Showing papers on "Maximum a posteriori estimation published in 2014"

Robust Point Matching via Vector Field Consensus

Abstract: In this paper, we propose an efficient algorithm, called vector field consensus, for establishing robust point correspondences between two sets of points. Our algorithm starts by creating a set of putative correspondences which can contain a very large number of false correspondences, or outliers, in addition to a limited number of true correspondences (inliers). Next, we solve for correspondence by interpolating a vector field between the two point sets, which involves estimating a consensus of inlier points whose matching follows a nonparametric geometrical constraint. We formulate this a maximum a posteriori (MAP) estimation of a Bayesian model with hidden/latent variables indicating whether matches in the putative set are outliers or inliers. We impose nonparametric geometrical constraints on the correspondence, as a prior distribution, using Tikhonov regularizers in a reproducing kernel Hilbert space. MAP estimation is performed by the EM algorithm which by also estimating the variance of the prior model (initialized to a large value) is able to obtain good estimates very quickly (e.g., avoiding many of the local minima inherent in this formulation). We illustrate this method on data sets in 2D and 3D and demonstrate that it is robust to a very large number of outliers (even up to 90%). We also show that in the special case where there is an underlying parametric geometrical model (e.g., the epipolar line constraint) that we obtain better results than standard alternatives like RANSAC if a large number of outliers are present. This suggests a two-stage strategy, where we use our nonparametric model to reduce the size of the putative set and then apply a parametric variant of our approach to estimate the geometric parameters. Our algorithm is computationally efficient and we provide code for others to use it. In addition, our approach is general and can be applied to other problems, such as learning with a badly corrupted training data set.

Maximum Likelihood Estimation.

Abstract: In a previous lecture, we estimated the relationship between dependent and explanatory variables using linear regression. But what if a linear relationship is not an appropriate assumption for our model? One widely used alternative is maximum likelihood estimation, which involves specifying a class of distributions, indexed by unknown parameters, and then using the data to pin down these parameter values. The benefit relative to linear regression is that it allows more flexibility in the probabilistic relationships between variables. Here we illustrate maximum likelihood by replicating Daniel Treisman’s (2016) paper, Russia’s Billionaires, which connects the number of billionaires in a country to its economic characteristics. The paper concludes that Russia has a higher number of billionaires than economic factors such as market size and tax rate predict.

Bridging Viterbi and posterior decoding: a generalized risk approach to hidden path inference based on hidden Markov models

Abstract: Motivated by the unceasing interest in hidden Markov models (HMMs), this paper reexamines hidden path inference in these models, using primarily a risk-based framework. While the most common maximum a posteriori (MAP), or Viterbi, path estimator and the minimum error, or Posterior Decoder (PD) have long been around, other path estimators, or decoders, have been either only hinted at or applied more recently and in dedicated applications generally unfamiliar to the statistical learning community. Over a decade ago, however, a family of algorithmically defined decoders aiming to hybridize the two standard ones was proposed elsewhere. The present paper gives a careful analysis of this hybridization approach, identifies several problems and issues with it and other previously proposed approaches, and proposes practical resolutions of those. Furthermore, simple modifications of the classical criteria for hidden path recognition are shown to lead to a new class of decoders. Dynamic programming algorithms to compute these decoders in the usual forward-backward manner are presented. A particularly interesting subclass of such estimators can be also viewed as hybrids of the MAP and PD estimators. Similar to previously proposed MAP-PD hybrids, the new class is parameterized by a small number of tunable parameters. Unlike their algorithmic predecessors, the new risk-based decoders are more clearly interpretable, and, most importantly, work "out-of-the box" in practice, which is demonstrated on some real bioinformatics tasks and data. Some further generalizations and applications are discussed in the conclusion.

Stereovision-Based Estimation of Relative Dynamics Between Noncooperative Satellites: Theory and Experiments

Abstract: Estimating the relative pose and motion of cooperative satellites using on-board sensors is a challenging problem. When the satellites are noncooperative, the problem becomes even more complicated, as there might be poor a priori information about the motion and structure of the target satellite. In this paper, the mentioned problem is solved by using only visual sensors, which measurements are processed through robust filtering algorithms. Using two cameras mounted on a chaser satellite, the relative state with respect to a target satellite, including the position, attitude, and rotational and translational velocities, is estimated. The new approach employs a stereoscopic vision system for tracking a set of feature points on the target spacecraft. The perspective projection of these points on the two cameras constitutes the observation model of an iterated extended Kalman filter (IEKF) estimation scheme. Using new theoretical results, the information contained in the visual data is quantified using the Fisher information matrix. It is shown that, even in the noncooperative case, there is information that can be extracted pertaining to the relative attitude and target structure. Finally, a method is proposed for rendering the relative motion filtering algorithm robust to uncertainties in the target's inertia tensor. This is accomplished by endowing the IEKF with a maximum a posteriori identification scheme for determining the most probable inertia tensor from several available hypotheses. The performance of the new filtering algorithm is validated by Monte-Carlo simulations. Also a preliminary experimental evaluation is provided.

Convex Optimization, Shape Constraints, Compound Decisions, and Empirical Bayes Rules

Abstract: Estimation of mixture densities for the classical Gaussian compound decision problem and their associated (empirical) Bayes rules is considered from two new perspectives. The first, motivated by Brown and Greenshtein, introduces a nonparametric maximum likelihood estimator of the mixture density subject to a monotonicity constraint on the resulting Bayes rule. The second, motivated by Jiang and Zhang, proposes a new approach to computing the Kiefer–Wolfowitz nonparametric maximum likelihood estimator for mixtures. In contrast to prior methods for these problems, our new approaches are cast as convex optimization problems that can be efficiently solved by modern interior point methods. In particular, we show that the reformulation of the Kiefer–Wolfowitz estimator as a convex optimization problem reduces the computational effort by several orders of magnitude for typical problems, by comparison to prior EM-algorithm based methods, and thus greatly expands the practical applicability of the resulting method...

Improving SAR-Based Urban Change Detection by Combining MAP-MRF Classifier and Nonlocal Means Similarity Weights

Abstract: In remote sensing change detection, Markov random field (MRF) has been used successfully to model the prior probability using class-labels dependencies MRF has played an important role in the detection of complex urban changes using optical images However, the preservation of details in urban change analysis turns out to be a highly complex task if multitemporal SAR images with their speckle are to be used Here, the ability of MRF to preserve geometric details and to combat speckle effect at the same time becomes questionable Blob-region phenomenon and fine structures removal are common consequences of the application of traditional MRF-based change detection algorithm To overcome these limitations, the iterated conditional modes (ICM) framework for the optimization of the maximum a posteriori (MAP-MRF) criterion function is extended to include a nonlocal probability maximization step This probability model, which characterizes the relationship between pixels’ class-labels in a nonlocal scale, has the potential to preserve spatial details and to reduce speckle effects Two multitemporal SAR datasets were used to assess the proposed algorithm Experimental results using three density functions [ie, the log normal (LN), generalized Gaussian (GG), and normal distributions (ND)] have demonstrated the efficiency of the proposed approach in terms of detail preservation and noise suppression Compared with the traditional MRF algorithm, the proposed approach proved to be less-sensitive to the value of the contextual parameter and the chosen density function The proposed approach has also shown less sensitivity to the quality of the initial change map when compared with the ICM algorithm

Eliminating conditionally independent sets in factor graphs: A unifying perspective based on smart factors

Abstract: Factor graphs are a general estimation framework that has been widely used in computer vision and robotics. In several classes of problems a natural partition arises among variables involved in the estimation. A subset of the variables are actually of interest for the user: we call those target variables. The remaining variables are essential for the formulation of the optimization problem underlying maximum a posteriori (MAP) estimation; however these variables, that we call support variables, are not strictly required as output of the estimation problem. In this paper, we propose a systematic way to abstract support variables, defining optimization problems that are only defined over the set of target variables. This abstraction naturally leads to the definition of smart factors, which correspond to constraints among target variables. We show that this perspective unifies the treatment of heterogeneous problems, ranging from structureless bundle adjustment to robust estimation in SLAM. Moreover, it enables to exploit the underlying structure of the optimization problem and the treatment of degenerate instances, enhancing both computational efficiency and robustness.

Maximum a posteriori probability estimation for online surveillance video synopsis

Abstract: To reduce human efforts in browsing long surveillance videos, synopsis videos are proposed. Traditional synopsis video generation applying optimization on video tubes is very time consuming and infeasible for real-time online generation. This dilemma significantly reduces the feasibility of synopsis video generation in practical situations. To solve this problem, the synopsis video generation problem is formulated as a maximum a posteriori probability (MAP) estimation problem in this paper, where the positions and appearing frames of video objects are chronologically rearranged in real time without the need to know their complete trajectories. Moreover, a synopsis table is employed with MAP estimation to decide the temporal locations of the incoming foreground objects in the synopsis video without needing an optimization procedure. As a result, the computational complexity of the proposed video synopsis generation method can be significantly reduced. Furthermore, as it does not require prescreening the entire video, this approach can be applied on online streaming videos.

C-KLAM: Constrained keyframe-based localization and mapping

Abstract: In this paper, we present C-KLAM, a Maximum A Posteriori (MAP) estimator-based keyframe approach for SLAM. As opposed to many existing keyframe-based SLAM approaches, that discard information from non-keyframes for reducing the computational complexity, the proposed C-KLAM presents a novel, elegant, and computationally-efficient technique for incorporating most of this information, resulting in improved estimation accuracy. To achieve this, C-KLAM projects both proprioceptive and exteroceptive information from the nonkeyframes to the keyframes, using marginalization, while maintaining the sparse structure of the associated information matrix, resulting in fast and efficient solutions. The performance of CKLAM has been tested in both simulations and experimentally, using visual and inertial measurements, to demonstrate that it achieves performance comparable to that of the computationallyintensive batch MAP-based 3D SLAM, that uses all available measurement information.

Denoising diffusion-weighted magnitude MR images using rank and edge constraints.

Abstract: Purpose To improve signal-to-noise ratio for diffusion-weighted magnetic resonance images. Methods A new method is proposed for denoising diffusion-weighted magnitude images. The proposed method formulates the denoising problem as an maximum a posteriori} estimation problem based on Rician/noncentral χ likelihood models, incorporating an edge prior and a low-rank model. The resulting optimization problem is solved efficiently using a half-quadratic method with an alternating minimization scheme. Results The performance of the proposed method has been validated using simulated and experimental data. Diffusion-weighted images and noisy data were simulated based on the diffusion tensor imaging model and Rician/noncentral χ distributions. The simulation study (with known gold standard) shows substantial improvements in single-to-noise ratio and diffusion tensor estimation after denoising. In vivo diffusion imaging data at different b-values were acquired. Based on the experimental data, qualitative improvement in image quality and quantitative improvement in diffusion tensor estimation were demonstrated. Additionally, the proposed method is shown to outperform one of the state-of-the-art nonlocal means-based denoising algorithms, both qualitatively and quantitatively. Conclusion The single-to-noise ratio of diffusion-weighted images can be effectively improved with rank and edge constraints, resulting in an improvement in diffusion parameter estimation accuracy. Magn Reson Med 71:1272–1284, 2014. © 2013 Wiley Periodicals, Inc.

Error Rate Bounds and Iterative Weighted Majority Voting for Crowdsourcing

Abstract: Crowdsourcing has become an eective and popular tool for human-powered computation to label large datasets. Since the workers can be unreliable, it is common in crowdsourcing to assign multiple workers to one task, and to aggregate the labels in order to obtain results of high quality. In this paper, we provide nite-sample exponential bounds on the error rate (in probability and in expectation) of general aggregation rules under the Dawid-Skene crowdsourcing model. The bounds are derived for multi-class labeling, and can be used to analyze many aggregation methods, including majority voting, weighted majority voting and the oracle Maximum A Posteriori (MAP) rule. We show that the oracle MAP rule approximately optimizes our upper bound on the mean error rate of weighted majority voting in certain setting. We propose an iterative weighted majority voting (IWMV) method that optimizes the error rate bound and approximates the oracle MAP rule. Its one step version has a provable theoretical guarantee on the error rate. The IWMV method is intuitive and computationally simple. Experimental results on simulated and real data show that IWMV performs at least on par with the state-of-the-art methods, and it has a much lower computational cost (around one hundred times faster) than the state-of-the-art methods.

A Prototype Software Package to Retrieve Soil Moisture From Sentinel-1 Data by Using a Bayesian Multitemporal Algorithm

Abstract: The Sentinel-1 mission will offer the opportunity to obtain C-band radar data characterized by short revisit time, thus allowing for the generation of frequent soil moisture maps. This work presents a prototype software implementing a multitemporal approach to the problem of soil moisture retrieval using Synthetic Aperture Radar (SAR) data. The approach exploits the short revisit time of Sentinel-1 data by assuming the availability of a time series of SAR images that is integrated within a retrieval algorithm based on the Bayesian maximum a posteriori probability statistical criterion. The paper focuses on the combination of on-line and off-line processing that has been designed in order to decrease the time necessary to produce a soil moisture map, which may be a critical aspect of multitemporal approaches. It describes also the optimization of the algorithm carried out to find the set of algorithm parameters that allow obtaining the best tradeoff between accuracy of the estimates and computational efficiency. A set of simulations of C-band SAR data, produced by applying a well-established radar-backscattering model, is used to perform the optimization. The designed system is tested on a series of ERS-1 SAR data acquired on February-April 1994 in Central Italy with a revisit time of three days. The results indicate that the temporal trend of estimated soil moisture is consistent with the succession of rain events occurred throughout the period of ERS-1 acquisitions over the observed geographic area.

Empirical Bayes posterior concentration in sparse high-dimensional linear models

Abstract: We propose a new empirical Bayes approach for inference in the $p \gg n$ normal linear model. The novelty is the use of data in the prior in two ways, for centering and regularization. Under suitable sparsity assumptions, we establish a variety of concentration rate results for the empirical Bayes posterior distribution, relevant for both estimation and model selection. Computation is straightforward and fast, and simulation results demonstrate the strong finite-sample performance of the empirical Bayes model selection procedure.

Stochastic Digital Backpropagation

Abstract: In this paper, we propose a novel detector for single-channel long-haul coherent optical communications, termed stochastic digital backpropagation (SDBP), which takes into account noise from the optical amplifiers in addition to handling deterministic linear and nonlinear impairments. We discuss the design approach behind this detector, which is based on the maximum a posteriori (MAP) principle. As closed-form expressions of the MAP detector are not tractable for coherent optical transmission, we employ the framework of Bayesian graphical models, which allows a numerical evaluation of the proposed detector. Through simulations, we observe that by accounting for nonlinear signal—noise interactions, we achieve a significant improvement in system reach with SDBP over digital backpropagation (DBP) for systems with periodic inline optical dispersion compensation. In uncompensated links with high symbol rates, the performance difference in terms of system reach for SDBP over DBP is small. In the absence of noise, the proposed detector is equivalent to the well-known DBP detector.

Bayesian estimation based on progressive Type-II censoring from two-parameter bathtub-shaped lifetime model: an Markov chain Monte Carlo approach

Abstract: In this paper, maximum likelihood and Bayes estimators of the parameters, reliability and hazard functions have been obtained for two-parameter bathtub-shaped lifetime distribution when sample is available from progressive Type-II censoring scheme. The Markov chain Monte Carlo (MCMC) method is used to compute the Bayes estimates of the model parameters. It has been assumed that the parameters have gamma priors and they are independently distributed. Gibbs within the Metropolis–Hasting algorithm has been applied to generate MCMC samples from the posterior density function. Based on the generated samples, the Bayes estimates and highest posterior density credible intervals of the unknown parameters as well as reliability and hazard functions have been computed. The results of Bayes estimators are obtained under both the balanced-squared error loss and balanced linear-exponential (BLINEX) loss. Moreover, based on the asymptotic normality of the maximum likelihood estimators the approximate confidence interva...

A Fast and Scalable Method for A-Optimal Design of Experiments for Infinite-dimensional Bayesian Nonlinear Inverse Problems

Abstract: We address the problem of optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by PDEs. The goal is to find a placement of sensors, at which experimental data are collected, so as to minimize the uncertainty in the inferred parameter field. We formulate the OED objective function by generalizing the classical A-optimal experimental design criterion using the expected value of the trace of the posterior covariance. We seek a method that solves the OED problem at a cost (measured in the number of forward PDE solves) that is independent of both the parameter and sensor dimensions. To facilitate this, we construct a Gaussian approximation to the posterior at the maximum a posteriori probability (MAP) point, and use the resulting covariance operator to define the OED objective function. We use randomized trace estimation to compute the trace of this (implicitly defined) covariance operator. The resulting OED problem includes as constraints the PDEs characterizing the MAP point, and the PDEs describing the action of the covariance operator to vectors. The sparsity of the sensor configurations is controlled using sparsifying penalty functions. We elaborate our OED method for the problem of determining the sensor placement to best infer the coefficient of an elliptic PDE. Adjoint methods are used to compute the gradient of the PDE-constrained OED objective function. We provide numerical results for inference of the permeability field in a porous medium flow problem, and demonstrate that the number of PDE solves required for the evaluation of the OED objective function and its gradient is essentially independent of both the parameter and sensor dimensions. The number of quasi-Newton iterations for computing an OED also exhibits the same dimension invariance properties.

Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte Carlo

Abstract: We consider the Riemann manifold Hamiltonian Monte Carlo (RMHMC) method for solving statistical inverse problems governed by partial differential equations (PDEs). The Bayesian framework is employed to cast the inverse problem into the task of statistical inference whose solution is the posterior distribution in infinite dimensional parameter space conditional upon observation data and Gaussian prior measure. We discretize both the likelihood and the prior using the H1-conforming finite element method together with a matrix transfer technique. The power of the RMHMC method is that it exploits the geometric structure induced by the PDE constraints of the underlying inverse problem. Consequently, each RMHMC posterior sample is almost uncorrelated/independent from the others providing statistically efficient Markov chain simulation. However this statistical efficiency comes at a computational cost. This motivates us to consider computationally more efficient strategies for RMHMC. At the heart of our construction is the fact that for Gaussian error structures the Fisher information matrix coincides with the Gauss–Newton Hessian. We exploit this fact in considering a computationally simplified RMHMC method combining state-of-the-art adjoint techniques and the superiority of the RMHMC method. Specifically, we first form the Gauss–Newton Hessian at the maximum a posteriori point and then use it as a fixed constant metric tensor throughout RMHMC simulation. This eliminates the need for the computationally costly differential geometric Christoffel symbols, which in turn greatly reduces computational effort at a corresponding loss of sampling efficiency. We further reduce the cost of forming the Fisher information matrix by using a low rank approximation via a randomized singular value decomposition technique. This is efficient since a small number of Hessian-vector products are required. The Hessian-vector product in turn requires only two extra PDE solves using the adjoint technique. Various numerical results up to 1025 parameters are presented to demonstrate the ability of the RMHMC method in exploring the geometric structure of the problem to propose (almost) uncorrelated/independent samples that are far away from each other, and yet the acceptance rate is almost unity. The results also suggest that for the PDE models considered the proposed fixed metric RMHMC can attain almost as high a quality performance as the original RMHMC, i.e. generating (almost) uncorrelated/independent samples, while being two orders of magnitude less computationally expensive.

MRF-Based Fast HEVC Inter CU Decision With the Variance of Absolute Differences

Abstract: The newly developed High Efficiency Video Coding (HEVC) Standard has improved video coding performance significantly in comparison to its predecessors. However, more intensive computation complexity is introduced by implementing a number of new coding tools. In this paper, a fast coding unit (CU) decision based on Markov random field (MRF) is proposed for HEVC inter frames. First, it is observed that the variance of the absolute difference (VAD) is proportional with the rate-distortion (R-D) cost. The VAD based feature is designed for the CU selection. Second, the decision of CU splittings is modeled as an MRF inference problem, which can be optimized by the Graphcut algorithm. Third, a maximum a posteriori (MAP) approach based on the R-D cost is conducted to evaluate whether the unsplit CUs should be further split or not. Experimental results show that the proposed algorithm can achieve about 53% reduction of the coding time with negligible coding performance degradation, which outperforms the state-of-the-art algorithms significantly.

Distribution-Aware Sampling and Weighted Model Counting for SAT

Abstract: Given a CNF formula and a weight for each assignment of values to variables, two natural problems are weighted model counting and distribution-aware sampling of satisfying assignments. Both problems have a wide variety of important applications. Due to the inherent complexity of the exact versions of the problems, interest has focused on solving them approximately. Prior work in this area scaled only to small problems in practice, or failed to provide strong theoretical guarantees, or employed a computationally-expensive maximum a posteriori probability (MAP) oracle that assumes prior knowledge of a factored representation of the weight distribution. We present a novel approach that works with a black-box oracle for weights of assignments and requires only an {\NP}-oracle (in practice, a SAT-solver) to solve both the counting and sampling problems. Our approach works under mild assumptions on the distribution of weights of satisfying assignments, provides strong theoretical guarantees, and scales to problems involving several thousand variables. We also show that the assumptions can be significantly relaxed while improving computational efficiency if a factored representation of the weights is known.

Bayesian structural inference for hidden processes

Abstract: We introduce a Bayesian approach to discovering patterns in structurally complex processes. The proposed method of Bayesian structural inference (BSI) relies on a set of candidate unifilar hidden Markov model (uHMM) topologies for inference of process structure from a data series. We employ a recently developed exact enumeration of topological e-machines. (A sequel then removes the topological restriction.) This subset of the uHMM topologies has the added benefit that inferred models are guaranteed to be e-machines, irrespective of estimated transition probabilities. Properties of e-machines and uHMMs allow for the derivation of analytic expressions for estimating transition probabilities, inferring start states, and comparing the posterior probability of candidate model topologies, despite process internal structure being only indirectly present in data. We demonstrate BSI's effectiveness in estimating a process's randomness, as reflected by the Shannon entropy rate, and its structure, as quantified by the statistical complexity. We also compare using the posterior distribution over candidate models and the single, maximum a posteriori model for point estimation and show that the former more accurately reflects uncertainty in estimated values. We apply BSI to in-class examples of finite- and infinite-order Markov processes, as well to an out-of-class, infinite-state hidden process.

Adaptive total variation-based spectral deconvolution with the split Bregman method

Abstract: Spectroscopic data often suffer from common problems of band overlap and noise This paper presents a maximum a posteriori (MAP)-based algorithm for the band overlap problem In the MAP framework, the likelihood probability density function (PDF) is constructed with Gaussian noise assumed, and the prior PDF is constructed with adaptive total variation (ATV) regularization The split Bregman iteration algorithm is employed to optimize the ATV spectral deconvolution model and accelerate the speed of the spectral deconvolution The main advantage of this algorithm is that it can obtain peak structure information as well as suppress noise simultaneity Simulated and real spectra experiments manifest that this algorithm can satisfactorily recover the overlap peaks as well as suppress noise and are robust to the regularization parameter

Multi-Contrast Multi-Atlas Parcellation of Diffusion Tensor Imaging of the Human Brain

Abstract: In this paper, we propose a novel method for parcellating the human brain into 193 anatomical structures based on diffusion tensor images (DTIs). This was accomplished in the setting of multi-contrast diffeomorphic likelihood fusion using multiple DTI atlases. DTI images are modeled as high dimensional fields, with each voxel exhibiting a vector valued feature comprising of mean diffusivity (MD), fractional anisotropy (FA), and fiber angle. For each structure, the probability distribution of each element in the feature vector is modeled as a mixture of Gaussians, the parameters of which are estimated from the labeled atlases. The structure-specific feature vector is then used to parcellate the test image. For each atlas, a likelihood is iteratively computed based on the structure-specific vector feature. The likelihoods from multiple atlases are then fused. The updating and fusing of the likelihoods is achieved based on the expectation-maximization (EM) algorithm for maximum a posteriori (MAP) estimation problems. We first demonstrate the performance of the algorithm by examining the parcellation accuracy of 18 structures from 25 subjects with a varying degree of structural abnormality. Dice values ranging 0.8–0.9 were obtained. In addition, strong correlation was found between the volume size of the automated and the manual parcellation. Then, we present scan-rescan reproducibility based on another dataset of 16 DTI images – an average of 3.73%, 1.91%, and 1.79% for volume, mean FA, and mean MD respectively. Finally, the range of anatomical variability in the normal population was quantified for each structure.

Bayesian View Synthesis and Image-Based Rendering Principles

Abstract: In this paper, we address the problem of synthesizing novel views from a set of input images. State of the art methods, such as the Unstructured Lumigraph, have been using heuristics to combine information from the original views, often using an explicit or implicit approximation of the scene geometry. While the proposed heuristics have been largely explored and proven to work effectively, a Bayesian formulation was recently introduced, formalizing some of the previously proposed heuristics, pointing out which physical phenomena could lie behind each. However, some important heuristics were still not taken into account and lack proper formalization. We contribute a new physics-based generative model and the corresponding Maximum a Posteriori estimate, providing the desired unification between heuristics-based methods and a Bayesian formulation. The key point is to systematically consider the error induced by the uncertainty in the geometric proxy. We provide an extensive discussion, analyzing how the obtained equations explain the heuristics developed in previous methods. Furthermore, we show that our novel Bayesian model significantly improves the quality of novel views, in particular if the scene geometry estimate is inaccurate.

The moments of the Gompertz distribution and maximum likelihood estimation of its parameters

Abstract: The Gompertz distribution is widely used to describe the distribution of adult deaths. Previous works concentrated on formulating approximate relationships to characterise it. However, using the generalised integro-exponential function, exact formulas can be derived for its moment-generating function and central moments. Based on the exact central moments, higher accuracy approximations can be defined for them. In demographic or actuarial applications, maximum likelihood estimation is often used to determine the parameters of the Gompertz distribution. By solving the maximum likelihood estimates analytically, the dimension of the optimisation problem can be reduced to one both in the case of discrete and continuous data. Monte Carlo experiments show that by ML estimation, higher accuracy estimates can be acquired than by the method of moments.

A Variational Bayes Framework for Sparse Adaptive Estimation

Abstract: Recently, a number of mostly l1-norm regularized least-squares-type deterministic algorithms have been proposed to address the problem of sparse adaptive signal estimation and system identification. From a Bayesian perspective, this task is equivalent to maximum a posteriori probability estimation under a sparsity promoting heavy-tailed prior for the parameters of interest. Following a different approach, this paper develops a unifying framework of sparse variational Bayes algorithms that employ heavy-tailed priors in conjugate hierarchical form to facilitate posterior inference. The resulting fully automated variational schemes are first presented in a batch iterative form. Then, it is shown that by properly exploiting the structure of the batch estimation task, new sparse adaptive variational Bayes algorithms can be derived, which have the ability to impose and track sparsity during real-time processing in a time-varying environment. The most important feature of the proposed algorithms is that they completely eliminate the need for computationally costly parameter fine-tuning, a necessary ingredient of sparse adaptive deterministic algorithms. Extensive simulation results are provided to demonstrate the effectiveness of the new sparse adaptive variational Bayes algorithms against state-of-the-art deterministic techniques for adaptive channel estimation. The results show that the proposed algorithms are numerically robust and exhibit in general superior estimation performance compared to their deterministic counterparts.

Maximum a posteriori estimates in linear inverse problems with log-concave priors are proper Bayes estimators

Abstract: A frequent matter of debate in Bayesian inversion is the question of which of the two principal point-estimators, the maximum a posteriori (MAP) or the conditional mean (CM) estimate, is to be preferred. As the MAP estimate corresponds to the solution given by variational regularization techniques, this is also a constant matter of debate between the two research areas. Following a theoretical argument—the Bayes cost formalism—the CM estimate is classically preferred for being the Bayes estimator for the mean squared error cost, while the MAP estimate is classically discredited for being only asymptotically the Bayes estimator for the uniform cost function. In this article we present recent theoretical and computational observations that challenge this point of view, in particular for high-dimensional sparsity-promoting Bayesian inversion. Using Bregman distances, we present new, proper convex Bayes cost functions for which the MAP estimator is the Bayes estimator. We complement this finding with results that correct further common misconceptions about MAP estimates. In total, we aim to rehabilitate MAP estimates in linear inverse problems with log-concave priors as proper Bayes estimators.

Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring

Abstract: In this paper, the statistical inference of the unknown parameters of a two-parameter inverse Weibull (IW) distribution based on the progressive type-II censored sample has been considered. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, hence the approximate MLEs are proposed, which are in explicit forms. The Bayes and generalized Bayes estimators for the IW parameters and the reliability function based on the squared error and Linex loss functions are provided. The Bayes and generalized Bayes estimators cannot be obtained explicitly, hence Lindley's approximation is used to obtain the Bayes and generalized Bayes estimators. Furthermore, the highest posterior density credible intervals of the unknown parameters based on Gibbs sampling technique are computed, and using an optimality criterion the optimal censoring scheme has been suggested. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, two data sets have been analysed for...