Robust Inference of Trees

This paper is concerned with the reliable inference of optimal tree-approximations to the dependency structure of an unknown distribution generating data. The traditional approach to the problem measures the dependency strength between random variables by the index called mutual information. In this paper reliability is achieved by Walley's imprecise Dirichlet model, which generalizes Bayesian learning with Dirichlet priors. Adopting the imprecise Dirichlet model results in posterior interval expectation for mutual information, and in a set of plausible trees consistent with the data. Reliable inference about the actual tree is achieved by focusing on the substructure common to all the plausible trees. We develop an exact algorithm that infers the substructure in time O(m^4), m being the number of random variables. The new algorithm is applied to a set of data sampled from a known distribution. The method is shown to reliably infer edges of the actual tree even when the data are very scarce, unlike the traditional approach. Finally, we provide lower and upper credibility limits for mutual information under the imprecise Dirichlet model. These enable the previous developments to be extended to a full inferential method for trees.Comment: 26 pages, 7 figure

Robust Estimators under the Imprecise Dirichlet Model

Walley's Imprecise Dirichlet Model (IDM) for categorical data overcomes several fundamental problems which other approaches to uncertainty suffer from. Yet, to be useful in practice, one needs efficient ways for computing the imprecise=robust sets or intervals. The main objective of this work is to derive exact, conservative, and approximate, robust and credible interval estimates under the IDM for a large class of statistical estimators, including the entropy and mutual information.Comment: 16 LaTeX page

Robust estimation of risks from small samples

Data-driven risk analysis involves the inference of probability distributions from measured or simulated data. In the case of a highly reliable system, such as the electricity grid, the amount of relevant data is often exceedingly limited, but the impact of estimation errors may be very large. This paper presents a robust nonparametric Bayesian method to infer possible underlying distributions. The method obtains rigorous error bounds even for small samples taken from ill-behaved distributions. The approach taken has a natural interpretation in terms of the intervals between ordered observations, where allocation of probability mass across intervals is well-specified, but the location of that mass within each interval is unconstrained. This formulation gives rise to a straightforward computational resampling method: Bayesian Interval Sampling. In a comparison with common alternative approaches, it is shown to satisfy strict error bounds even for ill-behaved distributions.Comment: 13 pages, 3 figures; supplementary information provided. A revised version of this manuscript has been accepted for publication in Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Science

Generalized Bayesian inference under prior-data conflict

This thesis is concerned with the generalisation of Bayesian inference towards the use of imprecise or interval probability, with a focus on model behaviour in case of prior-data conflict. Bayesian inference is one of the main approaches to statistical inference. It requires to express (subjective) knowledge on the parameter(s) of interest not incorporated in the data by a so-called prior distribution. All inferences are then based on the so-called posterior distribution, the subsumption of prior knowledge and the information in the data calculated via Bayes' Rule. The adequate choice of priors has always been an intensive matter of debate in the Bayesian literature. While a considerable part of the literature is concerned with so-called non-informative priors aiming to eliminate (or, at least, to standardise) the influence of priors on posterior inferences, inclusion of specific prior information into the model may be necessary if data are scarce, or do not contain much information about the parameter(s) of interest; also, shrinkage estimators, common in frequentist approaches, can be considered as Bayesian estimators based on informative priors. When substantial information is used to elicit the prior distribution through, e.g, an expert's assessment, and the sample size is not large enough to eliminate the influence of the prior, prior-data conflict can occur, i.e., information from outlier-free data suggests parameter values which are surprising from the viewpoint of prior information, and it may not be clear whether the prior specifications or the integrity of the data collecting method (the measurement procedure could, e.g., be systematically biased) should be questioned. In any case, such a conflict should be reflected in the posterior, leading to very cautious inferences, and most statisticians would thus expect to observe, e.g., wider credibility intervals for parameters in case of prior-data conflict. However, at least when modelling is based on conjugate priors, prior-data conflict is in most cases completely averaged out, giving a false certainty in posterior inferences. Here, imprecise or interval probability methods offer sound strategies to counter this issue, by mapping parameter uncertainty over sets of priors resp. posteriors instead of over single distributions. This approach is supported by recent research in economics, risk analysis and artificial intelligence, corroborating the multi-dimensional nature of uncertainty and concluding that standard probability theory as founded on Kolmogorov's or de Finetti's framework may be too restrictive, being appropriate only for describing one dimension, namely ideal stochastic phenomena. The thesis studies how to efficiently describe sets of priors in the setting of samples from an exponential family. Models are developed that offer enough flexibility to express a wide range of (partial) prior information, give reasonably cautious inferences in case of prior-data conflict while resulting in more precise inferences when prior and data agree well, and still remain easily tractable in order to be useful for statistical practice. Applications in various areas, e.g. common-cause failure modeling and Bayesian linear regression, are explored, and the developed approach is compared to other imprecise probability models.Das Thema dieser Dissertation ist die Generalisierung der Bayes-Inferenz durch die Verwendung von unscharfen oder intervallwertigen Wahrscheinlichkeiten. Ein besonderer Fokus liegt dabei auf dem Modellverhalten in dem Fall, dass Vorwissen und beobachtete Daten in Konflikt stehen. Die Bayes-Inferenz ist einer der Hauptansätze zur Herleitung von statistischen Inferenzmethoden. In diesem Ansatz muss (eventuell subjektives) Vorwissen über die Modellparameter in einer sogenannten Priori-Verteilung (kurz: Priori) erfasst werden. Alle Inferenzaussagen basieren dann auf der sogenannten Posteriori-Verteilung (kurz: Posteriori), welche mittels des Satzes von Bayes berechnet wird und das Vorwissen und die Informationen in den Daten zusammenfasst. Wie eine Priori-Verteilung in der Praxis zu wählen sei, ist dabei stark umstritten. Ein großer Teil der Literatur befasst sich mit der Bestimmung von sogenannten nichtinformativen Prioris. Diese zielen darauf ab, den Einfluss der Priori auf die Posteriori zu eliminieren oder zumindest zu standardisieren. Falls jedoch nur wenige Daten zur Verfügung stehen, oder diese nur wenige Informationen in Bezug auf die Modellparameter bereitstellen, kann es hingegen nötig sein, spezifische Priori-Informationen in ein Modell einzubeziehen. Außerdem können sogenannte Shrinkage-Schätzer, die in frequentistischen Ansätzen häufig zum Einsatz kommen, als Bayes-Schätzer mit informativen Prioris angesehen werden. Wenn spezifisches Vorwissen zur Bestimmung einer Priori genutzt wird (beispielsweise durch eine Befragung eines Experten), aber die Stichprobengröße nicht ausreicht, um eine solche informative Priori zu überstimmen, kann sich ein Konflikt zwischen Priori und Daten ergeben. Dieser kann sich darin äußern, dass die beobachtete (und von eventuellen Ausreißern bereinigte) Stichprobe Parameterwerte impliziert, die aus Sicht der Priori äußerst überraschend und unerwartet sind. In solch einem Fall kann es unklar sein, ob eher das Vorwissen oder eher die Validität der Datenerhebung in Zweifel gezogen werden sollen. (Es könnten beispielsweise Messfehler, Kodierfehler oder eine Stichprobenverzerrung durch selection bias vorliegen.) Zweifellos sollte sich ein solcher Konflikt in der Posteriori widerspiegeln und eher vorsichtige Inferenzaussagen nach sich ziehen; die meisten Statistiker würden daher davon ausgehen, dass sich in solchen Fällen breitere Posteriori-Kredibilitätsintervalle für die Modellparameter ergeben. Bei Modellen, die auf der Wahl einer bestimmten parametrischen Form der Priori basieren, welche die Berechnung der Posteriori wesentlich vereinfachen (sogenannte konjugierte Priori-Verteilungen), wird ein solcher Konflikt jedoch einfach ausgemittelt. Dann werden Inferenzaussagen, die auf einer solchen Posteriori basieren, den Anwender in falscher Sicherheit wiegen. In dieser problematischen Situation können Intervallwahrscheinlichkeits-Methoden einen fundierten Ausweg bieten, indem Unsicherheit über die Modellparameter mittels Mengen von Prioris beziehungsweise Posterioris ausgedrückt wird. Neuere Erkenntnisse aus Risikoforschung, Ökonometrie und der Forschung zu künstlicher Intelligenz, die die Existenz von verschiedenen Arten von Unsicherheit nahelegen, unterstützen einen solchen Modellansatz, der auf der Feststellung aufbaut, dass die auf den Ansätzen von Kolmogorov oder de Finetti basierende übliche Wahrscheinlichkeitsrechung zu restriktiv ist, um diesen mehrdimensionalen Charakter von Unsicherheit adäquat einzubeziehen. Tatsächlich kann in diesen Ansätzen nur eine der Dimensionen von Unsicherheit modelliert werden, nämlich die der idealen Stochastizität. In der vorgelegten Dissertation wird untersucht, wie sich Mengen von Prioris für Stichproben aus Exponentialfamilien effizient beschreiben lassen. Wir entwickeln Modelle, die eine ausreichende Flexibilität gewährleisten, sodass eine Vielfalt von Ausprägungen von partiellem Vorwissen beschrieben werden kann. Diese Modelle führen zu vorsichtigen Inferenzaussagen, wenn ein Konflikt zwischen Priori und Daten besteht, und ermöglichen dennoch präzisere Aussagen für den Fall, dass Priori und Daten im Wesentlichen übereinstimmen, ohne dabei die Einsatzmöglichkeiten in der statistischen Praxis durch eine zu hohe Komplexität in der Anwendung zu erschweren. Wir ermitteln die allgemeinen Inferenzeigenschaften dieser Modelle, die sich durch einen klaren und nachvollziehbaren Zusammenhang zwischen Modellunsicherheit und der Präzision von Inferenzaussagen auszeichnen, und untersuchen Anwendungen in verschiedenen Bereichen, unter anderem in sogenannten common-cause-failure-Modellen und in der linearen Bayes-Regression. Zudem werden die in dieser Dissertation entwickelten Modelle mit anderen Intervallwahrscheinlichkeits-Modellen verglichen und deren jeweiligen Stärken und Schwächen diskutiert, insbesondere in Bezug auf die Präzision von Inferenzaussagen bei einem Konflikt von Vorwissen und beobachteten Daten

Learning from samples using coherent lower previsions

Het hoofdonderwerp van dit werk is het afleiden, voorstellen en bestuderen van voorspellende en parametrische gevolgtrekkingsmodellen die gebaseerd zijn op de theorie van coherente onderprevisies. Een belangrijk nevenonderwerp is het vinden en bespreken van extreme onderwaarschijnlijkheden. In het hoofdstuk ‘Modeling uncertainty’ geef ik een inleidend overzicht van de theorie van coherente onderprevisies ─ ook wel theorie van imprecieze waarschijnlijkheden genoemd ─ en de ideeën waarop ze gestoeld is. Deze theorie stelt ons in staat onzekerheid expressiever ─ en voorzichtiger ─ te beschrijven. Dit overzicht is origineel in de zin dat ze meer dan andere inleidingen vertrekt van de intuitieve theorie van coherente verzamelingen van begeerlijke gokken. Ik toon in het hoofdstuk ‘Extreme lower probabilities’ hoe we de meest extreme vormen van onzekerheid kunnen vinden die gemodelleerd kunnen worden met onderwaarschijnlijkheden. Elke andere onzekerheidstoestand beschrijfbaar met onderwaarschijnlijkheden kan geformuleerd worden in termen van deze extreme modellen. Het belang van de door mij bekomen en uitgebreid besproken resultaten in dit domein is voorlopig voornamelijk theoretisch. Het hoofdstuk ‘Inference models’ behandelt leren uit monsters komende uit een eindige, categorische verzameling. De belangrijkste basisveronderstelling die ik maak is dat het bemonsteringsproces omwisselbaar is, waarvoor ik een nieuwe definitie geef in termen van begeerlijke gokken. Mijn onderzoek naar de gevolgen van deze veronderstelling leidt ons naar enkele belangrijke representatiestellingen: onzekerheid over (on)eindige rijen monsters kan gemodelleerd worden in termen van categorie-aantallen (-frequenties). Ik bouw hier op voort om voor twee populaire gevolgtrekkingsmodellen voor categorische data ─ het voorspellende imprecies Dirichlet-multinomiaalmodel en het parametrische imprecies Dirichletmodel ─ een verhelderende afleiding te geven, louter vertrekkende van enkele grondbeginselen; deze modellen pas ik toe op speltheorie en het leren van Markov-ketens. In het laatste hoofdstuk, ‘Inference models for exponential families’, verbreed ik de blik tot niet-categorische exponentiële-familie-bemonsteringsmodellen; voorbeelden zijn normale bemonstering en Poisson-bemonstering. Eerst onderwerp ik de exponentiële families en de aanverwante toegevoegde parametrische en voorspellende previsies aan een grondig onderzoek. Deze aanverwante previsies worden gebruikt in de klassieke Bayesiaanse gevolgtrekkingsmodellen gebaseerd op toegevoegd updaten. Ze dienen als grondslag voor de nieuwe, door mij voorgestelde imprecieze-waarschijnlijkheidsgevolgtrekkingsmodellen. In vergelijking met de klassieke Bayesiaanse aanpak, laat de mijne toe om voorzichtiger te zijn bij de beschrijving van onze kennis over het bemonsteringsmodel; deze voorzichtigheid wordt weerspiegeld door het op deze modellen gebaseerd gedrag (getrokken besluiten, gemaakte voorspellingen, genomen beslissingen). Ik toon ten slotte hoe de voorgestelde gevolgtrekkingsmodellen gebruikt kunnen worden voor classificatie door de naïeve credale classificator.This thesis's main subject is deriving, proposing, and studying predictive and parametric inference models that are based on the theory of coherent lower previsions. One important side subject also appears: obtaining and discussing extreme lower probabilities. In the chapter ‘Modeling uncertainty’, I give an introductory overview of the theory of coherent lower previsions ─ also called the theory of imprecise probabilities ─ and its underlying ideas. This theory allows us to give a more expressive ─ and a more cautious ─ description of uncertainty. This overview is original in the sense that ─ more than other introductions ─ it is based on the intuitive theory of coherent sets of desirable gambles. I show in the chapter ‘Extreme lower probabilities’ how to obtain the most extreme forms of uncertainty that can be modeled using lower probabilities. Every other state of uncertainty describable by lower probabilities can be formulated in terms of these extreme ones. The importance of the results in this area obtained and extensively discussed by me is currently mostly theoretical. The chapter ‘Inference models’ treats learning from samples from a finite, categorical space. My most basic assumption about the sampling process is that it is exchangeable, for which I give a novel definition in terms of desirable gambles. My investigation of the consequences of this assumption leads us to some important representation theorems: uncertainty about (in)finite sample sequences can be modeled entirely in terms of category counts (frequencies). I build on this to give an elucidating derivation from first principles for two popular inference models for categorical data ─ the predictive imprecise Dirichlet-multinomial model and the parametric imprecise Dirichlet model; I apply these models to game theory and learning Markov chains. In the last chapter, ‘Inference models for exponential families’, I enlarge the scope to exponential family sampling models; examples are normal sampling and Poisson sampling. I first thoroughly investigate exponential families and the related conjugate parametric and predictive previsions used in classical Bayesian inference models based on conjugate updating. These previsions serve as a basis for the new imprecise-probabilistic inference models I propose. Compared to the classical Bayesian approach, mine allows to be much more cautious when trying to express what we know about the sampling model; this caution is reflected in behavior (conclusions drawn, predictions made, decisions made) based on these models. Lastly, I show how the proposed inference models can be used for classification with the naive credal classifier

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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Loss Distribution Approach for Operational Risk Capital Modelling under Basel II: Combining Different Data Sources for Risk Estimation

The management of operational risk in the banking industry has undergone significant changes over the last decade due to substantial changes in operational risk environment. Globalization, deregulation, the use of complex financial products and changes in information technology have resulted in exposure to new risks very different from market and credit risks. In response, Basel Committee for banking Supervision has developed a regulatory framework, referred to as Basel II, that introduced operational risk category and corresponding capital requirements. Over the past five years, major banks in most parts of the world have received accreditation under the Basel II Advanced Measurement Approach (AMA) by adopting the loss distribution approach (LDA) despite there being a number of unresolved methodological challenges in its implementation. Different approaches and methods are still under hot debate. In this paper, we review methods proposed in the literature for combining different data sources (internal data, external data and scenario analysis) which is one of the regulatory requirement for AMA

Extraction of decision rules via imprecise probabilities

"This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of General Systems on 2017, available online: https://www.tandfonline.com/doi/full/10.1080/03081079.2017.1312359"Data analysis techniques can be applied to discover important relations among features. This is the main objective of the Information Root Node Variation (IRNV) technique, a new method to extract knowledge from data via decision trees. The decision trees used by the original method were built using classic split criteria. The performance of new split criteria based on imprecise probabilities and uncertainty measures, called credal split criteria, differs significantly from the performance obtained using the classic criteria. This paper extends the IRNV method using two credal split criteria: one based on a mathematical parametric model, and other one based on a non-parametric model. The performance of the method is analyzed using a case study of traffic accident data to identify patterns related to the severity of an accident. We found that a larger number of rules is generated, significantly supplementing the information obtained using the classic split criteria.This work has been supported by the Spanish "Ministerio de Economia y Competitividad" [Project number TEC2015-69496-R] and FEDER funds.Abellán, J.; López-Maldonado, G.; Garach, L.; Castellano, JG. (2017). Extraction of decision rules via imprecise probabilities. International Journal of General Systems. 46(4):313-331. https://doi.org/10.1080/03081079.2017.1312359S313331464Abellan, J., & Bosse, E. (2018). Drawbacks of Uncertainty Measures Based on the Pignistic Transformation. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48(3), 382-388. doi:10.1109/tsmc.2016.2597267Abellán, J., & Klir, G. J. (2005). Additivity of uncertainty measures on credal sets. International Journal of General Systems, 34(6), 691-713. doi:10.1080/03081070500396915Abellán, J., & Masegosa, A. R. (2010). An ensemble method using credal decision trees. European Journal of Operational Research, 205(1), 218-226. doi:10.1016/j.ejor.2009.12.003(2003). International Journal of Intelligent Systems, 18(12). doi:10.1002/int.v18:12Abellán, J., Klir, G. J., & Moral, S. (2006). Disaggregated total uncertainty measure for credal sets. International Journal of General Systems, 35(1), 29-44. doi:10.1080/03081070500473490Abellán, J., Baker, R. M., & Coolen, F. P. A. (2011). Maximising entropy on the nonparametric predictive inference model for multinomial data. European Journal of Operational Research, 212(1), 112-122. doi:10.1016/j.ejor.2011.01.020Abellán, J., López, G., & de Oña, J. (2013). Analysis of traffic accident severity using Decision Rules via Decision Trees. Expert Systems with Applications, 40(15), 6047-6054. doi:10.1016/j.eswa.2013.05.027Abellán, J., Baker, R. M., Coolen, F. P. A., Crossman, R. J., & Masegosa, A. R. (2014). Classification with decision trees from a nonparametric predictive inference perspective. Computational Statistics & Data Analysis, 71, 789-802. doi:10.1016/j.csda.2013.02.009Alkhalid, A., Amin, T., Chikalov, I., Hussain, S., Moshkov, M., & Zielosko, B. (2013). Optimization and analysis of decision trees and rules: dynamic programming approach. International Journal of General Systems, 42(6), 614-634. doi:10.1080/03081079.2013.798902Chang, L.-Y., & Chien, J.-T. (2013). Analysis of driver injury severity in truck-involved accidents using a non-parametric classification tree model. Safety Science, 51(1), 17-22. doi:10.1016/j.ssci.2012.06.017Chang, L.-Y., & Wang, H.-W. (2006). Analysis of traffic injury severity: An application of non-parametric classification tree techniques. Accident Analysis & Prevention, 38(5), 1019-1027. doi:10.1016/j.aap.2006.04.009DE CAMPOS, L. M., HUETE, J. F., & MORAL, S. (1994). PROBABILITY INTERVALS: A TOOL FOR UNCERTAIN REASONING. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 02(02), 167-196. doi:10.1142/s0218488594000146DGT. 2011b.Spanish Road Safety Strategy 2011–2020, 222 p. Madrid: Traffic General Directorate.Dolques, X., Le Ber, F., Huchard, M., & Grac, C. (2016). Performance-friendly rule extraction in large water data-sets with AOC posets and relational concept analysis. International Journal of General Systems, 45(2), 187-210. doi:10.1080/03081079.2015.1072927Gray, R. C., Quddus, M. A., & Evans, A. (2008). Injury severity analysis of accidents involving young male drivers in Great Britain. Journal of Safety Research, 39(5), 483-495. doi:10.1016/j.jsr.2008.07.003Guo, J., & Chankong, V. (2002). Rough set-based approach to rule generation and rule induction. International Journal of General Systems, 31(6), 601-617. doi:10.1080/0308107021000034353Huang, H., Chin, H. C., & Haque, M. M. (2008). Severity of driver injury and vehicle damage in traffic crashes at intersections: A Bayesian hierarchical analysis. Accident Analysis & Prevention, 40(1), 45-54. doi:10.1016/j.aap.2007.04.002Kashani, A. T., & Mohaymany, A. S. (2011). Analysis of the traffic injury severity on two-lane, two-way rural roads based on classification tree models. Safety Science, 49(10), 1314-1320. doi:10.1016/j.ssci.2011.04.019Li, X., & Yu, L. (2016). Decision making under various types of uncertainty. International Journal of General Systems, 45(3), 251-252. doi:10.1080/03081079.2015.1086574Mantas, C. J., & Abellán, J. (2014). Analysis and extension of decision trees based on imprecise probabilities: Application on noisy data. Expert Systems with Applications, 41(5), 2514-2525. doi:10.1016/j.eswa.2013.09.050Mayhew, D. R., Simpson, H. M., & Pak, A. (2003). Changes in collision rates among novice drivers during the first months of driving. 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Bayesian semiparametric stochastic volatility modeling

This paper extends the existing fully parametric Bayesian literature on stochastic volatility to allow for more general return distributions. Instead of specifying a particular distribution for the return innovation, we use nonparametric Bayesian methods to flexibly model the skewness and kurtosis of the distribution while continuing to model the dynamics of volatility with a parametric structure. Our semiparametric Bayesian approach provides a full characterization of parametric and distributional uncertainty. We present a Markov chain Monte Carlo sampling approach to estimation with theoretical and computational issues for simulation from the posterior predictive distributions. The new model is assessed based on simulation evidence, an empirical example, and comparison to parametric models.Econometric models ; Stochastic analysis