Generalized Bayesian MARS: Tools for Emulating Stochastic Computer Models

The multivariate adaptive regression spline (MARS) approach of Friedman (1991) and its Bayesian counterpart (Francom et al. 2018) are effective approaches for the emulation of computer models. The traditional assumption of Gaussian errors limits the usefulness of MARS, and many popular alternatives, when dealing with stochastic computer models. We propose a generalized Bayesian MARS (GBMARS) framework which admits the broad class of generalized hyperbolic distributions as the induced likelihood function. This allows us to develop tools for the emulation of stochastic simulators which are parsimonious, scalable, interpretable and require minimal tuning, while providing powerful predictive and uncertainty quantification capabilities. GBMARS is capable of robust regression with t distributions, quantile regression with asymmetric Laplace distributions and a general form of "Normal-Wald" regression in which the shape of the error distribution and the structure of the mean function are learned simultaneously. We demonstrate the effectiveness of GBMARS on various stochastic computer models and we show that it compares favorably to several popular alternatives

Pseudo Label Selection is a Decision Problem

Pseudo-Labeling is a simple and effective approach to semi-supervised learning. It requires criteria that guide the selection of pseudo-labeled data. The latter have been shown to crucially affect pseudo-labeling's generalization performance. Several such criteria exist and were proven to work reasonably well in practice. However, their performance often depends on the initial model fit on labeled data. Early overfitting can be propagated to the final model by choosing instances with overconfident but wrong predictions, often called confirmation bias. In two recent works, we demonstrate that pseudo-label selection (PLS) can be naturally embedded into decision theory. This paves the way for BPLS, a Bayesian framework for PLS that mitigates the issue of confirmation bias. At its heart is a novel selection criterion: an analytical approximation of the posterior predictive of pseudo-samples and labeled data. We derive this selection criterion by proving Bayes-optimality of this "pseudo posterior predictive". We empirically assess BPLS for generalized linear, non-parametric generalized additive models and Bayesian neural networks on simulated and real-world data. When faced with data prone to overfitting and thus a high chance of confirmation bias, BPLS outperforms traditional PLS methods. The decision-theoretic embedding further allows us to render PLS more robust towards the involved modeling assumptions. To achieve this goal, we introduce a multi-objective utility function. We demonstrate that the latter can be constructed to account for different sources of uncertainty and explore three examples: model selection, accumulation of errors and covariate shift.Comment: Accepted for presentation at the 46th German Conference on Artificial Intelligenc

Bayesian Inference in Processing Experimental Data: Principles and Basic Applications

This report introduces general ideas and some basic methods of the Bayesian probability theory applied to physics measurements. Our aim is to make the reader familiar, through examples rather than rigorous formalism, with concepts such as: model comparison (including the automatic Ockham's Razor filter provided by the Bayesian approach); parametric inference; quantification of the uncertainty about the value of physical quantities, also taking into account systematic effects; role of marginalization; posterior characterization; predictive distributions; hierarchical modelling and hyperparameters; Gaussian approximation of the posterior and recovery of conventional methods, especially maximum likelihood and chi-square fits under well defined conditions; conjugate priors, transformation invariance and maximum entropy motivated priors; Monte Carlo estimates of expectation, including a short introduction to Markov Chain Monte Carlo methods.Comment: 40 pages, 2 figures, invited paper for Reports on Progress in Physic

Ensemble learning method for hidden markov models.

For complex classification systems, data are gathered from various sources and potentially have different representations. Thus, data may have large intra-class variations. In fact, modeling each data class with a single model might lead to poor generalization. The classification error can be more severe for temporal data where each sample is represented by a sequence of observations. Thus, there is a need for building a classification system that takes into account the variations within each class in the data. This dissertation introduces an ensemble learning method for temporal data that uses a mixture of Hidden Markov Model (HMM) classifiers. We hypothesize that the data are generated by K models, each of which reacts a particular trend in the data. Model identification could be achieved through clustering in the feature space or in the parameters space. However, this approach is inappropriate in the context of sequential data. The proposed approach is based on clustering in the log-likelihood space, and has two main steps. First, one HMM is fit to each of the N individual sequences. For each fitted model, we evaluate the log-likelihood of each sequence. This will result in an N-by-N log-likelihood distance matrix that will be partitioned into K groups using a relational clustering algorithm. In the second step, we learn the parameters of one HMM per group. We propose using and optimizing various training approaches for the different K groups depending on their size and homogeneity. In particular, we investigate the maximum likelihood (ML), the minimum classification error (MCE) based discriminative, and the Variational Bayesian (VB) training approaches. Finally, to test a new sequence, its likelihood is computed in all the models and a final confidence value is assigned by combining the multiple models outputs using a decision level fusion method such as an artificial neural network or a hierarchical mixture of experts. Our approach was evaluated on two real-world applications: (1) identification of Cardio-Pulmonary Resuscitation (CPR) scenes in video simulating medical crises; and (2) landmine detection using Ground Penetrating Radar (GPR). Results on both applications show that the proposed method can identify meaningful and coherent HMM mixture components that describe different properties of the data. Each HMM mixture component models a group of data that share common attributes. The results indicate that the proposed method outperforms the baseline HMM that uses one model for each class in the data

On the usefulness of imprecise Bayesianism in chemical kinetics

International audienceBayesian methods are growing ever more popular in chemical kinetics. The reasons for this and general challenges related to kinetic parameter estimation are shortly reviewed. Most authors content themselves with using one single (mostly uniform) prior distribution. The goal of this paper is to go into some serious issues this raises. The problems of confusing knowledge and ignorance and of reparametrisation are examined. The legitimacy of a probabilistic Ockham’s razor is called into question. A synthetic example involving two reaction models was used to illustrate how merging the parameter space volume with the model accuracy into a single number might be unwise. Robust Bayesian analysis appears to be a simple andstraightforward way to avoid the problems mentioned throughout this article