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We provide a new unifying view, including all existing proper probabilistic sparse approximations for Gaussian process regression. Our approach relies on expressing the effective prior which the methods are using. This allows new insights to be gained, and highlights the relationship between existing methods. It also allows for a clear theoretically(More)
We present a new sparse Gaussian Process (GP) model for regression. The key novel idea is to sparsify the spectral representation of the GP. This leads to a simple, practical algorithm for regression tasks. We compare the achievable trade-offs between predictive accuracy and computational requirements, and show that these are typically superior to existing(More)
We describe a new Bayesian click-through rate (CTR) prediction algorithm used for Sponsored Search in Microsoft's Bing search engine. The algorithm is based on a probit regression model that maps discrete or real-valued input features to probabilities. It maintains Gaussian beliefs over weights of the model and performs Gaussian online updates derived from(More)
We consider the problem of multi-step ahead prediction in time series analysis using the non-parametric Gaussian process model.-step ahead forecasting of a discrete-time non-linear dynamic system can be performed by doing repeated one-step ahead predictions. For a state-space model of the form is based on the point estimates of the previous outputs. In this(More)
The Gaussian process latent variable model (GP-LVM) is a generative approach to nonlinear low dimensional embedding, that provides a smooth probabilistic mapping from latent to data space. It is also a non-linear generalization of probabilistic PCA (PPCA) (Tipping & Bishop, 1999). While most approaches to non-linear dimensionality methods focus on(More)
Online advertising allows advertisers to only bid and pay for measurable user responses, such as clicks on ads. As a consequence, click prediction systems are central to most online advertising systems. With over 750 million daily active users and over 1 million active advertisers, predicting clicks on Facebook ads is a challenging machine learning task. In(More)
This work shows how to leverage causal inference to understand the behavior of complex learning systems interacting with their environment and predict the consequences of changes to the system. Such predictions allow both humans and algorithms to select the changes that would have improved the system performance. This work is illustrated by experiments on(More)
The object of Bayesian modelling is the predictive distribution, which in a forecasting scenario enables evaluation of forecasted values and their uncertainties. In this paper we focus on reliably estimating the predictive mean and variance of forecasted values using Bayesian kernel based models such as the Gaussian Process and the Relevance Vector Machine.(More)
The Relevance Vector Machine (RVM) is a sparse approximate Bayesian kernel method. It provides full predictive distributions for test cases. However, the predictive uncertainties have the unintuitive property, that they get smaller the further you move away from the training cases. We give a thorough analysis. Inspired by the analogy to non-degenerate(More)
This Chapter presents the PASCAL 1 Evaluating Predictive Uncertainty Challenge, introduces the contributed Chapters by the participants who obtained outstanding results, and provides a discussion with some lessons to be learnt. The Challenge was set up to evaluate the ability of Machine Learning algorithms to provide good " probabilis-tic predictions " ,(More)