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Rozprawy doktorskie na temat „Bayesian estimation”

Autor: Grafiati

Data publikacji: 4 czerwca 2021

Data aktualizacji: 25 lipca 2025

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1

Rademeyer, Estian. "Bayesian kernel density estimation." Diss., University of Pretoria, 2017. http://hdl.handle.net/2263/64692.

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This dissertation investigates the performance of two-class classi cation credit scoring data sets with low default ratios. The standard two-class parametric Gaussian and naive Bayes (NB), as well as the non-parametric Parzen classi ers are extended, using Bayes' rule, to include either a class imbalance or a Bernoulli prior. This is done with the aim of addressing the low default probability problem. Furthermore, the performance of Parzen classi cation with Silverman and Minimum Leave-one-out Entropy (MLE) Gaussian kernel bandwidth estimation is also investigated. It is shown that the n

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2

Weiss, Yair. "Bayesian motion estimation and segmentation." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/9354.

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Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Brain and Cognitive Sciences, 1998.<br>Includes bibliographical references (leaves 195-204).<br>Estimating motion in scenes containing multiple moving objects remains a difficult problem in computer vision yet is solved effortlessly by humans. In this thesis we present a computational investigation of this astonishing performance in human vision. The method we use throughout is to formulate a small number of assumptions and see the extent to which the optimal interpretation given these assumptions corresponds to the human percept.

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Bouda, Milan. "Bayesian Estimation of DSGE Models." Doctoral thesis, Vysoká škola ekonomická v Praze, 2012. http://www.nusl.cz/ntk/nusl-200007.

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Thesis is dedicated to Bayesian Estimation of DSGE Models. Firstly, the history of DSGE modeling is outlined as well as development of this macroeconometric field in the Czech Republic and in the rest of the world. Secondly, the comprehensive DSGE framework is described in detail. It means that everyone is able to specify or estimate arbitrary DSGE model according to this framework. Thesis contains two empirical studies. The first study describes derivation of the New Keynesian DSGE Model and its estimation using Bayesian techniques. This model is estimated with three different Taylor rules an

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4

Pramanik, Santanu. "The Bayesian and approximate Bayesian methods in small area estimation." College Park, Md.: University of Maryland, 2008. http://hdl.handle.net/1903/8856.

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Thesis (Ph. D.) -- University of Maryland, College Park, 2008.<br>Thesis research directed by: Joint Program in Survey Methodology. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

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5

Campolieti, Michele. "Bayesian estimation of discrete duration models." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape16/PQDD_0001/NQ27884.pdf.

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6

Hissmann, Michael. "Bayesian estimation for white light interferometry." Berlin Pro Business, 2005. http://shop.pro-business.com/product_info.php?products_id=357.

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7

Makarava, Natallia. "Bayesian estimation of self-similarity exponent." Phd thesis, Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2013/6409/.

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Estimation of the self-similarity exponent has attracted growing interest in recent decades and became a research subject in various fields and disciplines. Real-world data exhibiting self-similar behavior and/or parametrized by self-similarity exponent (in particular Hurst exponent) have been collected in different fields ranging from finance and human sciencies to hydrologic and traffic networks. Such rich classes of possible applications obligates researchers to investigate qualitatively new methods for estimation of the self-similarity exponent as well as identification of long-range d

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8

Graham, Matthew Corwin 1986. "Robust Bayesian state estimation and mapping." Thesis, Massachusetts Institute of Technology, 2015. http://hdl.handle.net/1721.1/98678.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2015.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 135-146).<br>Virtually all robotic and autonomous systems rely on navigation and mapping algorithms (e.g. the Kalman filter or simultaneous localization and mapping (SLAM)) to determine their location in the world. Unfortunately, these algorithms are not robust to outliers and even a single faulty measurement can cause a catastrophic failure of the navigation system. This thesis proposes several novel rob

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9

Vega-Brown, Will (William Robert). "Predictive parameter estimation for Bayesian filtering." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/81715.

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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2013.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 113-117).<br>In this thesis, I develop CELLO, an algorithm for predicting the covariances of any Gaussian model used to account for uncertainty in a complex system. The primary motivation for this work is state estimation; often, complex raw sensor measurements are processed into low dimensional observations of a vehicle state. I argue that the covariance of these observations can be well-modelled as a function of the r

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10

Xing, Guan. "LASSOING MIXTURES AND BAYESIAN ROBUST ESTIMATION." Case Western Reserve University School of Graduate Studies / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=case1164135815.

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11

Kan, Chengzhang <1997&gt. "Term structure models and Bayesian estimation." Master's Degree Thesis, Università Ca' Foscari Venezia, 2022. http://hdl.handle.net/10579/22033.

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This paper shall cover: 1)foundational results of term structure: interest rate, bond, swap, financial products. equity...a review of the literature on term structure models and a review on Bayesian modelling of term structure. 2)recent advances of term structure, especially affine process and term structure of equity. 3)Bayesian estimation of affine term structure model. 4)Bayesian paradigm (pro and con), dealing with the choice of a statistical paradigm and decision theory, the uncertainty problem relating to behaviour and economic facts.

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12

Ferroni, Filippo. "Essay on Bayesian Estimation of DSGE Models." Doctoral thesis, Universitat Pompeu Fabra, 2009. http://hdl.handle.net/10803/7397.

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Esta tesis presenta tres diferentes experimentos de política utilizando estimaciones Bayesianas de modelos DSGE. En la primera parte, se quiere demostrar que una política fiscal contracíclica es un instrumento importante para la estabilidad macroeconómica. Este resultado es robusto a diferentes controles. En la segunda parte, se demuestra las variaciones de las estimaciones de los parámetros estructurales según la descomposición ciclo-tendencia, si en uno o en dos estadios. Resulta que con un procedimiento a dos estadios la volatilidad del PIB es explicada mayormente por shocks nominales, mien

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13

Jain, Achin. "Software defect content estimation: A Bayesian approach." Thesis, University of Ottawa (Canada), 2005. http://hdl.handle.net/10393/26932.

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Software inspection is a method to detect errors in software artefacts early in the development cycle. At the end of the inspection process the inspectors need to make a decision whether the inspected artefact is of sufficient quality or not. Several methods have been proposed to assist in making this decision like capture recapture methods and Bayesian approach. In this study these methods have been analyzed and compared and a new Bayesian approach for software inspection is proposed. All of the estimation models rely on an underlying assumption that the inspectors are independent. However, t

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14

Armstrong, Helen School of Mathematics UNSW. "Bayesian estimation of decomposable Gaussian graphical models." Awarded by:University of New South Wales. School of Mathematics, 2005. http://handle.unsw.edu.au/1959.4/24295.

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This thesis explains to statisticians what graphical models are and how to use them for statistical inference; in particular, how to use decomposable graphical models for efficient inference in covariance selection and multivariate regression problems. The first aim of the thesis is to show that decomposable graphical models are worth using within a Bayesian framework. The second aim is to make the techniques of graphical models fully accessible to statisticians. To achieve these aims the thesis makes a number of statistical contributions. First, it proposes a new prior for decomposable graphs

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15

Werthmüller, Dieter. "Bayesian estimation of resistivities from seismic velocities." Thesis, University of Edinburgh, 2014. http://hdl.handle.net/1842/8932.

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I address the problem of finding a background model for the estimation of resistivities in the earth from controlled-source electromagnetic (CSEM) data by using seismic data and well logs as constraints. Estimation of resistivities is normally done by trial-and-error, in a process called “inversion”, by finding a model of the earth whose responses match the data to within an acceptable error; what comes out of the inversion is what is put into the model by the geophysicist: it does not come out of the data directly. The premise underlying this thesis is that an earth model can be found that sa

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16

Wang, Ya Li. "Interactions between gaussian processes and bayesian estimation." Doctoral thesis, Université Laval, 2014. http://hdl.handle.net/20.500.11794/25377.

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L’apprentissage (machine) de modèle et l’estimation d’état sont cruciaux pour interpréter les phénomènes sous-jacents à de nombreuses applications du monde réel. Toutefois, il est souvent difficile d’apprendre le modèle d’un système et de capturer les états latents, efficacement et avec précision, en raison du fait que la connaissance du monde est généralement incertaine. Au cours des dernières années, les approches d’estimation et de modélisation bayésiennes ont été extensivement étudiées afin que l’incertain soit réduit élégamment et de manière flexible. Dans la pratique cependant, différent

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17

Kim, Jae-yoon. "Essays on DSGE Models and Bayesian Estimation." Diss., Virginia Tech, 2018. http://hdl.handle.net/10919/83515.

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This thesis explores the theory and practice of sovereignty. I begin with a conceptual analysis of sovereignty, examining its theological roots in contrast with its later influence in contestations over political authority. Theological debates surrounding God’s sovereignty dealt not with the question of legitimacy, which would become important for political sovereignty, but instead with the limits of his ability. Read as an ontological capacity, sovereignty is coterminous with an existent’s activity in the world. As lived, this capacity is regularly limited by the ways in which space is produc

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18

Cao, Di. "BAYESIAN ADAPTIVE ESTIMATION OF HIGH DIMENSIONAL VECTORS." The Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1408874038.

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19

White, Staci A. "Quantifying Model Error in Bayesian Parameter Estimation." The Ohio State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=osu1433771825.

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20

Callahan, Margaret D. "Bayesian Parameter Estimation and Inference Across Scales." Case Western Reserve University School of Graduate Studies / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=case1459523006.

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21

Malsiner-Walli, Gertraud, Sylvia Frühwirth-Schnatter, and Bettina Grün. "Identifying mixtures of mixtures using Bayesian estimation." Taylor & Francis, 2017. http://dx.doi.org/10.1080/10618600.2016.1200472.

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The use of a finite mixture of normal distributions in model-based clustering allows to capture non-Gaussian data clusters. However, identifying the clusters from the normal components is challenging and in general either achieved by imposing constraints on the model or by using post-processing procedures. Within the Bayesian framework we propose a different approach based on sparse finite mixtures to achieve identifiability. We specify a hierarchical prior where the hyperparameters are carefully selected such that they are reflective of the cluster structure aimed at. In addition, this

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22

Chan, Kwokleung. "Bayesian learning in classification and density estimation /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC IP addresses, 2002. http://wwwlib.umi.com/cr/ucsd/fullcit?p3061619.

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23

Oleson, Jacob J. "Bayesian spatial models for small area estimation /." free to MU campus, to others for purchase, 2002. http://wwwlib.umi.com/cr/mo/fullcit?p3052203.

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24

Masoero, Lorenzo. "Genomic variety estimation with Bayesian nonparametric hierarchies." Thesis, Massachusetts Institute of Technology, 2019. https://hdl.handle.net/1721.1/121737.

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Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 75-83).<br>The recent availability of large genomic studies, with tens of thousands of observations, opens up the intriguing possibility to investigate and understand the effect of rare genetic variants in biological human evolution as well as their impact in the developement of rare diseases. To do so, it is imperative to develop a statistical framework to assess what fraction of the overall

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25

Mai, The Tien. "PAC-Bayesian estimation of low-rank matrices." Electronic Thesis or Diss., Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLG001.

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Les deux premi`eres parties de cette th`ese 'etudient respectivement des estimateurs pseudo-bay'esiens dans les probl`emes de compl'etion de matrices, et de tomographie quantique. Dans chaque probl`eme, on propose une loi a priori qui induit des matrices de faible rang. On 'etudie les performances statistiques: dans chacun des deux cas, on prouve des vitesses de convergence pour nos estimateurs. Notre analyse repose essentiellement sur des in'egalit'es PAC-Bay'esiennes. On propose aussi un algorithme MCMC pour impl'ementer notre estimateur. On teste ensuite ses performances sur des donn'ees si

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26

Keim, Michelle. "Bayesian information retrieval /." Thesis, Connect to this title online; UW restricted, 1997. http://hdl.handle.net/1773/8937.

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27

KAWAGUCHI, Nobuo, and Seigo ITO. "Bayesian Based Location Estimation System Using Wireless LAN." IEEE, 2005. http://hdl.handle.net/2237/15455.

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28

Kanemura, Atsunori. "Inversive and synthetical Bayesian methods for image estimation." 京都大学 (Kyoto University), 2009. http://hdl.handle.net/2433/126471.

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29

Konrad, Janusz. "Bayesian estimation of motion fields from image sequences." Thesis, McGill University, 1989. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74225.

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This thesis addresses the ill-posed problem of estimating two-dimensional motion in time-varying images. The approach proposed here uses the theory of stochastic processes at the formulation and solution stages. Independent Gaussian random variables are used to model the relationship between motion fields and images, and vector and binary Markov random fields are used to model motion and motion discontinuity fields, respectively. These models, combined using Bayes rule, result in Gibbsian a posteriori probability distribution from which the Maximum A Posteriori Probability (MAP) and the Minimu

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30

Hadzagic, Melita. "Bayesian approaches to trajectory estimation in maritime surveillance." Thesis, McGill University, 2010. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=94978.

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In maritime surveillance, multi-sensor data differ to a great extent in their temporal resolution. Additionally, due to multi-level security and information management processing, many contact reports arrive hours after observations. This makes the contact report data usually available for batch processing. The dissimilar multi-source information environment results in contact reports with heteroscedastic and correlated errors (i.e. measurement errors characterized by normal probability distributions with non-constant and non-diagonal covariance matrices), while the obtained measuremen

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31

Nasios, Nikolaos. "Bayesian learning for parametric and kernel density estimation." Thesis, University of York, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.428460.

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Al, Hakmani Rahab. "Bayesian Estimation of Mixture IRT Models using NUTS." OpenSIUC, 2018. https://opensiuc.lib.siu.edu/dissertations/1641.

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The No-U-Turn Sampler (NUTS) is a relatively new Markov chain Monte Carlo (MCMC) algorithm that avoids the random walk behavior that common MCMC algorithms such as Gibbs sampling or Metropolis Hastings usually exhibit. Given the fact that NUTS can efficiently explore the entire space of the target distribution, the sampler converges to high-dimensional target distributions more quickly than other MCMC algorithms and is hence less computational expensive. The focus of this study is on applying NUTS to one of the complex IRT models, specifically the two-parameter mixture IRT (Mix2PL) model, and

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33

Sonogashira, Motoharu. "Variational Bayesian Image Restoration with Transformation Parameter Estimation." Kyoto University, 2018. http://hdl.handle.net/2433/232409.

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34

Majidi, Mohammad Hassan. "Bayesian estimation of discrete signals with local dependencies." Thesis, Supélec, 2014. http://www.theses.fr/2014SUPL0014/document.

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L'objectif de cette thèse est d'étudier le problème de la détection de données dans le système de communication sans fil, à la fois pour le cas de l'information d'état de canal parfaite et imparfaite au niveau du récepteur. Comme on le sait, la complexité de MLSE est exponentielle en la mémoire de canal et la cardinalité de l'alphabet symbole est rapidement ingérable, ce qui force à recourir à des approches sousoptimales. Par conséquent, en premier lieu, nous proposons une nouvelle égalisation itérative lorsque le canal est inconnu à l'émetteur et parfaitement connu au niveau du récepteur. Ce

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35

Mazzetta, Chiara. "Bayesian estimation of temporal dynamics in population ecology." Thesis, University of Cambridge, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.613030.

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36

Nicoloutsopoulos, Dimitrios. "Parametric and Bayesian non-parametric estimation of copulas." Thesis, University College London (University of London), 2005. http://discovery.ucl.ac.uk/1445722/.

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This thesis studies parametric and non-parametric methods of cop ula estimation with special focus on the Archimedean class of copu las. The first part proposes an estimation procedure which is indepen dent of the marginal distributions and performs well for one-parame ter or two-parameter families of copulas, where traditional methods give questionable results especially for small sample sizes. In the sec ond part we follow a Bayesian methodology and represent the copula density as a random piecewise constant, function. Under the presence of some data, we set up a probability distribution ove

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37

Wong, Jackie Siaw Tze. "Bayesian estimation and model comparison for mortality forecasting." Thesis, University of Southampton, 2017. https://eprints.soton.ac.uk/415627/.

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The ability to perform mortality forecasting accurately is of considerable interest for a wide variety of applications to avoid adverse costs. The recent decline in mortality poses a major challenge to various institutions in their attempts to forecast mortality within acceptable risk margins. The ultimate aim of our project is to develop a methodology to produce accurate mortality forecasts, with carefully calibrated probabilistic intervals to quantify the uncertainty encountered during the forecasts. Bayesian methodology is mainly implemented throughout the thesis for various benefits, but p

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38

Torrence, Robert Billington. "Bayesian Parameter Estimation on Three Models of Influenza." Thesis, Virginia Tech, 2017. http://hdl.handle.net/10919/77611.

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Mathematical models of viral infections have been informing virology research for years. Estimating parameter values for these models can lead to understanding of biological values. This has been successful in HIV modeling for the estimation of values such as the lifetime of infected CD8 T-Cells. However, estimating these values is notoriously difficult, especially for highly complex models. We use Bayesian inference and Monte Carlo Markov Chain methods to estimate the underlying densities of the parameters (assumed to be continuous random variables) for three models of influenza. We discuss t

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39

Hotti, Alexandra. "Bayesian insurance pricing using informative prior estimation techniques." Thesis, KTH, Skolan för elektroteknik och datavetenskap (EECS), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-286312.

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Large, well-established insurance companies build statistical pricing models based on customer claim data. Due to their long experience and large amounts of data, they can predict their future expected claim losses accurately. In contrast, small newly formed insurance start-ups do not have access to such data. Instead, a start-up’s pricing model’s initial parameters can be set by directly estimating the risk premium tariff’s parameters in a non-statistical manner. However, this approach results in a pricing model that cannot be adjusted based on new claim data through classical frequentist ins

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40

Fan, Hang. "Estimation of Species Tree Using Approximate Bayesian Computation." The Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1281732679.

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41

Roth, Michael. "Advanced Kalman Filtering Approaches to Bayesian State Estimation." Doctoral thesis, Linköpings universitet, Reglerteknik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-134867.

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Bayesian state estimation is a flexible framework to address relevant problems at the heart of existing and upcoming technologies. Application examples are obstacle tracking for driverless cars and indoor navigation using smartphone sensor data. Unfortunately, the mathematical solutions of the underlying theory cannot be translated to computer code in general. Therefore, this thesis discusses algorithms and approximations that are related to the Kalman filter (KF). Four scientific articles and an introduction with the relevant background on Bayesian state estimation theory and algorithms are i

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42

Lee, Suhwon. "Nonparametric bayesian density estimation with intrinsic autoregressive priors /." free to MU campus, to others for purchase, 2003. http://wwwlib.umi.com/cr/mo/fullcit?p3115565.

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43

Gorynin, Ivan. "Bayesian state estimation in partially observable Markov processes." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLL009/document.

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Cette thèse porte sur l'estimation bayésienne d'état dans les séries temporelles modélisées à l'aide des variables latentes hybrides, c'est-à-dire dont la densité admet une composante discrète-finie et une composante continue. Des algorithmes généraux d'estimation des variables d'états dans les modèles de Markov partiellement observés à états hybrides sont proposés et comparés avec les méthodes de Monte-Carlo séquentielles sur un plan théorique et appliqué. Le résultat principal est que ces algorithmes permettent de réduire significativement le coût de calcul par rapport aux méthodes de Monte-

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44

Gorynin, Ivan. "Bayesian state estimation in partially observable Markov processes." Electronic Thesis or Diss., Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLL009.

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Cette thèse porte sur l'estimation bayésienne d'état dans les séries temporelles modélisées à l'aide des variables latentes hybrides, c'est-à-dire dont la densité admet une composante discrète-finie et une composante continue. Des algorithmes généraux d'estimation des variables d'états dans les modèles de Markov partiellement observés à états hybrides sont proposés et comparés avec les méthodes de Monte-Carlo séquentielles sur un plan théorique et appliqué. Le résultat principal est que ces algorithmes permettent de réduire significativement le coût de calcul par rapport aux méthodes de Monte-

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45

Filho, Paulo Cilas Marques. "Análise bayesiana de densidades aleatórias simples." Universidade de São Paulo, 2011. http://www.teses.usp.br/teses/disponiveis/45/45133/tde-25052012-184549/.

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Definimos, a partir de uma partição de um intervalo limitado da reta real formada por subintervalos, uma distribuição a priori sobre uma classe de densidades em relação à medida de Lebesgue construindo uma densidade aleatória cujas realizações são funções simples não negativas que assumem um valor constante em cada subintervalo da partição e possuem integral unitária. Utilizamos tais densidades aleatórias simples na análise bayesiana de um conjunto de observáveis absolutamente contínuos e provamos que a distribuição a priori é fechada sob amostragem. Exploramos as distribuições a priori e a po

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46

Stephens, Matthew. "Bayesian methods for mixtures of normal distributions." Thesis, University of Oxford, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.242056.

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47

Neto, Anselmo Ramalho Pitombeira. "Dynamic bayesian statistical models for the estimation of the origin-destination matrix." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=14698.

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In transportation planning, one of the first steps is to estimate the travel demand. A product of the estimation process is the so-called origin-destination matrix (OD matrix), whose entries correspond to the number of trips between pairs of zones in a geographic region in a reference time period. Traditionally, the OD matrix has been estimated through direct methods, such as home-based surveys, road-side interviews and license plate automatic recognition. These direct methods require large samples to achieve a target statistical error, which may be technically or economically infeasible. Alte

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Lewis, John Robert. "Bayesian Restricted Likelihood Methods." The Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1407505392.

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Merkle, Edgar C. "Bayesian estimation of factor analysis models with incomplete data." Connect to resource, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1126895149.

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Thesis (Ph. D.)--Ohio State University, 2005.<br>Title from first page of PDF file. Document formatted into pages; contains xi, 106 p.; also includes graphics. Includes bibliographical references (p. 103-106). Available online via OhioLINK's ETD Center

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Lee, Deok-Jin. "Nonlinear bayesian filtering with applications to estimation and navigation." Texas A&M University, 2004. http://hdl.handle.net/1969.1/2269.

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In principle, general approaches to optimal nonlinear filtering can be described in a unified way from the recursive Bayesian approach. The central idea to this recur- sive Bayesian estimation is to determine the probability density function of the state vector of the nonlinear systems conditioned on the available measurements. However, the optimal exact solution to this Bayesian filtering problem is intractable since it requires an infinite dimensional process. For practical nonlinear filtering applications approximate solutions are required. Recently efficient and accurate approximate non- l

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