This work focuses on the mixed membership models for multivariate categorical data widely used for analyzing survey responses and population genetics data. These grade of membership (GoM) models offer rich modeling power but present significant estimation challenges for high-dimensional polytomous data. Popular existing approaches, such as Bayesian MCMC inference, are not scalable and lack theoretical guarantees in high-dimensional settings. To address this, we first observe that data from this model can be reformulated as a three-way (quasi-)tensor, with many subjects responding to many items with varying numbers of categories. We introduce a novel and simple approach that flattens the three-way quasi-tensor into a "fat" matrix, and then perform a singular value decomposition of it to estimate parameters by exploiting the singular subspace geometry. Our fast spectral method can accommodate a broad range of data distributions with arbitrarily locally dependent noise, which we formalize as the generalized-GoM models. We establish finite-sample entrywise error bounds for the generalized-GoM model parameters. This is supported by a new sharp two-to-infinity singular subspace perturbation theory for locally dependent and flexibly distributed noise, a contribution of independent interest. Simulations and applications to data in political surveys, population genetics, and single-cell sequencing demonstrate our method's superior performance.

How to properly fuse information from complex sources is still an open problem. Lots of methods have been put forward to provide a effective solution in fusing intricate information. Among them, Dempster-Shafer evidences theory (DSET) is one of the representatives, it is widely used to handle uncertain information. Based on DSET, a completely new method to fuse information from different sources based on pignistic transformation and Z-numbers is proposed in this paper which is able to handle separate situations of information and keeps high accuracy in producing rational and correct judgments on actual situations. Besides, in order to illustrate the superiority of the proposed method, some numerical examples and application are also provided to verify the validity and robustness of it.

Strong artificial intelligence (AI) is envisioned to possess general cognitive abilities and scientific creativity comparable to human intelligence, encompassing both knowledge acquisition and problem-solving. While remarkable progress has been made in weak AI, the realization of strong AI remains a topic of intense debate and critical examination. In this paper, we explore pivotal innovations in the history of astronomy and physics, focusing on the discovery of Neptune and the concept of scientific revolutions as perceived by philosophers of science. Building on these insights, we introduce a simple theoretical and statistical framework of weak beliefs, termed the Transformational Belief (TB) framework, designed as a foundation for modeling scientific creativity. Through selected illustrative examples in statistical science, we demonstrate the TB framework's potential as a promising foundation for understanding, analyzing, and even fostering creativity -- paving the way toward the development of strong AI. We conclude with reflections on future research directions and potential advancements.

We provide a comparative analysis of the deduction, induction, and abduction formulas used in Probabilistic Logic Networks (PLN) and the Non-Axiomatic Reasoning System (NARS), two uncertain reasoning frameworks aimed at AGI. One difference between the two systems is that, at the level of individual inference rules, PLN directly leverages both term and relationship probabilities, whereas NARS only leverages relationship frequencies and has no simple analogue of term probabilities. Thus we focus here on scenarios where there is high uncertainty about term probabilities, and explore how this uncertainty influences the comparative inferential conclusions of the two systems. We compare the product of strength and confidence ($s\times c$) in PLN against the product of frequency and confidence ($f\times c$) in NARS (quantities we refer to as measuring the "power" of an uncertain statement) in cases of high term probability uncertainty, using heuristic analyses and elementary numerical computations. We find that in many practical situations with high term probability uncertainty, PLN and NARS formulas give very similar results for the power of an inference conclusion, even though they sometimes come to these similar numbers in quite different ways.

Local causal discovery aims to learn and distinguish the direct causes and effects of a target variable from observed data. Existing constraint-based local causal discovery methods use AND or OR rules in constructing the local causal skeleton, but using either rule alone is prone to produce cascading errors in the learned local causal skeleton, and thus impacting the inference of local causal relationships. On the other hand, directly applying score-based global causal discovery methods to local causal discovery may randomly return incorrect results due to the existence of local equivalence classes. To address the above issues, we propose a Hybrid Local Causal Discovery algorithm, called HLCD. Specifically, HLCD initially utilizes a constraint-based approach combined with the OR rule to obtain a candidate skeleton and then employs a score-based method to eliminate redundant portions in the candidate skeleton. Furthermore, during the local causal orientation phase, HLCD distinguishes between V-structures and equivalence classes by comparing the local structure scores between the two, thereby avoiding orientation interference caused by local equivalence classes. We conducted extensive experiments with seven state-of-the-art competitors on 14 benchmark Bayesian network datasets, and the experimental results demonstrate that HLCD significantly outperforms existing local causal discovery algorithms.

The signature kernel is a kernel between time series of arbitrary length and comes with strong theoretical guarantees from stochastic analysis. It has found applications in machine learning such as covariance functions for Gaussian processes. A strength of the underlying signature features is that they provide a structured global description of a time series. However, this property can quickly become a curse when local information is essential and forgetting is required; so far this has only been addressed with ad-hoc methods such as slicing the time series into subsegments. To overcome this, we propose a principled, data-driven approach by introducing a novel forgetting mechanism for signatures. This allows the model to dynamically adapt its context length to focus on more recent information. To achieve this, we revisit the recently introduced Random Fourier Signature Features, and develop Random Fourier Decayed Signature Features (RFDSF) with Gaussian processes (GPs). This results in a Bayesian time series forecasting algorithm with variational inference, that offers a scalable probabilistic algorithm that processes and transforms a time series into a joint predictive distribution over time steps in one pass using recurrence. For example, processing a sequence of length $10^4$ steps in $\approx 10^{-2}$ seconds and in $< 1\text{GB}$ of GPU memory. We demonstrate that it outperforms other GP-based alternatives and competes with state-of-the-art probabilistic time series forecasting algorithms.

We present a likelihood-free probabilistic inversion method based on normalizing flows for high-dimensional inverse problems. The proposed method is composed of two complementary networks: a summary network for data compression and an inference network for parameter estimation. The summary network encodes raw observations into a fixed-size vector of summary features, while the inference network generates samples of the approximate posterior distribution of the model parameters based on these summary features. The posterior samples are produced in a deep generative fashion by sampling from a latent Gaussian distribution and passing these samples through an invertible transformation. We construct this invertible transformation by sequentially alternating conditional invertible neural network and conditional neural spline flow layers. The summary and inference networks are trained simultaneously. We apply the proposed method to an inversion problem in groundwater hydrology to estimate the posterior distribution of the log-conductivity field conditioned on spatially sparse time-series observations of the system's hydraulic head responses.The conductivity field is represented with 706 degrees of freedom in the considered problem.The comparison with the likelihood-based iterative ensemble smoother PEST-IES method demonstrates that the proposed method accurately estimates the parameter posterior distribution and the observations' predictive posterior distribution at a fraction of the inference time of PEST-IES.

We study learning of human preferences from a limited comparison feedback. This task is ubiquitous in machine learning. Its applications such as reinforcement learning from human feedback, have been transformational. We formulate this problem as learning a Plackett-Luce model over a universe of $N$ choices from $K$-way comparison feedback, where typically $K \ll N$. Our solution is the D-optimal design for the Plackett-Luce objective. The design defines a data logging policy that elicits comparison feedback for a small collection of optimally chosen points from all ${N \choose K}$ feasible subsets. The main algorithmic challenge in this work is that even fast methods for solving D-optimal designs would have $O({N \choose K})$ time complexity. To address this issue, we propose a randomized Frank-Wolfe (FW) algorithm that solves the linear maximization sub-problems in the FW method on randomly chosen variables. We analyze the algorithm, and evaluate it empirically on synthetic and open-source NLP datasets.

The optimization on the structure of process of information management under uncertain environment has attracted lots of attention from researchers around the world. Nevertheless, how to obtain accurate and rational evaluation from assessments produced by experts is still an open problem. Specially, intuitionistic fuzzy set provides an effective solution in handling indeterminate information. And Yager proposes a novel method for fusion of probabilistic evidence to handle uncertain and conflicting information lately which is called soft likelihood function. This paper devises a novel framework of soft likelihood function based on information volume of fuzzy membership and credibility measure for extracting truly useful and valuable information from uncertainty. An application is provided to verify the validity and correctness of the proposed framework. Besides, the comparisons with other existing methods further demonstrate the superiority of the novel framework of soft likelihood function.

Weakly supervised temporal action localization (WS-TAL) is a task of targeting at localizing complete action instances and categorizing them with video-level labels. Action-background ambiguity, primarily caused by background noise resulting from aggregation and intra-action variation, is a significant challenge for existing WS-TAL methods. In this paper, we introduce a hybrid multi-head attention (HMHA) module and generalized uncertainty-based evidential fusion (GUEF) module to address the problem. The proposed HMHA effectively enhances RGB and optical flow features by filtering redundant information and adjusting their feature distribution to better align with the WS-TAL task. Additionally, the proposed GUEF adaptively eliminates the interference of background noise by fusing snippet-level evidences to refine uncertainty measurement and select superior foreground feature information, which enables the model to concentrate on integral action instances to achieve better action localization and classification performance. Experimental results conducted on the THUMOS14 dataset demonstrate that our method outperforms state-of-the-art methods. Our code is available in \url{https://github.com/heyuanpengpku/GUEF/tree/main}.