Bayesian model averaging, obtained as the expectation of a likelihood function by a posterior distribution, has been widely used for prediction, evaluation of uncertainty, and model selection. Various approaches have been developed to efficiently capture the information in the posterior distribution; one such approach is the optimization of a set of models simultaneously with interaction to ensure the diversity of the individual models in the same way as ensemble learning. A representative approach is particle variational inference (PVI), which uses an ensemble of models as an empirical approximation for the posterior distribution. PVI iteratively updates each model with a repulsion force to ensure the diversity of the optimized models. However, despite its promising performance, a theoretical understanding of this repulsion and its association with the generalization ability remains unclear. In this paper, we tackle this problem in light of PAC-Bayesian analysis. First, we provide a new second-order Jensen inequality, which has the repulsion term based on the loss function. Thanks to the repulsion term, it is tighter than the standard Jensen inequality. Then, we derive a novel generalization error bound and show that it can be reduced by enhancing the diversity of models. Finally, we derive a new PVI that optimizes the generalization error bound directly. Numerical experiments demonstrate that the performance of the proposed PVI compares favorably with existing methods in the experiment.

This paper establishes a central limit theorem under the assumption that conditional variances can vary in a largely unstructured history-dependent way across experiments subject only to the restriction that they lie in a fixed interval. Limits take a novel and tractable form, and are expressed in terms of oscillating Brownian motion. A second contribution is application of this result to a class of multi-armed bandit problems where the decision-maker is loss averse.

Sampling from a continuous distribution in high dimensions is a fundamental problem in algorithm design. As sampling serves as a key subroutine in a variety of tasks in machine learning [AdFDJ03], statistical methods [RC99], and scientific computing [Liu01], it is an important undertaking to understand the complexity of sampling from families of distributions arising in applications. The more restricted problem of sampling from a particular family of distributions, which we call "well-conditioned distributions," has garnered a substantial amount of recent research effort from the algorithmic learning and statistics communities. This specific family is interesting for a number of reasons. First of all, it is practically relevant: Bayesian methods have found increasing use in machine learning applications [Bar12], and many distributions arising from these methods are wellconditioned, such as multivariate Gaussians, mixture models with small separation, and densities arising from Bayesian logistic regression with a Gaussian prior [DCWY18].

Despite its wide use, recent studies have revealed unexpected and undesirable properties of neural autoregressive sequence models trained with maximum likelihood, such as an unreasonably high affinity to short sequences after training and to infinitely long sequences at decoding time. We propose to study these phenomena by investigating how the modes, or local maxima, of a distribution are maintained throughout the full learning chain of the ground-truth, empirical, learned and decoding-induced distributions, via the newly proposed mode recovery cost. We design a tractable testbed where we build three types of ground-truth distributions: (1) an LSTM based structured distribution, (2) an unstructured distribution where probability of a sequence does not depend on its content, and (3) a product of these two which we call a semi-structured distribution. Our study reveals both expected and unexpected findings. First, starting with data collection, mode recovery cost strongly relies on the ground-truth distribution and is most costly with the semi-structured distribution. Second, after learning, mode recovery cost from the ground-truth distribution may increase or decrease compared to data collection, with the largest cost degradation occurring with the semi-structured ground-truth distribution. Finally, the ability of the decoding-induced distribution to recover modes from the learned distribution is highly impacted by the choices made earlier in the learning chain. We conclude that future research must consider the entire learning chain in order to fully understand the potentials and perils and to further improve neural autoregressive sequence models.

Normalizing flows are generative models that provide tractable density estimation by transforming a simple base distribution into a complex target distribution. However, this technique cannot directly model data supported on an unknown low-dimensional manifold, a common occurrence in real-world domains such as image data. Recent attempts to remedy this limitation have introduced geometric complications that defeat a central benefit of normalizing flows: exact density estimation. We recover this benefit with Conformal Embedding Flows, a framework for designing flows that learn manifolds with tractable densities. We argue that composing a standard flow with a trainable conformal embedding is the most natural way to model manifold-supported data. To this end, we present a series of conformal building blocks and apply them in experiments with real-world and synthetic data to demonstrate that flows can model manifold-supported distributions without sacrificing tractable likelihoods.

Independent component analysis provides a principled framework for unsupervised representation learning, with solid theory on the identifiability of the latent code that generated the data, given only observations of mixtures thereof. Unfortunately, when the mixing is nonlinear, the model is provably nonidentifiable, since statistical independence alone does not sufficiently constrain the problem. Identifiability can be recovered in settings where additional, typically observed variables are included in the generative process. We investigate an alternative path and consider instead including assumptions reflecting the principle of independent causal mechanisms exploited in the field of causality. Specifically, our approach is motivated by thinking of each source as independently influencing the mixing process. This gives rise to a framework which we term independent mechanism analysis. We provide theoretical and empirical evidence that our approach circumvents a number of nonidentifiability issues arising in nonlinear blind source separation.

Given a set of samples, the parameters of a GM are determined in such a way as to best fit the samples in a maximum likelihood way. Solutions for equally weighted samples are readily available, expectation-maximization (EM) based methods being the most prevalent because of low computational requirements and ease of implementation. So it comes as a surprise that GM estimation for weighted samples is hard to find in literature. It might be even more surprising that the standard reference [1] gives incorrect results, see Figure 1. 2. Context Applications for sample-to-density function approximation include clustering of unlabled data [2, 3], multi-target tracking [4, 5], group tracking [6], multilateration [7, 8], and arbitrary density representation in nonlinear filters [9, 10]. A popular basic solution to this is the k-means algorithm. It does not find a complete density representation, only the means of the individual clusters. The k-means algorithm uses hard sample-tomean associations, therefore yields merely approximate solutions but can be computationally optimized using k-d trees [11, 12]. Moreover, the global optimum can be found deterministically [13], therefore it can be used to provide an initial guess for more elaborate algorithms. A sample-to-density approximation that is optimal in a maximum likelihood sense can be searched with numerical optimization techniques such as the Newton algorithm that has quadratic convergence but high computational demand per iteration, quasi-Newton methods, the method of scoring, or the conjugate gradient method with slower convergence but less computational effort per iteration [14].

Model discovery aims at autonomously discovering differential equations underlying a dataset. Approaches based on Physics Informed Neural Networks (PINNs) have shown great promise, but a fully-differentiable model which explicitly learns the equation has remained elusive. In this paper we propose such an approach by combining neural network based surrogates with Sparse Bayesian Learning (SBL). We start by reinterpreting PINNs as multitask models, applying multitask learning using uncertainty, and show that this leads to a natural framework for including Bayesian regression techniques. We then construct a robust model discovery algorithm by using SBL, which we showcase on various datasets. Concurrently, the multitask approach allows the use of probabilistic approximators, and we show a proof of concept using normalizing flows to directly learn a density model from single particle data. Our work expands PINNs to various types of neural network architectures, and connects neural network-based surrogates to the rich field of Bayesian parameter inference.

Linear mixed models (LMM) (Laird and Ware, 1982) are widely used in longitudinal data analysis as they incorporate random effects to deal with group-specific heterogeneity. Data involving repeated observations of the same variables are common in epidemiology, medical statistics and many other fields. Likelihood-based methods are often used to make inference for (generalized) linear mixed models (Bates et al., 2000; Gumedze and Dunne, 2011). Schelldorfer et al. (2011) and Groll and Tutz (2014) introduced separately the L1-penalized estimation for high-dimensional linear mixed models. Fong et al. (2010) argued that for small sample sizes likelihood-based inference can be unreliable with variance components being difficult to estimate and suggested to use the Bayesian method. When the random effects distribution is misspecified, the resulting maximum likelihood estimators are inconsistent and biased (Neuhaus et al., 1992; Heagerty and Kurland, 2001; Litière et al., 2008). Fahrmeir and Lang (2001) presented a fully Bayesian inference via Markov Chain Monte Carlo (MCMC) simulation in generalized additive and semiparametric mixed models. Rosa et al. (2003) described a normal/independent residual distributions for robust inference and suggested also the Bayesian framework. Bayesian inference for mixed models can be conducted with for example BayesX, a program with MCMC simulation techniques (Lang and Brezger, 2000).

Hawkes processes are a class of point processes that have the ability to model the self- and mutual-exciting phenomena. Although the classic Hawkes processes cover a wide range of applications, their expressive ability is limited due to three key hypotheses: parametric, linear and homogeneous. Recent work has attempted to address these limitations separately. This work aims to overcome all three assumptions simultaneously by proposing the flexible state-switching Hawkes processes: a flexible, nonlinear and nonhomogeneous variant where a state process is incorporated to interact with the point processes. The proposed model empowers Hawkes processes to be applied to time-varying systems. For inference, we utilize the latent variable augmentation technique to design two efficient Bayesian inference algorithms: Gibbs sampler and mean-field variational inference, with analytical iterative updates to estimate the posterior. In experiments, our model achieves superior performance compared to the state-of-the-art competitors.