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1.
Bayes’ rule specifies how to obtain a posterior from a class of hypotheses endowed with a prior and the observed data. There are three fundamental ways to use this posterior for predicting the future: marginalization (integration over the hypotheses w.r.t. the posterior), MAP (taking the a posteriori most probable hypothesis), and stochastic model selection (selecting a hypothesis at random according to the posterior distribution). If the hypothesis class is countable, and contains the data generating distribution (this is termed the “realizable case”), strong consistency theorems are known for the former two methods in a sequential prediction framework, asserting almost sure convergence of the predictions to the truth as well as loss bounds. We prove corresponding results for stochastic model selection, for both discrete and continuous observation spaces. As a main technical tool, we will use the concept of a potential: this quantity, which is always positive, measures the total possible amount of future prediction errors. Precisely, in each time step, the expected potential decrease upper bounds the expected error. We introduce the entropy potential of a hypothesis class as its worst-case entropy, with regard to the true distribution. Our results are proven within a general stochastic online prediction framework, that comprises both online classification and prediction of non-i.i.d. sequences.  相似文献   

2.
A statistical Bayesian framework is used to solve the inverse problem and develop the posterior distributions of parameters for a density-driven groundwater flow model. This Bayesian approach is implemented using a Markov Chain Monte Carlo (MCMC) sampling method. Three sets of data pertaining to the location of the freshwater–seawater transition zone exist for the site, including chemistry data, hydraulic head data and newly collected magnetotelluric (MT) data. A sequential conditioning approach is implemented where the chemistry data and MT-converted salinity are combined as a single data set and are used to first condition the parameter distributions. The head data are subsequently used as a second conditioning data set where the posterior distribution developed by the first conditioning is used as a prior for this second conditioning. Results of this analysis indicate that conditioning on the available data sets yields dramatic reduction of uncertainty compared to unconditioned simulations, especially for the recharge–conductivity ratio. This ratio controls the location of the transition zone, and the conditioning results in a smaller range of variability compared to the distribution used in previous modelling of the site. Using the conditioned distributions to solve the density-driven flow problem in a stochastic framework (i.e., model parameters are randomly sampled from the posterior distributions) results in a range of output flow fields that is much narrower than the previous model. The ensemble mean of these solutions and the uncertainty bounds expressed by the mean ± one standard deviation lie within the uncertainty bounds of the original model. For the case study shown here, the effect of conditioning data is dominant over the effect of prior information.  相似文献   

3.
Environmental modeling often requires combining prior knowledge with information obtained from data. The robust Bayesian approach makes it possible to consider ambiguity in this prior knowledge. Describing such ambiguity using sets of probability distributions defined by the Density Ratio Class has important conceptual advantages over alternative robust formulations. Earlier studies showed that the Density Ratio Class is invariant under Bayesian inference and marginalization. We prove that (i) the Density Ratio Class is also invariant under propagation through deterministic models, whereas (ii) predictions of a stochastic model with parameters defined by a Density Ratio Class are embedded in a Density Ratio Class. These invariance properties make it possible to describe sequential learning and prediction under a unified framework. We developed numerical algorithms to minimize the additional computational burden relative to the use of single priors. Practical feasibility of these methods is demonstrated by their application to a simple ecological model.  相似文献   

4.
We examine a general Bayesian framework for constructing on-line prediction algorithms in the experts setting. These algorithms predict the bits of an unknown Boolean sequence using the advice of a finite set of experts. In this framework we use probabilistic assumptions on the unknown sequence to motivate prediction strategies. However, the relative bounds that we prove on the number of prediction mistakes made by these strategies hold for any sequence. The Bayesian framework provides a unified derivation and analysis of previously known prediction strategies, such as the Weighted Majority and Binomial Weighting algorithms. Furthermore, it provides a principled way of automatically adapting the parameters of Weighted Majority to the sequence, in contrast to previous ad hoc doubling techniques. Finally, we discuss the generalization of our methods to algorithms making randomized predictions. Received February 5, 1997; revised July 17, 1997.  相似文献   

5.
We consider the problem of smoothing a sequence of noisy observations using a fixed class of models. Via a deterministic analysis, we obtain necessary and sufficient conditions on the noise sequence and model class that ensure that a class of natural estimators gives near-optimal smoothing. In the case of i.i.d. random noise, we show that the accuracy of these estimators depends on a measure of complexity of the model class involving covering numbers. Our formulation and results are quite general and are related to a number of problems in learning, prediction, and estimation. As a special case, we consider an application to output smoothing for certain classes of linear and nonlinear systems. The performance of output smoothing is given in terms of natural complexity parameters of the model class, such as bounds on the order of linear systems, the l1-norm of the impulse response of stable linear systems, or the memory of a Lipschitz nonlinear system satisfying a fading memory condition.  相似文献   

6.
We present a new approach to shape-based segmentation and tracking of deformable anatomical structures in medical images, and validate this approach by detecting and tracking the endocardial contour in an echocardiographic image sequence. To this end, some global prior shape knowledge of the endocardial boundary is captured by a prototype template with a set of predefined global and local deformations to take into account its inherent natural variability over time. In this deformable model-based Bayesian segmentation, the data likelihood model relies on an accurate statistical modelling of the grey level distribution of each class present in the ultrasound image. The parameters of this distribution mixture are given by a preliminary iterative estimation step. This estimation scheme relies on a Markov Random Field prior model, and takes into account the imaging process as well as the distribution shape of each class present in the image. Then the detection and the tracking problem is stated in a Bayesian framework, where it ends up as a cost function minimisation problem for each image of the sequence. In our application, this energy optimisation problem is efficiently solved by a genetic algorithm combined with a steepest ascent procedure. This technique has been successfully applied on synthetic images, and on a real echocardiographic image sequence.  相似文献   

7.
8.
We consider the problem of smoothing a sequence of noisy observations using a fixed class of models. Via a deterministic analysis, we obtain necessary and sufficient conditions on the noise sequence and model class that ensure that a class of natural estimators gives near-optimal smoothing. In the case of i.i.d. random noise, we show that the accuracy of these estimators depends on a measure of complexity of the model class involving covering numbers. Our formulation and results are quite general and are related to a number of problems in learning, prediction, and estimation. As a special case, we consider an application to output smoothing for certain classes of linear and nonlinear systems. The performance of output smoothing is given in terms of natural complexity parameters of the model class, such as bounds on the order of linear systems, the -norm of the impulse response of stable linear systems, or the memory of a Lipschitz nonlinear system satisfying a fading memory condition.  相似文献   

9.
A Bayesian model of learning to learn by sampling from multiple tasks is presented. The multiple tasks are themselves generated by sampling from a distribution over an environment of related tasks. Such an environment is shown to be naturally modelled within a Bayesian context by the concept of an objective prior distribution. It is argued that for many common machine learning problems, although in general we do not know the true (objective) prior for the problem, we do have some idea of a set of possible priors to which the true prior belongs. It is shown that under these circumstances a learner can use Bayesian inference to learn the true prior by learning sufficiently many tasks from the environment. In addition, bounds are given on the amount of information required to learn a task when it is simultaneously learnt with several other tasks. The bounds show that if the learner has little knowledge of the true prior, but the dimensionality of the true prior is small, then sampling multiple tasks is highly advantageous. The theory is applied to the problem of learning a common feature set or equivalently a low-dimensional-representation (LDR) for an environment of related tasks.  相似文献   

10.
Smoothing spline ANOVA (SSANOVA) provides an approach to semiparametric function estimation based on an ANOVA type of decomposition. Wahba et al. (1995) decomposed the regression function based on a tensor sum decomposition of inner product spaces into orthogonal subspaces, so the effects of the estimated functions from each subspace can be viewed independently. Recent research related to smoothing spline ANOVA focuses on either frequentist approaches or a Bayesian framework for variable selection and prediction. In our approach, we seek “objective” priors especially suited to estimation. The prior for linear terms including level effects is a variant of the Zellner–Siow prior (Zellner and Siow, 1980), and the prior for a smooth effect is specified in terms of effective degrees of freedom. We study this fully Bayesian SSANOVA model for Gaussian response variables, and the method is illustrated with a real data set.  相似文献   

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