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We present a framework for approximate Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained due to computational constraints, which is becoming increasingly common for applications of complex…
Methods for inference and simulation of linearly constrained Gaussian Markov Random Fields (GMRF) are computationally prohibitive when the number of constraints is large. In some cases, such as for intrinsic GMRFs, they may even be…
In this paper we first describe the class of log-Gaussian Cox processes (LGCPs) as models for spatial and spatio-temporal point process data. We discuss inference, with a particular focus on the computational challenges of likelihood-based…
We present a novel statistical inference framework for convex empirical risk minimization, using approximate stochastic Newton steps. The proposed algorithm is based on the notion of finite differences and allows the approximation of a…
We derive upper bounds on the Wasserstein distance ($W_1$), with respect to $\sup$-norm, between any continuous $\mathbb{R}^d$ valued random field indexed by the $n$-sphere and the Gaussian, based on Stein's method. We develop a novel…
Gaussian processes are powerful non-parametric probabilistic models for stochastic functions. However, the direct implementation entails a complexity that is computationally intractable when the number of observations is large, especially…
Examples with bound information on the regression function and density abound in many real applications. We propose a novel approach for estimating such functions by incorporating the prior knowledge on the bounds. Specially, a Gaussian…
We consider the problem of inferring a latent function in a probabilistic model of data. When dependencies of the latent function are specified by a Gaussian process and the data likelihood is complex, efficient computation often involve…
Bayesian learning using Gaussian processes provides a foundational framework for making decisions in a manner that balances what is known with what could be learned by gathering data. In this dissertation, we develop techniques for…
This paper introduces a method to approximate Gaussian process regression by representing the problem as a stochastic differential equation and using variational inference to approximate solutions. The approximations are compared with full…
We propose a lower bound on the log marginal likelihood of Gaussian process regression models that can be computed without matrix factorisation of the full kernel matrix. We show that approximate maximum likelihood learning of model…
Inference for GP models with non-Gaussian noises is computationally expensive when dealing with large datasets. Many recent inference methods approximate the posterior distribution with a simpler distribution defined on a small number of…
Random fields are useful mathematical tools for representing natural phenomena with complex dependence structures in space and/or time. In particular, the Gaussian random field is commonly used due to its attractive properties and…
This Ph.D. thesis explores approximations and regularity for the Heston stochastic volatility model through three interconnected works. The first work focuses on developing high-order weak approximations for the Cox-Ingersoll-Ross (CIR)…
The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline…
A Gaussian Cox process is a popular model for point process data, in which the intensity function is a transformation of a Gaussian process. Posterior inference of this intensity function involves an intractable integral (i.e., the…
We introduce new Gaussian Process (GP) high-order approximations to linear operations that are frequently used in various numerical methods. Our method employs the kernel-based GP regression modeling, a non-parametric Bayesian approach to…
Deep Gaussian processes provide a flexible approach to probabilistic modelling of data using either supervised or unsupervised learning. For tractable inference approximations to the marginal likelihood of the model must be made. The…
It is common in nature to see aggregation of objects in space. Exploring the mechanism associated with the locations of such clustered observations can be essential to understanding the phenomenon, such as the source of spatial…
We introduce Gaussian orthogonal latent factor processes for modeling and predicting large correlated data. To handle the computational challenge, we first decompose the likelihood function of the Gaussian random field with a…