net penalties have all been extended to the proportional hazards model (Tibshirani, 1997; Gui and Li, 2005; Zou and Hastie, 2005; Simon et al., 2011).

Fuzzy inference engine, as one of the most important components of fuzzy systems, can obtain some meaningful outputs from fuzzy sets on input space and fuzzy rule base using fuzzy logic inference methods. In order to enhance the computational efficiency of fuzzy inference engine in multi-input-single-output (MISO) fuzzy systems, this paper aims mainly to investigate three MISO fuzzy hierarchial inference engines based on fuzzy implications satisfying the law of importation with aggregation functions (LIA). We firstly find some aggregation functions for well-known fuzzy implications such that they satisfy (LIA) with them. For a given aggregation function, the fuzzy implication which satisfies (LIA) with this aggregation function is then characterized. Finally, we construct three fuzzy hierarchical inference engines in MISO fuzzy systems applying aforementioned theoretical developments.

This paper provides both an introduction to and a detailed overview of the principles and practice of classifier calibration. A well-calibrated classifier correctly quantifies the level of uncertainty or confidence associated with its instance-wise predictions. This is essential for critical applications, optimal decision making, cost-sensitive classification, and for some types of context change. Calibration research has a rich history which predates the birth of machine learning as an academic field by decades. However, a recent increase in the interest on calibration has led to new methods and the extension from binary to the multiclass setting. The space of options and issues to consider is large, and navigating it requires the right set of concepts and tools. We provide both introductory material and up-to-date technical details of the main concepts and methods, including proper scoring rules and other evaluation metrics, visualisation approaches, a comprehensive account of post-hoc calibration methods for binary and multiclass classification, and several advanced topics.

Information field theory (IFT), the information theory for fields, is a mathematical framework for signal reconstruction and non-parametric inverse problems. Here, fields denote physical quantities that change continuously as a function of space (and time) and information theory refers to Bayesian probabilistic logic equipped with the associated entropic information measures. Reconstructing a signal with IFT is a computational problem similar to training a generative neural network (GNN). In this paper, the inference in IFT is reformulated in terms of GNN training and the cross-fertilization of numerical variational inference methods used in IFT and machine learning are discussed. The discussion suggests that IFT inference can be regarded as a specific form of artificial intelligence. In contrast to classical neural networks, IFT based GNNs can operate without pre-training thanks to incorporating expert knowledge into their architecture.

Independent component analysis is intended to recover the unknown components as independent as possible from their linear mixtures. This technique has been widely used in many fields, such as data analysis, signal processing, and machine learning. In this paper, we present a novel boosting-based algorithm for independent component analysis. Our algorithm fills the gap in the nonparametric independent component analysis by introducing boosting to maximum likelihood estimation. A variety of experiments validate its performance compared with many of the presently known algorithms.