Academic literature on the topic 'Bayesian non-Parametric model'

In this paper, the non-parametric bootstrap and non-parametric Bayesian bootstrap methods are applied for parameter estimation in the binary logistic regression model. A real data study and a simulation study are conducted to compare the Nonparametric bootstrap, Non-parametric Bayesian bootstrap and the maximum likelihood methods. Study results shows that three methods are all effective ways for parameter estimation in the binary logistic regression model. In small sample case, the non-parametric Bayesian bootstrap method performs relatively better than the non-parametric bootstrap and the maximum likelihood method for parameter estimation in the binary logistic regression model.

With the development of technology and the widespread collection of data, high-dimensional data analysis has become a research hotspot in many fields. Traditional parameter methods often face problems such as dimensional disasters in high-dimensional data analysis. Non parametric methods have broad application prospects in high-dimensional data because they do not rely on specific parameter distribution assumptions. The Bayesian rule is more suitable for dealing with noise and outliers in high-dimensional data because it takes uncertainty into account. Therefore, it is of great significance to combine non parametric methods with Bayesian methods for application research in high-dimensional data analysis. In this paper, the nonparametric Bayesian method was applied to the analysis of high-dimensional data, and the Dirichlet process Mixture model was used to cluster high-dimensional data. The regression analysis of high-dimensional data was carried out through the prediction model of nonparametric Bayesian regression. In this paper, the nonparametric Bayesian method based on Bayesian sparse linear model was used for feature selection of high-dimensional data. In order to determine the superiority of nonparametric Bayesian methods in high-dimensional data analysis, this paper conducted experiments on nonparametric Bayesian methods and traditional parametric methods in high-dimensional data analysis from five aspects of cluster analysis, classification analysis, regression analysis, feature selection and anomaly detection, and evaluated them through multiple indicators. This article explored the application of non parametric Bayesian methods in high-dimensional data analysis from these aspects through simulation experiments. The experimental results show that the clustering accuracy of the non parametric Bayesian clustering algorithm was 0.93, and the accuracy of the non parametric Bayesian classification algorithm was between 0.93 and 0.99; the coefficient of determination of nonparametric Bayesian regression algorithm was 0.98; the F1 values of non parametric Bayesian methods in anomaly detection ranged from 0.86 to 0.91, which was superior to traditional methods. Non parametric Bayesian methods have broad application prospects in high-dimensional data analysis, and can be applied in multiple fields such as clustering, classification, regression, etc.

AbstractWe develop quantile functions from regression models in order to derive risk margin and to evaluate capital in non-life insurance applications. By utilizing the entire range of conditional quantile functions, especially higher quantile levels, we detail how quantile regression is capable of providing an accurate estimation of risk margin and an overview of implied capital based on the historical volatility of a general insurers loss portfolio. Two modeling frameworks are considered based around parametric and non-parametric regression models which we develop specifically in this insurance setting. In the parametric framework, quantile functions are derived using several distributions including the flexible generalized beta (GB2) distribution family, asymmetric Laplace (AL) distribution and power-Pareto (PP) distribution. In these parametric model based quantile regressions, we detail two basic formulations. The first involves embedding the quantile regression loss function from the nonparameteric setting into the argument of the kernel of a parametric data likelihood model, this is well known to naturally lead to the AL parametric model case. The second formulation we utilize in the parametric setting adopts an alternative quantile regression formulation in which we assume a structural expression for the regression trend and volatility functions which act to modify a base quantile function in order to produce the conditional data quantile function. This second approach allows a range of flexible parametric models to be considered with different tail behaviors. We demonstrate how to perform estimation of the resulting parametric models under a Bayesian regression framework. To achieve this, we design Markov chain Monte Carlo (MCMC) sampling strategies for the resulting Bayesian posterior quantile regression models. In the non-parametric framework, we construct quantile functions by minimizing an asymmetrically weighted loss function and estimate the parameters under the AL proxy distribution to resemble the minimization process. This quantile regression model is contrasted to the parametric AL mean regression model and both are expressed as a scale mixture of uniform distributions to facilitate efficient implementation. The models are extended to adopt dynamic mean, variance and skewness and applied to analyze two real loss reserve data sets to perform inference and discuss interesting features of quantile regression for risk margin calculations.

The classical Gumbel probability distribution is modified in order to study the failure times of a given system. Bayesian estimates of the reliability function under five different parametric priors and the square error loss are studied. The Bayesian reliability estimate under the non-parametric kernel density prior is compared with those under the parametric priors and numerical computations are given to study their effectiveness.

The primary objective of this study is to employ semi-parametric regression techniques in the balanced longitudinal data model. Where the parametric regression models are plagued by the problem of strict constraints, while non-parametric regression models, despite their flexibility, suffer from the problem of the curse of dimensionality. Consequently, semi-parametric regression presents a suitable solution to address the problems in parametric and non-parametric regression methods. The advantage of this model is that it contains all the positive properties included in the previous two models such as containing strict restrictions in its parametric component, complete flexibility in its non-parametric component, and clarity of the interaction between its parametric and non-parametric components. According to the above, two methods were used to estimate a semi-parametric balanced longitudinal data model. The first is the Bayesian estimating method; the second is the Speckman method, which estimated the unknown nonparametric smoothing function by employing the kernel smoothing Nadaraya & Watson method. The Aim was to make a comparison between the Bayesian estimation method and the classical estimation method. Based on simulation experiments conducted on three different sample sizes (50, 100, and 200), it was concluded that the Bayes method is best at the variance levels (1,5). In contrast, the Profile least square method is best at the variance level (10).

Cette thèse porte sur l’apprentissage statistique et l’analyse de données multi-dimensionnelles. Elle se focalise particulièrement sur l’apprentissage non supervisé de modèles génératifs pour la classification automatique. Nous étudions les modèles de mélanges Gaussians, aussi bien dans le contexte d’estimation par maximum de vraisemblance via l’algorithme EM, que dans le contexte Bayésien d’estimation par Maximum A Posteriori via des techniques d’échantillonnage par Monte Carlo. Nous considérons principalement les modèles de mélange parcimonieux qui reposent sur une décomposition spectrale de la matrice de covariance et qui offre un cadre flexible notamment pour les problèmes de classification en grande dimension. Ensuite, nous investiguons les mélanges Bayésiens non-paramétriques qui se basent sur des processus généraux flexibles comme le processus de Dirichlet et le Processus du Restaurant Chinois. Cette formulation non-paramétrique des modèles est pertinente aussi bien pour l’apprentissage du modèle, que pour la question difficile du choix de modèle. Nous proposons de nouveaux modèles de mélanges Bayésiens non-paramétriques parcimonieux et dérivons une technique d’échantillonnage par Monte Carlo dans laquelle le modèle de mélange et son nombre de composantes sont appris simultanément à partir des données. La sélection de la structure du modèle est effectuée en utilisant le facteur de Bayes. Ces modèles, par leur formulation non-paramétrique et parcimonieuse, sont utiles pour les problèmes d’analyse de masses de données lorsque le nombre de classe est indéterminé et augmente avec les données, et lorsque la dimension est grande. Les modèles proposés validés sur des données simulées et des jeux de données réelles standard. Ensuite, ils sont appliqués sur un problème réel difficile de structuration automatique de données bioacoustiques complexes issues de signaux de chant de baleine. Enfin, nous ouvrons des perspectives Markoviennes via les processus de Dirichlet hiérarchiques pour les modèles Markov cachés
This thesis focuses on statistical learning and multi-dimensional data analysis. It particularly focuses on unsupervised learning of generative models for model-based clustering. We study the Gaussians mixture models, in the context of maximum likelihood estimation via the EM algorithm, as well as in the Bayesian estimation context by maximum a posteriori via Markov Chain Monte Carlo (MCMC) sampling techniques. We mainly consider the parsimonious mixture models which are based on a spectral decomposition of the covariance matrix and provide a flexible framework particularly for the analysis of high-dimensional data. Then, we investigate non-parametric Bayesian mixtures which are based on general flexible processes such as the Dirichlet process and the Chinese Restaurant Process. This non-parametric model formulation is relevant for both learning the model, as well for dealing with the issue of model selection. We propose new Bayesian non-parametric parsimonious mixtures and derive a MCMC sampling technique where the mixture model and the number of mixture components are simultaneously learned from the data. The selection of the model structure is performed by using Bayes Factors. These models, by their non-parametric and sparse formulation, are useful for the analysis of large data sets when the number of classes is undetermined and increases with the data, and when the dimension is high. The models are validated on simulated data and standard real data sets. Then, they are applied to a real difficult problem of automatic structuring of complex bioacoustic data issued from whale song signals. Finally, we open Markovian perspectives via hierarchical Dirichlet processes hidden Markov models

Background: Ethiopia is a country with 9 ethnically-based administrative regions and 2 city administrations, often cited, among other things, with high fertility rates and rapid population growth rate. Despite the country’s effort in their reduction, they still remain high, especially at regional-level. To this end, the study of fertility in Ethiopia, particularly on its regions, where fertility variation and its repercussion are at boiling point, is paramount important. An easy way of finding different characteristics of a fertility distribution is to build a suitable model of fertility pattern through different mathematical curves. ASFR is worthwhile in this regard. In general, the age-specific fertility pattern is said to have a typical shape common to all human populations through years though many countries some from Africa has already started showing a deviation from this classical bell shaped curve. Some of existing models are therefore inadequate to describe patterns of many of the African countries including Ethiopia. In order to describe this shape (ASF curve), a number of parametric and non-parametric functions have been exploited in the developed world though fitting these models to curves of Africa in general and that of Ethiopian in particular data has not been undertaken yet. To accurately model fertility patterns in Ethiopia, a new mathematical model that is both easily used, and provides good fit for the data is required. Objective: The principal goals of this thesis are therefore fourfold: (1). to examine the pattern of ASFRs at country and regional level,in Ethiopia; (2). to propose a model that best captures various shapes of ASFRs at both country and regional level, and then compare the performance of the model with some existing ones; (3). to fit the proposed model using Hierarchical Bayesian techniques and show that this method is flexible enough for local estimates vis-´a-vis traditional formula, where the estimates might be very imprecise, due to low sample size; and (4). to compare the resulting estimates obtained with the non-hierarchical procedures, such as Bayesian and Maximum likelihood counterparts. Methodology: In this study, we proposed a four parametric parametric model, Skew Normal model, to fit the fertility schedules, and showed that it is flexible enough in capturing fertility patterns shown at country level and most regions of Ethiopia. In order to determine the performance of this proposed model, we conducted a preliminary analysis along with ten other commonly used parametric and non-parametric models in demographic literature, namely: Quadratic Spline function, Cubic Splines, Coale-Trussell function, Beta, Gamma, Hadwiger distribution, Polynomial models, the Adjusted Error Model, Gompertz curve, Skew Normal, and Peristera & Kostaki Model. The criterion followed in fitting these models was Nonlinear Regression with nonlinear least squares (nls) estimation. We used Akaike Information Criterion (AIC) as model selecction criterion. For many demographers, however, estimating regional-specific ASFR model and the associated uncertainty introduced due those factors can be difficult, especially in a situation where we have extremely varying sample size among different regions. Recently, it has been proposed that Hierarchical procedures might provide more reliable parameter estimates than Non-Hierarchical procedures, such as complete pooling and independence to make local/regional-level analyses. In this study, a Hierarchical Bayesian procedure was, therefore, formulated to explore the posterior distribution of model parameters (for generation of region-specific ASFR point estimates and uncertainty bound). Besides, other non-hierarchical approaches, namely Bayesian and the maximum likelihood methods, were also instrumented to estimate parameters and compare the result obtained using these approaches with Hierarchical Bayesian counterparts. Gibbs sampling along with MetropolisHastings argorithm in R (Development Core Team, 2005) was applied to draw the posterior samples for each parameter. Data augmentation method was also implemented to ease the sampling process. Sensitivity analysis, convergence diagnosis and model checking were also thoroughly conducted to ensure how robust our results are. In all cases, non-informative prior distributions for all regional vectors (parameters) were used in order to real the lack of knowledge about these random variables. Result: The results obtained from this preliminary analysis testified that the values of the Akaike Information criterion(AIC) for the proposed model, Skew Normal (SN), is lowest: in the capital, Addis Ababa, Dire Dawa, Harari, Affar, Gambela, Benshangul-Gumuz, and country level data as well. On the contrary, its value was also higher some of the models and lower the rest on the remain regions, namely: Tigray, Oromiya, Amhara, Somali and SNNP. This tells us that the proposed model was able to capturing the pattern of fertility at the empirical fertility data of Ethiopia and its regions better than the other existing models considered in 6 of the 11 regions. The result from the HBA indicates that most of the posterior means were much closer to the true fixed fertility values. They were also more precise and have lower uncertainty with narrower credible interval vis-´a-vis the other approaches, ML and Bayesian estimate analogues. Conclusion: From the preliminary analysis, it can be concluded that the proposed model was better to capture ASFR pattern at national level and its regions than the other existing common models considered. Following this result, we conducted inference and prediction on the model parameters using these three approaches: HBA, BA and ML methods. The overall result suggested several points. One such is that HBA was the best approach to implement for such a data as it gave more consistent, precise (the low uncertainty) than the other approaches. Generally, both ML method and Bayesian method can be used to analyze our model, but they can be applicable to different conditions. ML method can be applied when precise values of model parameters have been known, large sample size can be obtained in the test; and similarly, Bayesian method can be applied when uncertainties on the model parameters exist, prior knowledge on the model parameters are available, and few data is available in the study.
Background: L’Etiopia è una nazione divisa in 9 regioni amministrative (definite su base etnica) e due città. Si tratta di una nazione citata spesso come esempio di alta fecondità e rapida crescita demografica. Nonostante gli sforzi del governo, fecondità e cresita della popolazione rimangono elevati, specialmente a livello regionale. Pertanto, lo studio della fecondità in Etiopia e nelle sue regioni – caraterizzate da un’alta variabilità – è di vitale importanza. Un modo semplice di rilevare le diverse caratteristiche della distribuzione della feconditàè quello di costruire in modello adatto, specificando diverse funzioni matematiche. In questo senso, vale la pena concentrarsi sui tassi specifici di fecondità, i quali mostrano una precisa forma comune a tutte le popolazioni. Tuttavia, molti paesi mostrano una “simmetrizzazione” che molti modelli non riescono a cogliere adeguatamente. Pertanto, per cogliere questa la forma dei tassi specifici, sono stati utilizzati alcuni modelli parametrici ma l’uso di tali modelliè ancora molto limitato in Africa ed in Etiopia in particolare. Obiettivo: In questo lavoro si utilizza un nuovo modello per modellare la fecondità in Etiopia con quattro obiettivi specifici: (1). esaminare la forma dei tassi specifici per età dell’Etiopia a livello nazionale e regionale; (2). proporre un modello che colga al meglio le varie forme dei tassi specifici sia a livello nazionale che regionale. La performance del modello proposto verrà confrontata con quella di altri modelli esistenti; (3). adattare la funzione di fecondità proposta attraverso un modello gerarchico Bayesiano e mostrare che tale modelloè sufficientemente flessibile per stimare la fecondità delle singole regioni – dove le stime possono essere imprecise a causa di una bassa numerosità campionaria; (4). confrontare le stime ottenute con quelle fornite da metodi non gerarchici (massima verosimiglianza o Bayesiana semplice) Metodologia: In questo studio, proponiamo un modello a 4 parametri, la Normale Asimmetrica, per modellare i tassi specifici di fecondità. Si mostra che questo modello è sufficientemente flessibile per cogliere adeguatamente le forme dei tassi specifici a livello sia nazionale che regionale. Per valutare la performance del modello, si è condotta un’analisi preliminare confrontandolo con altri dieci modelli parametrici e non parametrici usati nella letteratura demografica: la funzione splie quadratica, la Cubic-Spline, i modelli di Coale e Trussel, Beta, Gamma, Hadwiger, polinomiale, Gompertz, Peristera-Kostaki e l’Adjustment Error Model. I modelli sono stati stimati usando i minimi quadrati non lineari (nls) e il Criterio d’Informazione di Akaike viene usato per determinarne la performance. Tuttavia, la stima per le singole regioni pu‘o risultare difficile in situazioni dove abbiamo un’alta variabilità della numerosità campionaria. Si propone, quindi di usare procedure gerarchiche che permettono di ottenere stime più affidabili rispetto ai modelli non gerarchici (“pooling” completo o “unpooling”) per l’analisi a livello regionale. In questo studia si formula un modello Bayesiano gerarchico ottenendo la distribuzione a posteriori dei parametri per i tassi di fecnodità specifici a livello regionale e relativa stima dell’incertezza. Altri metodi non gerarchici (Bayesiano semplice e massima verosimiglianza) vengono anch’essi usati per confronto. Gli algoritmi Gibbs Sampling e Metropolis-Hastings vengono usati per campionare dalla distribuzione a posteriori di ogni parametro. Anche il metodo del “Data Augmentation” viene utilizzato per ottenere le stime. La robustezza dei risultati viene controllata attraverso un’analisi di sensibilità e l’opportuna diagnostica della convergenza degli algoritmi viene riportata nel testo. In tutti i casi, si sono usate distribuzioni a priori non-informative. Risultati: I risutlati ottenuti dall’analisi preliminare mostrano che il modello Skew Normal ha il pi`u basso AIC nelle regioni Addis Ababa, Dire Dawa, Harari, Affar, Gambela, Benshangul-Gumuz e anche per le stime nazionali. Nelle altre regioni (Tigray, Oromiya, Amhara, Somali e SNNP) il modello Skew Normal non risulta il milgiore, ma comunque mostra un buon adattamento ai dati. Dunque, il modello Skew Normal risulta il migliore in 6 regioni su 11 e sui tassi specifici di tutto il paese. Conclusioni: Dunque, il modello Skew Normal risulta globalmente il migliore. Da questo risultato iniziale, siè partiti per costruire i modelli Gerachico Bayesiano, Bayesiano semplice e di massima verosimiglianza. Il risultato del confronto tra questi tre approcci è che il modello gerarchico fornisce stime più preciso rispetto agli altri.

Structured output prediction is a machine learning tasks in which an input object is not just assigned a single class, as in classification, but multiple, interdependent labels. This means that the presence or value of a given label affects the other labels, for instance in text labelling problems, where output labels are applied to each word, and their interdependencies must be modelled. Non-parametric Bayesian (NPB) techniques are probabilistic modelling techniques which have the interesting property of allowing model capacity to grow, in a controllable way, with data complexity, while maintaining the advantages of Bayesian modelling. In this thesis, we develop NPB algorithms to solve structured output problems. We first study a map-reduce implementation of a stochastic inference method designed for the infinite hidden Markov model, applied to a computational linguistics task, part-of-speech tagging. We show that mainstream map-reduce frameworks do not easily support highly iterative algorithms. The main contribution of this thesis consists in a conceptually novel discriminative model, GPstruct. It is motivated by labelling tasks, and combines attractive properties of conditional random fields (CRF), structured support vector machines, and Gaussian process (GP) classifiers. In probabilistic terms, GPstruct combines a CRF likelihood with a GP prior on factors; it can also be described as a Bayesian kernelized CRF. To train this model, we develop a Markov chain Monte Carlo algorithm based on elliptical slice sampling and investigate its properties. We then validate it on real data experiments, and explore two topologies: sequence output with text labelling tasks, and grid output with semantic segmentation of images. The latter case poses scalability issues, which are addressed using likelihood approximations and an ensemble method which allows distributed inference and prediction. The experimental validation demonstrates: (a) the model is flexible and its constituent parts are modular and easy to engineer; (b) predictive performance and, most crucially, the probabilistic calibration of predictions are better than or equal to that of competitor models, and (c) model hyperparameters can be learnt from data.

We provide a Frequentist-Bayesian hybrid test statistic in this dissertation for two testing problems. The first one is to design a test for the significant differences between non-parametric functions and the second one is to design a test allowing any departure of predictors of high dimensional X from constant. The implementation is also given in construction of the proposal test statistics for both problems. For the first testing problem, we consider the statistical difference among massive outcomes or signals to be of interest in many diverse fields including neurophysiology, imaging, engineering, and other related fields. However, such data often have nonlinear system, including to row/column patterns, having non-normal distribution, and other hard-to-identifying internal relationship, which lead to difficulties in testing the significance in difference between them for both unknown relationship and high-dimensionality. In this dissertation, we propose an Adaptive Bayes Sum Test capable of testing the significance between two nonlinear system basing on universal non-parametric mathematical decomposition/smoothing components. Our approach is developed from adapting the Bayes sum test statistic by Hart (2009). Any internal pattern is treated through Fourier transformation. Resampling techniques are applied to construct the empirical distribution of test statistic to reduce the effect of non-normal distribution. A simulation study suggests our approach performs better than the alternative method, the Adaptive Neyman Test by Fan and Lin (1998). The usefulness of our approach is demonstrated with an application in the identification of electronic chips as well as an application to test the change of pattern of precipitations. For the second testing problem, currently numerous statistical methods have been developed for analyzing high-dimensional data. These methods mainly focus on variable selection approach, but are limited for purpose of testing with high-dimensional data, and often are required to have explicit derivative likelihood functions. In this dissertation, we propose ``Hybrid Omnibus Test'' for high-dimensional data testing purpose with much less requirements. Our Hybrid Omnibus Test is developed under semi-parametric framework where likelihood function is no longer necessary. Our Hybrid Omnibus Test is a version of Freqentist-Bayesian hybrid score-type test for a functional generalized partial linear single index model, which has link being functional of predictors through a generalized partially linear single index. We propose an efficient score based on estimating equation to the mathematical difficulty in likelihood derivation and construct our Hybrid Omnibus Test. We compare our approach with a empirical likelihood ratio test and Bayesian inference based on Bayes factor using simulation study in terms of false positive rate and true positive rate. Our simulation results suggest that our approach outperforms in terms of false positive rate, true positive rate, and computation cost in high-dimensional case and low-dimensional case. The advantage of our approach is also demonstrated by published biological results with application to a genetic pathway data of type II diabetes.
Ph. D.

Dans cette thèse nous avons traité le problème de l'estimation non paramétrique des lois de probabilités. Dans un premier temps, nous avons supposé que la densité inconnue f a été approchée par un mélange de base B-spline quadratique. Puis, nous avons proposé un nouvel estimateur de la densité inconnue f basé sur les fonctions B-splines quadratiques, avec deux méthodes d'estimation. La première est base sur la méthode du maximum de vraisemblance et la deuxième est basée sur la méthode d'estimation Bayésienne MAP. Ensuite, nous avons généralisé notre étude d'estimation dans le cadre du mélange et nous avons proposé un nouvel estimateur du mélange de lois inconnues basé sur les deux méthodes d'estimation adaptées. Dans un deuxième temps, nous avons traité le problème de la segmentation statistique semi supervisée des images en se basant sur le modèle de Markov caché et les fonctions B-splines. Nous avons montré l'apport de l'hybridation du modèle de Markov caché et les fonctions B-splines en segmentation statistique bayésienne semi supervisée des images. Dans un troisième temps, nous avons présenté une approche de fusion basée sur la méthode de maximum de vraisemblance, à travers l'estimation non paramétrique des probabilités, pour chaque pixel de l'image. Nous avons ensuite appliqué cette approche sur des images multi-spectrales et multi-temporelles segmentées par notre algorithme non paramétrique et non supervisé
In this thesis we are treated the problem of nonparametric estimation probability distributions. At first, we assumed that the unknown density f was approximated by a basic mixture quadratic B-spline. Then, we proposed a new estimate of the unknown density function f based on quadratic B-splines, with two methods estimation. The first is based on the maximum likelihood method and the second is based on the Bayesian MAP estimation method. Then we have generalized our estimation study as part of the mixture and we have proposed a new estimator mixture of unknown distributions based on the adapted estimation of two methods. In a second time, we treated the problem of semi supervised statistical segmentation of images based on the hidden Markov model and the B-sline functions. We have shown the contribution of hybridization of the hidden Markov model and B-spline functions in unsupervised Bayesian statistical image segmentation. Thirdly, we presented a fusion approach based on the maximum likelihood method, through the nonparametric estimation of probabilities, for each pixel of the image. We then applied this approach to multi-spectral and multi-temporal images segmented by our nonparametric and unsupervised algorithm

This chapter examines the technical issues in modelling health state valuation data. Most measures of health define too many states to directly value all of them (e.g. SF-6D defines 18,000 health states). The solution has been to value a subset and by using modelling to predict the values of all states. This chapter reviews two approaches to modelling: one using multiattribute utility theory to determine health values given an assumed functional form; and the other is using statistical modelling of SF-6D preference data that are skewed, bimodal, and clustered by respondents. This chapter examines the selection of health states for valuation, data preparation, model specification, and techniques for modelling the data starting with ordinary least squares (OLS) and moving on to more complex techniques including Bayesian non-parametric and semi-parametric approaches, and a hybrid approach that combines cardinal preference data with the results of paired data from a discrete choice experiment.

Abstract A Bayesian specification test based on the encompassing principle for the case of partial observability is proposed. A structural parametric null model is compared against a nonparametric alternative model at the level of latent variables. A same observability process is introduced in both models. The comparison is made between the posterior measures of the non-Euclidean parameter (of the alternative model) in the extended and in the alternative models. The general development is illustrated with an example where a linear combination of a latent vector is only observed.

Abstract In a dose-ranging study the goal is to establish an initial dose of a new drug for patients from the relevant target population. This may be achieved by observing individual responses over a range of dose levels from a randomly selected sample from the patient population. When data from such studies is analysed it is important to acknowledge between patient variability. Typically this is done using a hierarchical model with a (parametric) non-linear first stage model. Here we assume that a parametric dose-response curve is unknown but that a monotonic dose-response relation exists. We propose two semiparametric forms and demonstrate their use with simulated data.

In this last chapter of the second section, the authors present probabilistic solutions to mobile robot localization that bring together the recursive filters introduced in chapter 4 and all the components and models already discussed in the preceding chapters. It presents the general, Bayesian framework for a probabilistic solution to localization and mapping. The problem is formally described as a graphical model (in particular a dynamic Bayesian network), and the characteristics that can be exploited to approach it efficiently are elaborated. Among parametric Bayesian estimators, the family of the Kalman filters is introduced with examples and practical applications. Then, the more modern non-parametric filters, mainly particle filters, are explained. Due to the diversity of filters available for localization, comparative tables are included.

Model validation and calibration chapter demonstrate key statistical tests that are useful to measure predictive power of risk models. It mainly assesses the critical steps, data input quality, and discriminatory power of the models in predicting default or loss. Model validation has been a key task for risk-focused management for internal management of risk across various business lines. Reliable rating systems require efficient validation strategies. This chapter explains power curve-fitting techniques to assess discriminatory power of predictive models, method for checking model errors, and estimation of model accuracy in great detail. The separation power check through information value and KS test and their utility in scorecard development has been elaborated. Steps in Hosmer–Lemeshow goodness-of-fit test pertaining to logistic model and other non-parametric validation checks like Akaike information criteria, Bayesian information criterion, Kendal’s tau are described in this chapter. An independent and objective validation of the predictive power and efficacy of valuation and risk models through statistical tests is an integral part of a robust risk management system.

Abstract Point processes are a class of models where the notion of a variable dimension is inherent. The main part of this discussion is concerned with the application of marked point processes as prior models in non-parametric Bayesian function estimation, reformulating and revising earlier joint work with Elja Arjas and listing some other related work (Section 2).

Abstract This paper proposes and discusses several Bayesian attempts at nonparametric and semiparametric density estimation. The main categories of these ideas are as follows: (1) Build a nonparametric prior around a given parametric model. We look at cases where the nonparametric part of the construction is a Dirichlet process or relatives thereof. (2) Express the density as an additive expansion of orthogonal basis functions, and place priors on the coefficients. Here attention is given to a certain robust Hermite expansion around the normal distribution. Multiplicative expansions are also considered. (3) Express the unknown density as locally being of a certain parametric form, then construct suitable local likelihood functions to express information content, and place local priors on the local parameters.

This paper studies large sample classical and Bayesian inference in a prototypical linear DSGE model and demonstrates that inference on the structural parameters based on a Gaussian likelihood is unaffected by departures from Gaussianity of the structural shocks. This surprising result is due to a cancellation in the asymptotic variance resulting into a generalized information equality for the block corresponding to the structural parameters. The underlying reason for the cancellation is the certainty equivalence property of the linear rational expectation model. The main implication of this result is that classical and Bayesian Gaussian inference achieve a semi-parametric efficiency bound and there is no need for a “sandwich-form” correction of the asymptotic variance of the structural parameters. Consequently, MLE-based confidence intervals and Bayesian credible sets of the deep parameters based on a Gaussian likelihood have correct asymptotic coverage even when the structural shocks are non-Gaussian. On the other hand, inference on the reduced-form parameters characterizing the volatility of the shocks is invalid whenever the structural shocks have a non-Gaussian density and the paper proposes a simple Metropolis-within-Gibbs algorithm that achieves correct large sample inference for the volatility parameters.