Canonical correlation analysis is a statistical technique -dating back at least to [1] - that is used to maximally correlate multiple datasets for joint analysis. The technique has become a fundamental tool in biomedical research where technological advances have led to a huge number of multi-omic datasets ([2]; [3]; [4]). Over the past two decades, limited sample sizes, growing dimensionality, and the search for meaningful biological interpretations, have led to the development of sparse canonical correlation analysis ([2]), where a sparsity assumption is imposed on the canonical correlation vectors. This work falls under the topic of the Bayesian estimation of sparse canonical corrlation vectors. Model-based approaches to canonical correlation analysis were developed in the mid 2000's (see e.g., [5]), and paved the way for a Bayesian treatment of canonical correlation analysis ([6];[7]) and sparse canonical correlation analysis ([8]). However an serious shortcoming of such a Bayesian treatment is that this approach naturally requires a complete specification of the joint distribution of the data, so as to specify the likelihood function. This requirement is a serious limitation in many applications, where the data generating process is poorly understood, for example, image data.

Bayesian neural networks attempt to combine the strong predictive performance of neural networks with formal quantification of uncertainty associated with the predictive output in the Bayesian framework. However, it remains unclear how to endow the parameters of the network with a prior distribution that is meaningful when lifted into the output space of the network. A possible solution is proposed that enables the user to posit an appropriate covariance function for the task at hand. Our approach constructs a prior distribution for the parameters of the network, called a ridgelet prior, that approximates the posited covariance structure in the output space of the network. The approach is rooted in the ridgelet transform and we establish both finite-sample-size error bounds and the consistency of the approximation of the covariance function in a limit where the number of hidden units is increased. Our experimental assessment is limited to a proof-of-concept, where we demonstrate that the ridgelet prior can out-perform an unstructured prior on regression problems for which an informative covariance function can be a priori provided.

Spatial count data models are used to explain and predict the frequency of phenomena such as traffic accidents in geographically distinct entities such as census tracts or road segments. These models are typically estimated using Bayesian Markov chain Monte Carlo (MCMC) simulation methods, which, however, are computationally expensive and do not scale well to large datasets. Variational Bayes (VB), a method from machine learning, addresses the shortcomings of MCMC by casting Bayesian estimation as an optimisation problem instead of a simulation problem. Considering all these advantages of VB, a VB method is derived for posterior inference in negative binomial models with unobserved parameter heterogeneity and spatial dependence. P\'olya-Gamma augmentation is used to deal with the non-conjugacy of the negative binomial likelihood and an integrated non-factorised specification of the variational distribution is adopted to capture posterior dependencies. The benefits of the proposed approach are demonstrated in a Monte Carlo study and an empirical application on estimating youth pedestrian injury counts in census tracts of New York City. The VB approach is around 45 to 50 times faster than MCMC on a regular eight-core processor in a simulation and an empirical study, while offering similar estimation and predictive accuracy. Conditional on the availability of computational resources, the embarrassingly parallel architecture of the proposed VB method can be exploited to further accelerate its estimation by up to 20 times.

We develop a goodness-of-fit (GOF) test for generative models of self-exciting processes by making a new connection to this problem with the classical statistical theory of Quasi-maximum-likelihood estimator (QMLE). We present a non-parametric self-normalizing statistic for the GOF test: the Generalized Score (GS) statistics, and explicitly capture the model misspecification when establishing the asymptotic distribution of the GS statistic. Numerical experiments based on simulation and real-data validate our theory and demonstrate the proposed GS test's good performance.

Despite the improvement of translation quality, neural machine translation (NMT) often suffers from the lack of diversity in its generation. In this paper, we propose to generate diverse translations by deriving a large number of possible models with Bayesian modelling and sampling models from them for inference. The possible models are obtained by applying concrete dropout to the NMT model and each of them has specific confidence for its prediction, which corresponds to a posterior model distribution under specific training data in the principle of Bayesian modeling. With variational inference, the posterior model distribution can be approximated with a variational distribution, from which the final models for inference are sampled. We conducted experiments on Chinese-English and English-German translation tasks and the results shows that our method makes a better trade-off between diversity and accuracy.

The interactions among the constituent members of a microbial community play a major role in determining the overall behavior of the community and the abundance levels of its members. These interactions can be modeled using a network whose nodes represent microbial taxa and edges represent pairwise interactions. A microbial network is a weighted graph that is constructed from a sample-taxa count matrix, and can be used to model co-occurrences and/or interactions of the constituent members of a microbial community. The nodes in this graph represent microbial taxa and the edges represent pairwise associations amongst these taxa. A microbial network is typically constructed from a sample-taxa count matrix that is obtained by sequencing multiple biological samples and identifying taxa counts. From large-scale microbiome studies, it is evident that microbial community compositions and interactions are impacted by environmental and/or host factors. Thus, it is not unreasonable to expect that a sample-taxa matrix generated as part of a large study involving multiple environmental or clinical parameters can be associated with more than one microbial network. However, to our knowledge, microbial network inference methods proposed thus far assume that the sample-taxa matrix is associated with a single network.

Bayesian inference without the access of likelihood, called likelihood-free inference, is highlighted in simulation to yield a more realistic simulation result. Recent research updates an approximate posterior sequentially with the cumulative simulation input-output pairs over inference rounds. This paper observes that previous algorithms with Monte-Carlo Markov Chain present low accuracy for inference on a simulation with a multi-modal posterior due to the mode collapse of MCMC. From the observation, we propose an implicit sampling method, Implicit Surrogate Proposal (ISP), to draw balanced simulation inputs at each round. The resolution of mode collapse comes from two mechanisms: 1) a flexible surrogate proposal density estimator and 2) a parallel explored samples to train the surrogate density model. We demonstrate that ISP outperforms the baseline algorithms in multi-modal simulations.

Machine learning models have traditionally been developed under the assumption that the training and test distributions match exactly. However, recent success in few-shot learning and related problems are encouraging signs that these models can be adapted to more realistic settings where train and test distributions differ. Unfortunately, there is severely limited theoretical support for these algorithms and little is known about the difficulty of these problems. In this work, we provide novel information-theoretic lower-bounds on minimax rates of convergence for algorithms that are trained on data from multiple sources and tested on novel data. Our bounds depend intuitively on the information shared between sources of data, and characterize the difficulty of learning in this setting for arbitrary algorithms. We demonstrate these bounds on a hierarchical Bayesian model of meta-learning, computing both upper and lower bounds on parameter estimation via maximum-a-posteriori inference.

The data drawn from biological, economic, and social systems are often confounded due to the presence of unmeasured variables. Prior work in causal discovery has focused on discrete search procedures for selecting acyclic directed mixed graphs (ADMGs), specifically ancestral ADMGs, that encode ordinary conditional independence constraints among the observed variables of the system. However, confounded systems also exhibit more general equality restrictions that cannot be represented via these graphs, placing a limit on the kinds of structures that can be learned using ancestral ADMGs. In this work, we derive differentiable algebraic constraints that fully characterize the space of ancestral ADMGs, as well as more general classes of ADMGs, arid ADMGs and bow-free ADMGs, that capture all equality restrictions on the observed variables. We use these constraints to cast causal discovery as a continuous optimization problem and design differentiable procedures to find the best fitting ADMG when the data comes from a confounded linear system of equations with correlated errors. We demonstrate the efficacy of our method through simulations and application to a protein expression dataset.

Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) approach that exhibits favourable exploration properties in high-dimensional models such as neural networks. Unfortunately, HMC has limited use in large-data regimes and little work has explored suitable approaches that aim to preserve the entire Hamiltonian. In our work, we introduce a new symmetric integration scheme for split HMC that does not rely on stochastic gradients. We show that our new formulation is more efficient than previous approaches and is easy to implement with a single GPU. As a result, we are able to perform full HMC over common deep learning architectures using entire data sets. In addition, when we compare with stochastic gradient MCMC, we show that our method achieves better performance in both accuracy and uncertainty quantification. Our approach demonstrates HMC as a feasible option when considering inference schemes for large-scale machine learning problems.