On the relationship between Bayesian error bars and the input data density

We investigate the dependence of Bayesian error bars on the distribution of data in input space. For generalized linear regression models we derive an upper bound on the error bars which shows that, in the neighbourhood of the data points, the error bars are substantially reduced from their prior values. For regions of high data density we also show that the contribution to the output variance due to the uncertainty in the weights can exhibit an approximate inverse proportionality to the probability density. Empirical results support these conclusions.

Regression with input-dependent noise: A Gaussian process treatment

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.

Regression with input-dependent noise: A Bayesian treatment

In most treatments of the regression problem it is assumed that the distribution of target data can be described by a deterministic function of the inputs, together with additive Gaussian noise having constant variance. The use of maximum likelihood to train such models then corresponds to the minimization of a sum-of-squares error function. In many applications a more realistic model would allow the noise variance itself to depend on the input variables. However, the use of maximum likelihood to train such models would give highly biased results. In this paper we show how a Bayesian treatment can allow for an input-dependent variance while overcoming the bias of maximum likelihood.

A scatterometer neural network sensor model with input noise

The ERS-1 satellite carries a scatterometer which measures the amount of radiation scattered back toward the satellite by the ocean's surface. These measurements can be used to infer wind vectors. The implementation of a neural network based forward model which maps wind vectors to radar backscatter is addressed. Input noise cannot be neglected. To account for this noise, a Bayesian framework is adopted. However, Markov Chain Monte Carlo sampling is too computationally expensive. Instead, gradient information is used with a non-linear optimisation algorithm to find the maximum em a posteriori probability values of the unknown variables. The resulting models are shown to compare well with the current operational model when visualised in the target space.

Neural network regression with input uncertainty

It is generally assumed when using Bayesian inference methods for neural networks that the input data contains no noise or corruption. For real-world (errors in variable) problems this is clearly an unsafe assumption. This paper presents a Bayesian neural network framework which allows for input noise given that some model of the noise process exists. In the limit where this noise process is small and symmetric it is shown, using the Laplace approximation, that there is an additional term to the usual Bayesian error bar which depends on the variance of the input noise process. Further, by treating the true (noiseless) input as a hidden variable and sampling this jointly with the network's weights, using Markov Chain Monte Carlo methods, it is demonstrated that it is possible to infer the unbiassed regression over the noiseless input.

A scatterometer neural network sensor model with input noise

The ERS-1 satellite carries a scatterometer which measures the amount of radiation scattered back toward the satellite by the ocean's surface. These measurements can be used to infer wind vectors. The implementation of a neural network based forward model which maps wind vectors to radar backscatter is addressed. Input noise cannot be neglected. To account for this noise, a Bayesian framework is adopted. However, Markov Chain Monte Carlo sampling is too computationally expensive. Instead, gradient information is used with a non-linear optimisation algorithm to find the maximum em a posteriori probability values of the unknown variables. The resulting models are shown to compare well with the current operational model when visualised in the target space.

A linguistic resurrection? Aller + infinitive in narrative past contexts

This article studies the go periphrasis emerging in Contemporary French narrations and compares it with the narrative go periphrasis found in Middle French.

Author identification, idiolect and linguistic uniqueness

For forty years linguists have talked about idiolect and the uniqueness of individual utterances. This article explores how far these two concepts can be used to answer certain questions about the authorship of written documents—for instance how similar can two student essays be before one begins to suspect plagiarism? The article examines two ways of measuring similarity: the proportion of shared vocabulary and the number and length of shared phrases, and illustrates with examples drawn from both actual criminal court cases and incidents of student plagiarism. The article ends by engaging with Solan and Tiersma's contribution to this volume and considering whether such forensic linguistic evidence would be acceptable in American courts as well as how it might successfully be presented to a lay audience.

Foreign direct investment and local input linkages in Malaysia

This article examines variations in local input linkages in foreign transnational corporations in Malaysia. The extent to which transnational corporations foster such linkages, particularly in a developing host economy, has become an important issue for policy makers and others concerned with the long-term benefits associated with foreign direct investment. This article employs a unique data set, covering inward investors in the electrical and electronics industry, and analyzes in detail the determinants of variations in local input uses. The article develops a model of local input linkages, based on a transaction-cost framework using firm-specific factors, such as nationality of ownership, the age of the plant and its technology, and the extent to which firms employ locally recruited managers and engineers. In addition, the impacts of various policy measures on local input levels are discussed, and also the importance of the original motivation for investing in Malaysia. The article demonstrates that policy initiatives that target particular outcomes, such as stimulating exports or technology transfer, will result in a greater beneficial impact on the host country economy than more generic subsidies.