Above are papers related to the following paper: [Paper_ID] Paper #141596. 	[Title] Variational Gaussian Process Classifiers [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #131042. 	[Title] Efficient Implementation of Gaussian Processes [Abstract] Neural networks and Bayesian inference provide a useful framework within which to solve regression problems. However their parameterization means that the Bayesian analysis of neural networks can be difficult. In this paper, we investigate a method for regression using Gaussian process priors which allows exact Bayesian analysis using matrix manipulations. We discuss the workings of the method in detail. We will also detail a range of mathematical and numerical techniques that are useful in applying Gaussian processes to general problems including efficient approximate matrix inversion methods developed by Skilling. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1104258. 	[Title] Regression with Input-dependent Noise: A Gaussian Process Treatment [Abstract] Technical Report NCRG/98/002, available from http://www.ncrg.aston.ac.uk/ To appear in Advances in Neural Information Processing Systems 10 eds. M. I. Jordan, M. J. Kearns and S. A. Solla. Lawrence Erlbaum (1998). Abstract Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #34315. 	[Title] Gaussian Processes for Bayesian Classification via Hybrid Monte Carlo [Abstract] The full Bayesian method for applying neural networks to a prediction problem is to set up the prior/hyperprior structure for the net and then perform the necessary integrals. However, these integrals are not tractable analytically, and Markov Chain Monte Carlo (MCMC) methods are slow, especially if the parameter space is high-dimensional. Using Gaussian processes we can approximate the weight space integral analytically, so that only a small number of hyperparameters need be integrated over by MCMC methods. We have applied this idea to classification problems, obtaining ex cellent results on the real-world problems investigated so far. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #24974. 	[Title] EVALUATION OF GAUSSIAN PROCESSES AND OTHER METHODS FOR NON-LINEAR REGRESSION [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #40125. 	[Title] Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification [Abstract] Technical Report No. 9702, Department of Statistics, University of Toronto Abstract. Gaussian processes are a natural way of defining prior distributions over functions of one or more input variables. In a simple nonparametric regression problem, where such a function gives the mean of a Gaussian distribution for an observed response, a Gaussian process model can easily be implemented using matrix computations that are feasible for datasets of up to about a thousand cases. Hyperparameters that define the covariance function of the Gaussian process can be sampled using Markov chain methods. Regression models where the noise has a t distribution and logistic or probit models for classification applications can be implemented by sampling as well for latent values underlying the observations. Software is now available that implements these methods using covariance functions with hierarchical parameterizations. Models defined in this way can discover high-level properties of the data, such as which inputs are relevant to predicting the response. Please summarise all the given information.

Paper #141596 has the title `Variational Gaussian Process Classifiers  
` and the abstract ` 
`.
Question: Which category should Paper #141596 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #141596 has the title `Variational Gaussian Process Classifiers  
` and the abstract ` 
`.
Question: Which category should Paper #141596 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Neural_Networks