Safe-Bayesian Generalized Linear Regression

We study generalized Bayesian inference under misspecification, i.e. when the model is 'wrong but useful'. Generalized Bayes equips the likelihood with a learning rate $\eta$. We show that for generalized linear models (GLMs), $\eta$-generalized Bayes concentrates around the best approximation of the truth within the model for specific $\eta \neq 1$, even under severely misspecified noise, as long as the tails of the true distribution are exponential. We derive MCMC samplers for generalized Bayesian lasso and logistic regression and give examples of both simulated and real-world data in which generalized Bayes substantially outperforms standard Bayes.Comment: Final version. Accepted to AISTATS 202

Safe-Bayesian Generalized Linear Regression

We study generalized Bayesian inference under misspecification, i.e. when the model is ‘wrong but useful’. Generalized Bayes equips the likelihood with a learning rate η. We show that for generalized linear models (GLMs), η-generalized Bayes concentrates around the best approximation of the truth within the model for specific ηeq1, even under severely misspecified noise, as long as the tails of the true distribution are exponential. We derive MCMC samplers for generalized Bayesian lasso and logistic regression and give examples of both simulated and real-world data in which generalized Bayes substantially outperforms standard Bayes

The Overlooked Potential of Generalized Linear Models in Astronomy - I: Binomial Regression

Revealing hidden patterns in astronomical data is often the path to fundamental scientific breakthroughs; meanwhile the complexity of scientific inquiry increases as more subtle relationships are sought. Contemporary data analysis problems often elude the capabilities of classical statistical techniques, suggesting the use of cutting edge statistical methods. In this light, astronomers have overlooked a whole family of statistical techniques for exploratory data analysis and robust regression, the so-called Generalized Linear Models (GLMs). In this paper -- the first in a series aimed at illustrating the power of these methods in astronomical applications -- we elucidate the potential of a particular class of GLMs for handling binary/binomial data, the so-called logit and probit regression techniques, from both a maximum likelihood and a Bayesian perspective. As a case in point, we present the use of these GLMs to explore the conditions of star formation activity and metal enrichment in primordial minihaloes from cosmological hydro-simulations including detailed chemistry, gas physics, and stellar feedback. We predict that for a dark mini-halo with metallicity $\approx 1.3 \times 10^{-4} Z_{\bigodot}$, an increase of $1.2 \times 10^{-2}$ in the gas molecular fraction, increases the probability of star formation occurrence by a factor of 75%. Finally, we highlight the use of receiver operating characteristic curves as a diagnostic for binary classifiers, and ultimately we use these to demonstrate the competitive predictive performance of GLMs against the popular technique of artificial neural networks.Comment: 20 pages, 10 figures, 3 tables, accepted for publication in Astronomy and Computin

Nonparametric Bayesian hazard rate models based on penalized splines

Extensions of the traditional Cox proportional hazard model, concerning the following features are often desirable in applications: Simultaneous nonparametric estimation of baseline hazard and usual fixed covariate effects, modelling and detection of time-varying covariate effects and nonlinear functional forms of metrical covariates, and inclusion of frailty components. In this paper, we develop Bayesian multiplicative hazard rate models for survival and event history data that can deal with these issues in a flexible and unified framework. Some simpler models, such as piecewise exponential models with a smoothed baseline hazard, are covered as special cases. Embedded in the counting process approach, nonparametric estimation of unknown nonlinear functional effects of time or covariates is based on Bayesian penalized splines. Inference is fully Bayesian and uses recent MCMC sampling schemes. Smoothing parameters are an integral part of the model and are estimated automatically. We investigate performance of our approach through simulation studies, and illustrate it with a real data application

Bayesian Optimisation for Safe Navigation under Localisation Uncertainty

In outdoor environments, mobile robots are required to navigate through terrain with varying characteristics, some of which might significantly affect the integrity of the platform. Ideally, the robot should be able to identify areas that are safe for navigation based on its own percepts about the environment while avoiding damage to itself. Bayesian optimisation (BO) has been successfully applied to the task of learning a model of terrain traversability while guiding the robot through more traversable areas. An issue, however, is that localisation uncertainty can end up guiding the robot to unsafe areas and distort the model being learnt. In this paper, we address this problem and present a novel method that allows BO to consider localisation uncertainty by applying a Gaussian process model for uncertain inputs as a prior. We evaluate the proposed method in simulation and in experiments with a real robot navigating over rough terrain and compare it against standard BO methods.Comment: To appear in the proceedings of the 18th International Symposium on Robotics Research (ISRR 2017