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Gaussian Process (GPs) models are a rich distribution over functions with inductive biases controlled by a kernel function. Learning occurs through the optimisation of kernel hyperparameters using the marginal likelihood as the objective. This classical approach known as Type-II maximum likelihood (ML-II) yields point estimates of the hyperparameters, and continues to be the default method for training GPs. However, this approach risks underestimating predictive uncertainty and is prone to overfitting especially when there are many hyperparameters. Furthermore, gradient based optimisation makes ML-II point estimates highly susceptible to the presence of local minima. This work presents an alternative learning procedure where the hyperparameters of the kernel function are marginalised using Nested Sampling (NS), a technique that is well suited to sample from complex, multi-modal distributions. We focus on regression tasks with the spectral mixture (SM) class of kernels and find that a principled approach to quantifying model uncertainty leads to substantial gains in predictive performance across a range of synthetic and benchmark data sets. In this context, nested sampling is also found to offer a speed advantage over Hamiltonian Monte Carlo (HMC), widely considered to be the gold-standard in MCMC based inference.

In this paper we describe a probabilistic method for estimating the position of an object along with its covariance matrix using neural networks. Our method is designed to be robust to outliers, have bounded gradients with respect to the network outputs, among other desirable properties. To achieve this we introduce a novel probability distribution inspired by the Huber loss. We also introduce a new way to parameterize positive definite matrices to ensure invariance to the choice of orientation for the coordinate system we regress over. We evaluate our method on popular body pose and facial landmark datasets and get performance on par or exceeding the performance of non-heatmap methods. Our code is available at github.com/Davmo049/Public_prob_regression_with_huber_distributions

We introduce a simple diagnostic test for assessing the overall or partial goodness of fit of linear regression. We propose to evaluate the sensitivity of the regression coefficient with respect to changes of the marginal distribution of covariates by comparing the so-called higher-order least squares with the usual least squares estimates. In spite of its simplicity, this strategy is extremely general and powerful, including high-dimensional settings. Specifically, we show that it allows to distinguish between confounded and unconfounded predictor variables as well as determining ancestor variables in linear structural equation models assuming some non-Gaussianity. Thus, we provide a test for partial goodness of fit.

Gaussian process regression (GPR) model is a popular nonparametric regression model. In GPR, features of the regression function such as varying degrees of smoothness and periodicities are modeled through combining various covarinace kernels, which are supposed to model certain effects. The covariance kernels have unknown parameters which are estimated by the EM-algorithm or Markov Chain Monte Carlo. The estimated parameters are keys to the inference of the features of the regression functions, but identifiability of these parameters has not been investigated. In this paper, we prove identifiability of covariance kernel parameters in two radial basis mixed kernel GPR and radial basis and periodic mixed kernel GPR. We also provide some examples about non-identifiable cases in such mixed kernel GPRs.

Gibbs sampling methods for mixture models are based on data augmentation schemes that account for the unobserved partition in the data. Conditional samplers are known to suffer from slow mixing in infinite mixtures, where some form of truncation, either deterministic or random, is required. In mixtures with random number of components, the exploration of parameter spaces of different dimensions can also be challenging. We tackle these issues by expressing the mixture components in the random order of appearance in an exchangeable sequence directed by the mixing distribution. We derive a sampler that is straightforward to implement for mixing distributions with tractable size-biased ordered weights. In infinite mixtures, no form of truncation is necessary. As for finite mixtures with random dimension, a simple updating of the number of components is obtained by a blocking argument, thus, easing challenges found in trans-dimensional moves via Metropolis-Hasting steps. Additionally, sampling occurs in the space of ordered partitions with blocks labelled in the least element order. This improves mixing and promotes a consistent labelling of mixture components throughout iterations. The performance of the proposed algorithm is evaluated on simulated data.

This paper presents an efficient variational inference framework for deriving a family of structured gaussian process regression network (SGPRN) models. The key idea is to incorporate auxiliary inducing variables in latent functions and jointly treats both the distributions of the inducing variables and hyper-parameters as variational parameters. Then we propose structured variable distributions and marginalize latent variables, which enables the decomposability of a tractable variational lower bound and leads to stochastic optimization. Our inference approach is able to model data in which outputs do not share a common input set with a computational complexity independent of the size of the inputs and outputs and thus easily handle datasets with missing values. We illustrate the performance of our method on synthetic data and real datasets and show that our model generally provides better imputation results on missing data than the state-of-the-art. We also provide a visualization approach for time-varying correlation across outputs in electrocoticography data and those estimates provide insight to understand the neural population dynamics.

Labelled networks are an important class of data, naturally appearing in numerous applications in science and engineering. A typical inference goal is to determine how the vertex labels (or features) affect the network's structure. In this work, we introduce a new generative model, the feature-first block model (FFBM), that facilitates the use of rich queries on labelled networks. We develop a Bayesian framework and devise a two-level Markov chain Monte Carlo approach to efficiently sample from the relevant posterior distribution of the FFBM parameters. This allows us to infer if and how the observed vertex-features affect macro-structure. We apply the proposed methods to a variety of network data to extract the most important features along which the vertices are partitioned. The main advantages of the proposed approach are that the whole feature-space is used automatically and that features can be rank-ordered implicitly according to impact.

High dimensional non-Gaussian time series data are increasingly encountered in a wide range of applications. Conventional estimation methods and technical tools are inadequate when it comes to ultra high dimensional and heavy-tailed data. We investigate robust estimation of high dimensional autoregressive models with fat-tailed innovation vectors by solving a regularized regression problem using convex robust loss function. As a significant improvement, the dimension can be allowed to increase exponentially with the sample size to ensure consistency under very mild moment conditions. To develop the consistency theory, we establish a new Bernstein type inequality for the sum of autoregressive models. Numerical results indicate a good performance of robust estimates.

There has been a rich development of vector autoregressive (VAR) models for modeling temporally correlated multivariate outcomes. However, the existing VAR literature has largely focused on single subject parametric analysis, with some recent extensions to multi-subject modeling with known subgroups. Motivated by the need for flexible Bayesian methods that can pool information across heterogeneous samples in an unsupervised manner, we develop a novel class of non-parametric Bayesian VAR models based on heterogeneous multi-subject data. In particular, we propose a product of Dirichlet process mixture priors that enables separate clustering at multiple scales, which result in partially overlapping clusters that provide greater flexibility. We develop several variants of the method to cater to varying levels of heterogeneity. We implement an efficient posterior computation scheme and illustrate posterior consistency properties under reasonable assumptions on the true density. Extensive numerical studies show distinct advantages over competing methods in terms of estimating model parameters and identifying the true clustering and sparsity structures. Our analysis of resting state fMRI data from the Human Connectome Project reveals biologically interpretable differences between distinct fluid intelligence groups, and reproducible parameter estimates. In contrast, single-subject VAR analyses followed by permutation testing result in negligible differences, which is biologically implausible.

Person re-identification is being widely used in the forensic, and security and surveillance system, but person re-identification is a challenging task in real life scenario. Hence, in this work, a new feature descriptor model has been proposed using a multilayer framework of Gaussian distribution model on pixel features, which include color moments, color space values and Schmid filter responses. An image of a person usually consists of distinct body regions, usually with differentiable clothing followed by local colors and texture patterns. Thus, the image is evaluated locally by dividing the image into overlapping regions. Each region is further fragmented into a set of local Gaussians on small patches. A global Gaussian encodes, these local Gaussians for each region creating a multi-level structure. Hence, the global picture of a person is described by local level information present in it, which is often ignored. Also, we have analyzed the efficiency of earlier metric learning methods on this descriptor. The performance of the descriptor is evaluated on four public available challenging datasets and the highest accuracy achieved on these datasets are compared with similar state-of-the-arts, which demonstrate the superior performance.