This paper gives an efficient Bayesian method for inferring the parameters of a Plackett-Luce ranking model. Such models are parameterised distributions over rankings of a finite set of objects, and have typically been studied and applied within the psychometric, sociometric and econometric literature. The inference scheme is an application of Power EP (expectation propagation). The scheme is robust and can be readily applied to large scale data sets. The inference algorithm extends to variations of the basic Plackett-Luce model, including partial rankings. We show a number of advantages of the EP approach over the traditional maximum likelihood method. We apply the method to aggregate rankings of NASCAR racing drivers over the 2002 season, and also to rankings of movie genres.

In this paper we address the issue of learning to rank for document retrieval using Thurstonian models based on sparse Gaussian processes. Thurstonian models represent each document for a given query as a probability distribution in a score space; these distributions over scores naturally give rise to distributions over document rankings. However, in general we do not have observed rankings with which to train the model; instead, each document in the training set is judged to have a particular relevance level: for example "Bad", "Fair", "Good", or "Excellent". The performance of the model is then evaluated using information retrieval (IR) metrics such as Normalised Discounted Cumulative Gain (NDCG). Recently Taylor et al. presented a method called SoftRank which allows the direct gradient optimisation of a smoothed version of NDCG using a Thurstonian model. In this approach, document scores are represented by the outputs of a neural network, and score distributions are created artificially by adding random noise to the scores. The SoftRank mechanism is a general one; it can be applied to different IR metrics, and make use of different underlying models. In this paper we extend the SoftRank framework to make use of the score uncertainties which are naturally provided by a Gaussian process (GP), which is a probabilistic non-linear regression model. We further develop the model by using sparse Gaussian process techniques, which give improved performance and efficiency, and show competitive results against baseline methods when tested on the publicly available LETOR OHSUMED data set. We also explore how the available uncertainty information can be used in prediction and how it affects model performance.

We provide a general framework for learning precise, compact, and fast representations of the Bayesian predictive distribution for a model. This framework is based on minimizing the KL divergence between the true predictive density and a suitable compact approximation. We consider various methods for doing this, both sampling based approximations, and deterministic approximations such as expectation propagation. These methods are tested on a mixture of Gaussians model for density estimation and on binary linear classification, with both synthetic data sets for visualization and several real data sets. Our results show significant reductions in prediction time and memory footprint.

We present a new Gaussian process (GP) regression model whose covariance is parameterized by the the locations of M pseudo-input points, which we learn by a gradient based optimization. We take M N, where N is the number of real data points, and hence obtain a sparse regression method which has O(M 2 N ) training cost and O(M 2 ) prediction cost per test case. We also find hyperparameters of the covariance function in the same joint optimization. The method can be viewed as a Bayesian regression model with particular input dependent noise. The method turns out to be closely related to several other sparse GP approaches, and we discuss the relation in detail. We finally demonstrate its performance on some large data sets, and make a direct comparison to other sparse GP methods. We show that our method can match full GP performance with small M , i.e. very sparse solutions, and it significantly outperforms other approaches in this regime.

We generalise the Gaussian process (GP) framework for regression by learning a nonlinear transformation of the GP outputs. This allows for non-Gaussian processes and non-Gaussian noise. The learning algo- rithm chooses a nonlinear transformation such that transformed data is well-modelled by a GP. This can be seen as including a preprocessing transformation as an integral part of the probabilistic modelling problem, rather than as an ad-hoc step. We demonstrate on several real regression problems that learning the transformation can lead to significantly better performance than using a regular GP, or a GP with a fixed transformation.