1,570 research outputs found

    Learning from samples using coherent lower previsions

    Get PDF
    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

    Approximate Nonlinear Bayesian Estimation Based on Lower and Upper Densities

    Get PDF
    Recursive calculation of the probability density function characterizing the state estimate of a nonlinear stochastic dynamic system in general cannot be performed exactly, since the type of the density changes with every processing step and the complexity increases. Hence, an approximation of the true density is required. Instead of using a single complicated approximating density, this paper is concerned with bounding the true density from below and from above by means of two simple densities. This provides a kind of guaranteed estimator with respect to the underlying true density, which requires a mechanism for ordering densities. Here, a partial ordering with respect to the cumulative distributions is employed. Based on this partial ordering, a modified Bayesian filter step is proposed, which recursively propagates lower and upper density bounds. A specific implementation for piecewise linear densities with finite support is used for demonstrating the performance of the new approach in simulations

    Measurement errors in body size of sea scallops (Placopecten magellanicus) and their effect on stock assessment models

    Get PDF
    Body-size measurement errors are usually ignored in stock assessments, but may be important when body-size data (e.g., from visual sur veys) are imprecise. We used experiments and models to quantify measurement errors and their effects on assessment models for sea scallops (Placopecten magellanicus). Errors in size data obscured modes from strong year classes and increased frequency and size of the largest and smallest sizes, potentially biasing growth, mortality, and biomass estimates. Modeling techniques for errors in age data proved useful for errors in size data. In terms of a goodness of model fit to the assessment data, it was more important to accommodate variance than bias. Models that accommodated size errors fitted size data substantially better. We recommend experimental quantification of errors along with a modeling approach that accommodates measurement errors because a direct algebraic approach was not robust and because error parameters were diff icult to estimate in our assessment model. The importance of measurement errors depends on many factors and should be evaluated on a case by case basis

    Reasoning with random sets: An agenda for the future

    Full text link
    In this paper, we discuss a potential agenda for future work in the theory of random sets and belief functions, touching upon a number of focal issues: the development of a fully-fledged theory of statistical reasoning with random sets, including the generalisation of logistic regression and of the classical laws of probability; the further development of the geometric approach to uncertainty, to include general random sets, a wider range of uncertainty measures and alternative geometric representations; the application of this new theory to high-impact areas such as climate change, machine learning and statistical learning theory.Comment: 94 pages, 17 figure

    Confirmation, Decision, and Evidential Probability

    Get PDF
    Henry Kyburg’s theory of Evidential Probability offers a neglected tool for approaching problems in confirmation theory and decision theory. I use Evidential Probability to examine some persistent problems within these areas of the philosophy of science. Formal tools in general and probability theory in particular have great promise for conceptual analysis in confirmation theory and decision theory, but they face many challenges. In each chapter, I apply Evidential Probability to a specific issue in confirmation theory or decision theory. In Chapter 1, I challenge the notion that Bayesian probability offers the best basis for a probabilistic theory of evidence. In Chapter 2, I criticise the conventional measures of quantities of evidence that use the degree of imprecision of imprecise probabilities. In Chapter 3, I develop an alternative to orthodox utility-maximizing decision theory using Kyburg’s system. In Chapter 4, I confront the orthodox notion that Nelson Goodman’s New Riddle of Induction makes purely formal theories of induction untenable. Finally, in Chapter 5, I defend probabilistic theories of inductive reasoning against John D. Norton’s recent collection of criticisms. My aim is the development of fresh perspectives on classic problems and contemporary debates. I both defend and exemplify a formal approach to the philosophy of science. I argue that Evidential Probability has great potential for clarifying our concepts of evidence and rationality
    • …
    corecore