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Outliers in discrete choice response data may result from misclassification and misreporting of the response variable and from choice behaviour that is inconsistent with modelling assumptions (e.g. random utility maximisation). In the presence of outliers, standard discrete choice models produce biased estimates and suffer from compromised predictive accuracy. Robust statistical models are less sensitive to outliers than standard non-robust models. This paper analyses two robust alternatives to the multinomial probit (MNP) model. The two models are robit models whose kernel error distributions are heavy-tailed t-distributions to moderate the influence of outliers. The first model is the multinomial robit (MNR) model, in which a generic degrees of freedom parameter controls the heavy-tailedness of the kernel error distribution. The second model, the generalised multinomial robit (Gen-MNR) model, is more flexible than MNR, as it allows for distinct heavy-tailedness in each dimension of the kernel error distribution. For both models, we derive Gibbs samplers for posterior inference. In a simulation study, we illustrate the excellent finite sample properties of the proposed Bayes estimators and show that MNR and Gen-MNR produce more accurate estimates if the choice data contain outliers through the lens of the non-robust MNP model. In a case study on transport mode choice behaviour, MNR and Gen-MNR outperform MNP by substantial margins in terms of in-sample fit and out-of-sample predictive accuracy. The case study also highlights differences in elasticity estimates across models.

In multilevel populations, there are two types of population means of an outcome variable ie, the average of all individual outcomes ignoring cluster membership and the average of cluster-specific means. To estimate the first mean, individuals can be sampled directly with simple random sampling or with two-stage sampling (TSS), that is, sampling clusters first, and then individuals within the sampled clusters. When cluster size varies in the population, three TSS schemes can be considered, ie, sampling clusters with probability proportional to cluster size and then sampling the same number of individuals per cluster; sampling clusters with equal probability and then sampling the same percentage of individuals per cluster; and sampling clusters with equal probability and then sampling the same number of individuals per cluster. Unbiased estimation of the average of all individual outcomes is discussed under each sampling scheme assuming cluster size to be informative. Furthermore, the three TSS schemes are compared in terms of efficiency with each other and with simple random sampling under the constraint of a fixed total sample size. The relative efficiency of the sampling schemes is shown to vary across different cluster size distributions. However, sampling clusters with probability proportional to size is the most efficient TSS scheme for many cluster size distributions. Model-based and design-based inference are compared and are shown to give similar results. The results are applied to the distribution of high school size in Italy and the distribution of patient list size for general practices in England.

This thesis consists of three papers on matching and weighting methods for causal inference. The first paper conducts a Monte Carlo simulation study to evaluate the performance of multivariate matching methods that select a subset of treatment and control observations. The matching methods studied are the widely used nearest neighbor matching with propensity score calipers, and the more recently proposed methods, optimal matching of an optimally chosen subset and optimal cardinality matching. The main findings are: (i) covariate balance, as measured by differences in means, variance ratios, Kolmogorov-Smirnov distances, and cross-match test statistics, is better with cardinality matching since by construction it satisfies balance requirements; (ii) for given levels of covariate balance, the matched samples are larger with cardinality matching than with the other methods; (iii) in terms of covariate distances, optimal subset matching performs best; (iv) treatment effect estimates from cardinality matching have lower RMSEs, provided strong requirements for balance, specifically, fine balance, or strength-k balance, plus close mean balance. In standard practice, a matched sample is considered to be balanced if the absolute differences in means of the covariates across treatment groups are smaller than 0.1 standard deviations. However, the simulation results suggest that stronger forms of balance should be pursued in order to remove systematic biases due to observed covariates when a difference in means treatment effect estimator is used. In particular, if the true outcome model is additive then marginal distributions should be balanced, and if the true outcome model is additive with interactions then low-dimensional joints should be balanced. The second paper focuses on longitudinal studies, where marginal structural models (MSMs) are widely used to estimate the effect of time-dependent treatments in the presence of time-dependent confounders. Under a sequential ignorability assumption, MSMs yield unbiased treatment effect estimates by weighting each observation by the inverse of the probability of their observed treatment sequence given their history of observed covariates. However, these probabilities are typically estimated by fitting a propensity score model, and the resulting weights can fail to adjust for observed covariates due to model misspecification. Also, these weights tend to yield very unstable estimates if the predicted probabilities of treatment are very close to zero, which is often the case in practice. To address both of these problems, instead of modeling the probabilities of treatment, a design-based approach is taken and weights of minimum variance that adjust for the covariates across all possible treatment histories are directly found. For this, the role of weighting in longitudinal studies of treatment effects is analyzed, and a convex optimization problem that can be solved efficiently is defined. Unlike standard methods, this approach makes evident to the investigator the limitations imposed by the data when estimating causal effects without extrapolating. A simulation study shows that this approach outperforms standard methods, providing less biased and more precise estimates of time-varying treatment effects in a variety of settings. The proposed method is used on Chilean educational data to estimate the cumulative effect of attending a private subsidized school, as opposed to a public school, on students’ university admission tests scores. The third paper is centered on observational studies with multi-valued treatments. Generalizing methods for matching and stratifying to accommodate multi-valued treatments has proven to be a complex task. A natural way to address confounding in this case is by weighting the observations, typically by the inverse probability of treatment weights (IPTW). As in the MSMs case, these weights can be highly variable and produce unstable estimates due to extreme weights. In addition, model misspecification, small sample sizes, and truncation of extreme weights can cause the weights to fail to adjust appropriately for observed confounders. The conditions the weights need to satisfy in order to provide close to unbiased treatment effect estimates with a reduced variability are determined and the convex optimization problem that can be solved in polynomial time to obtain them is defined. A simulation study with different settings is conducted to compare the proposed weighting scheme to IPTW, including generalized propensity score estimation methods that also consider explicitly the covariate balance problem in the probability estimation process. The applicability of the methods to continuous treatments is also tested. The results show that directly targeting balance with the weights, instead of focusing on estimating treatment assignment probabilities, provides the best results in terms of bias and root mean square error of the treatment effect estimator. The effects of the intensity level of the 2010 Chilean earthquake on posttraumatic stress disorder are estimated using the proposed methodology.

The prediction of an object’s future motion has traditionally been solved by applying kinematic or dynamic state-space models which describe the object’s motion as the evolution of its state over time. These models perform fairly well for short-term predictions but become either highly complex, inaccurate, or even systematically prone to failure if the task is long-term motion prediction. In an effort to generate an accurate estimate of an object’s future motion, this work investigates a statistical approach to facilitate learning an implicit motion model from previously observed samples. A motion sample is expressed as a mixed-feature variable and comprises information on the object’s kinematics as well as its relation to the immediate environment. In general, prediction is accompanied by uncertainty. For motion prediction, the probability of an estimate and especially its spatio-temporal uncertainty is of great interest. Further, the algorithm must be capable of dealing with multimodalities in ambiguous situations. In this thesis, a forecasting methodology based on an approximation of the joint probability density function (PDF) between the past and future motion is developed in which the prediction is treated in a rigorous statistical manner. Mixtures of distributions provide a sophisticated way to approximate unknown distributional shapes of highly complex multimodal PDFs arbitrary well. In order to work with mixed-feature variables, the component distributions are composed of a multivariate Gaussian and a factorized multivariate Bernoulli distribution. Using the convex combination of these non-normal component distributions, it is shown that the model is capable of capturing correlations between the mixed-feature variables. To overcome the disadvantages of a maximum likelihood (ML) parameter inference, a novel Bayesian extension in which all parameters are given prior distributions is presented – the variational Bayesian Bernoulli-Gaussian mixture model (VBBGMM). Bayesian modeling avoids overfitting, prevents singularities, facilitates the integration of prior knowledge and, moreover, automatically determines the model complexity. Parameters are estimated by variational inference, an analytical approximation of the posterior distribution. This thesis defines the VBBGMM and derives all essential equations (i.e., the variational parameter inference, the variational lower bound, the posterior predictive density, its conditional density, and its marginal density). The VBBGMM forecasting methodology is applied to long-term motion prediction. Instead of a single or multiple motion hypotheses, the algorithm estimates an entire PDF over the space of future motion of which concrete predictions can be evaluated. The mixed-feature component distribution facilitates the integration of a broad variety of diverse input and output signals (binary, categorical, scalar, and function curves). Depending on the definition of the output variables, the prediction can consist of maneuver probabilities or trajectory hypotheses, each associated with a spatio-temporal uncertainty. Further, a novel representation where the predicted future motion is modeled as complete spatio-temporal PDF is derived. Because every intersection has some special characteristics that affect driving behavior, a combination of a general model and intersection-specific models is presented. This combination significantly improves the prediction performance in combination with an online learning extension of the VBBGMM. All different types of output variables are evaluated in real world intersection scenarios. The maneuvers of oncoming vehicles at an intersection are analyzed using a categorical output. Further, the intersection-specific model extension is evaluated for this scenario as well. Continuous motion prediction is considered for the ego vehicle where both the trajectory-based output representation as well as the complete spatio-temporal PDF are evaluated.

This dissertation is a contribution towards combining logical and probabilistic reasoning. The work is based on the language P-log which combines a recently developed non-monotonic logic language, Answer Set Prolog, with the philosophy of causal Bayesian networks. The goal of this dissertation was to design and implement an inference engine for P-log and to develop a methodology of its use. As the result of this research work, we had built two P-log inference engines. Through the experiments on various examples, we have shown the advantages of using plog2.0, a system based on partial grounding algorithm and a concept of partial possible worlds. We introduced a new action description language NB which allows specifying non-deterministic actions as well as probabilities associated with these actions. We developed an encoding which maps systems written in NB to P-log programs. We presented systematic methods of representing probabilistic diagnosis and planning problems and algorithms of finding the solutions with P-log systems. Finally, we investigated the performance on these two applications and compared with other similar systems.

International audience; In the literature, several fault diagnosis methods, qualitative as well quantitative, are proposed. The main objective of these methods is in one hand, to allow detection, isolation and identification of faults ; and in the other hand to insure safety, reliability and availability of systems. This paper presents a diagnosis method based on the use of a new and suitable mathematical tool : bayesian networks. Their learning and inference capabilities allow to model complex processes by taking into account the uncertainty and the incompleteness of the provided knowledge. Furthermore, the graphical representation of causal relations existing between variables, events or physical phenomena makes bayesian networks easy to use and leads to models which can be understandable by even a non specialist of the modeled domain.

Wildfires have become one of the principal environmental problems in the Mediterranean basin. While fire plays an important role in most terrestrial plant ecosystems, the potential hazard that it represents for human lives and property has led to the application of fire exclusion policies that, in the long term, have caused severe damage, mainly due to the increase of fuel loadings in forested areas, in some forest systems. The lack of an easy solution to forest fire management highlights the importance of preventive tasks. The observed spatio-temporal pattern of wildfire occurrences may be idealized as a realization of some stochastic process. In particular, we may use a space–time point pattern approach for the analysis and inference process. We studied wildfires in Catalonia, a region in the north-east of the Iberian Peninsula, and we analyzed the spatio-temporal patterns produced by those wildfire incidences by considering the influence of covariates on trends in the intensity of wildfire locations. A total of 3,166 wildfires from 1994–2008 have been recorded. We specified spatio-temporal log-Gaussian Cox process models. Models were estimated using Bayesian inference for Gaussian Markov Random Field through the integrated nested Laplace approximation algorithm. The results of our analysis have provided statistical evidence that areas closer to humans have more human induced wildfires, areas farther have more naturally occurring wildfires. We believe the methods presented in this paper may contribute to the prevention and management of those wildfires which are not random in space or time.