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Ron Rubinstein - One of the best experts on this subject based on the ideXlab platform.
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analysis versus synthesis in signal priors
Inverse Problems, 2007Co-Authors: Michael Elad, Peyman Milanfar, Ron RubinsteinAbstract:The concept of prior Probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors Assign Probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. The algebraic similarity between the two suggests that they could be strongly related; however, in the absence of a detailed study, contradicting approaches have emerged. While the computationally intensive synthesis approach is receiving ever-increasing attention and is notably preferred, other works hypothesize that the two might actually be much closer, going as far as to suggest that one can approximate the other. In this paper we describe the two prior classes in detail, focusing on the distinction between them. We show that although in the simpler complete and undercomplete formulations the two approaches are equivalent, in their overcomplete formulation they depart. Focusing on the l1 case, we present a novel approach for comparing the two types of priors based on high-dimensional polytopal geometry. We arrive at a series of theoretical and numerical results establishing the existence of an unbridgeable gap between the two.
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analysis versus synthesis in signal priors
European Signal Processing Conference, 2006Co-Authors: Michael Elad, Peyman Milanfar, Ron RubinsteinAbstract:The concept of prior Probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors Assign Probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. In this paper we describe these two prior classes, focusing on the distinction between them. We show that although when reducing to the complete and under-complete formulations the two become equivalent, in their more interesting overcomplete formulation the two types depart. Focusing on the l1 denoising case, we present several ways of comparing the two types of priors, establishing the existence of an unbridgeable gap between them.
Michael Elad - One of the best experts on this subject based on the ideXlab platform.
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analysis versus synthesis in signal priors
Inverse Problems, 2007Co-Authors: Michael Elad, Peyman Milanfar, Ron RubinsteinAbstract:The concept of prior Probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors Assign Probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. The algebraic similarity between the two suggests that they could be strongly related; however, in the absence of a detailed study, contradicting approaches have emerged. While the computationally intensive synthesis approach is receiving ever-increasing attention and is notably preferred, other works hypothesize that the two might actually be much closer, going as far as to suggest that one can approximate the other. In this paper we describe the two prior classes in detail, focusing on the distinction between them. We show that although in the simpler complete and undercomplete formulations the two approaches are equivalent, in their overcomplete formulation they depart. Focusing on the l1 case, we present a novel approach for comparing the two types of priors based on high-dimensional polytopal geometry. We arrive at a series of theoretical and numerical results establishing the existence of an unbridgeable gap between the two.
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analysis versus synthesis in signal priors
European Signal Processing Conference, 2006Co-Authors: Michael Elad, Peyman Milanfar, Ron RubinsteinAbstract:The concept of prior Probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors Assign Probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. In this paper we describe these two prior classes, focusing on the distinction between them. We show that although when reducing to the complete and under-complete formulations the two become equivalent, in their more interesting overcomplete formulation the two types depart. Focusing on the l1 denoising case, we present several ways of comparing the two types of priors, establishing the existence of an unbridgeable gap between them.
Martin D Weinberg - One of the best experts on this subject based on the ideXlab platform.
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computing the bayes factor from a markov chain monte carlo simulation of the posterior distribution
Bayesian Analysis, 2012Co-Authors: Martin D WeinbergAbstract:Computation of the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally difficult. The often-used harmonic mean approximation uses the posterior directly but is unstably sensitive to samples with anomalously small values of the likelihood. The Laplace approximation is stable but makes strong, and often inappropriate, assumptions about the shape of the posterior distribution. It is useful, but not general. We need algorithms that apply to general distributions, like the harmonic mean approximation, but do not suffer from convergence and instability issues. Here, I argue that the marginal likelihood can be reliably computed from a posterior sample by careful attention to the numerics of the Probability integral. Posing the expression for the marginal likelihood as a Lebesgue integral, we may convert the harmonic mean approximation from a sample statistic to a quadrature rule. As a quadrature, the harmonic mean approximation suffers from enormous truncation error as consequence . This error is a direct consequence of poor coverage of the sample space; the posterior sample required for accurate computation of the marginal likelihood is much larger than that required to characterize the posterior distribution when using the harmonic mean approximation. In addition, I demonstrate that the integral expression for the harmonic-mean approximation converges slowly at best for high-dimensional problems with uninformative prior distributions. These observations lead to two computationally-modest families of quadrature algorithms that use the full generality sample posterior but without the instability. The first algorithm automatically eliminates the part of the sample that contributes large truncation error. The second algorithm uses the posterior sample to Assign Probability to a partition of the sample space and performs the marginal likelihood integral directly. This eliminates convergence issues. The first algorithm is analogous to standard quadrature but can only be applied for convergent problems. The second is a hybrid of cubature: it uses the posterior to discover and tessellate the subset of that sample space was explored and uses quantiles to compute a representative field value. Qualitatively, the first algorithm improves the harmonic mean approximation using numerical analysis, and the second algorithm is an adaptive version of the Laplace approximation. Neither algorithm makes strong assumptions about the shape of the posterior distribution and neither is sensitive to outliers. Based on numerical tests, we recommend a combined application of both algorithms as consistency check to achieve a reliable estimate of the marginal likelihood from a simulated posterior distribution.
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computing the bayes factor from a markov chain monte carlo simulation of the posterior distribution
Bayesian Analysis, 2012Co-Authors: Martin D WeinbergAbstract:Determining the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally challenging. The often-used harmonic mean approximation (HMA) makes no prior assumptions about the character of the distribution but tends to be inconsistent. The Laplace approximation is stable but makes strong, and often inappropriate, assumptions about the shape of the posterior distribution. Here, I argue that the marginal likelihood can be reliably computed from a posterior sample using Lebesgue integration theory in one of two ways: 1) when the HMA integral exists, compute the measure function numerically and analyze the resulting quadrature to control error; 2) compute the measure function numerically for the marginal likelihood integral itself using a space-partitioning tree, followed by quadrature. The first algorithm automatically eliminates the part of the sample that contributes large truncation error in the HMA. Moreover, it provides a simple graphical test for the existence of the HMA integral. The second algorithm uses the posterior sample to Assign Probability to a partition of the sample space and performs the marginal likelihood integral directly. It uses the posterior sample to discover and tessellate the subset of the sample space that was explored and uses quantiles to compute a representative field value. When integrating directly, this space may be trimmed to remove regions with low Probability density and thereby improve accuracy. This second algorithm is consistent for all proper distributions. Error analysis provides some diagnostics on the numerical condition of the results in both cases.
Peyman Milanfar - One of the best experts on this subject based on the ideXlab platform.
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analysis versus synthesis in signal priors
Inverse Problems, 2007Co-Authors: Michael Elad, Peyman Milanfar, Ron RubinsteinAbstract:The concept of prior Probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors Assign Probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. The algebraic similarity between the two suggests that they could be strongly related; however, in the absence of a detailed study, contradicting approaches have emerged. While the computationally intensive synthesis approach is receiving ever-increasing attention and is notably preferred, other works hypothesize that the two might actually be much closer, going as far as to suggest that one can approximate the other. In this paper we describe the two prior classes in detail, focusing on the distinction between them. We show that although in the simpler complete and undercomplete formulations the two approaches are equivalent, in their overcomplete formulation they depart. Focusing on the l1 case, we present a novel approach for comparing the two types of priors based on high-dimensional polytopal geometry. We arrive at a series of theoretical and numerical results establishing the existence of an unbridgeable gap between the two.
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analysis versus synthesis in signal priors
European Signal Processing Conference, 2006Co-Authors: Michael Elad, Peyman Milanfar, Ron RubinsteinAbstract:The concept of prior Probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors Assign Probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. In this paper we describe these two prior classes, focusing on the distinction between them. We show that although when reducing to the complete and under-complete formulations the two become equivalent, in their more interesting overcomplete formulation the two types depart. Focusing on the l1 denoising case, we present several ways of comparing the two types of priors, establishing the existence of an unbridgeable gap between them.
Justin Gatwood - One of the best experts on this subject based on the ideXlab platform.
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cost utility analysis of glaucoma medication adherence
Ophthalmology, 2020Co-Authors: Paula Anne Newmancasey, Mariam Salman, Justin GatwoodAbstract:Purpose The majority of patients with glaucoma do not take their medications as prescribed. Estimates of the cost-utility value of adherence to prescribed glaucoma medication are vital to implement potentially effective interventions. Design Cost-utility analysis using Monte Carlo microsimulations incorporating a series of Markov cycles (10 000 iterations per strategy). Participants Patients with glaucoma aged ≥40 years with a full lifetime horizon (up to 60 years). Methods The analysis estimated glaucomatous progression on the basis of data from the United Kingdom Glaucoma Treatment Study. Participants with glaucoma entered the model at age 40 years with a mean deviation in the better-seeing eye of −1.4±−1.9 decibels (dB) and −4.3±−3.4 dB in the worse-seeing eye. Participants whose glaucoma worsened each year accumulate −0.8 dB loss compared with −0.1 dB loss for those who remained stable. Data from the Glaucoma Laser Trial and the Tube versus Trabeculectomy Studies were used to Assign probabilities of worsening disease among treated patients. Claims data estimating rates of glaucoma medication adherence over 4 years were used to Assign Probability of adherence. Those with poor adherence were modeled as having outcomes similar to the placebo arm of the clinical trials. As patients’ mean deviation deteriorated, they transitioned between health states from mild (≥−6 dB), to moderate ( Main Outcome Measures Cost and quality-adjusted life year (QALY) of glaucoma medication adherence. Results Beginning at an initial glaucoma diagnosis at age 40 years, patients proceeded to single-eye blindness as early as 19 years among those who were nonadherent and 23 years for those remaining adherent. Total healthcare costs for adherent patients averaged $62 782 (standard deviation [SD], 34 107), and those for nonadherent patients averaged $52 722 (SD, 38 868). Nonadherent patients had a mean loss of 0.34 QALYs, resulting in a cost-effectiveness ratio of $29 600 per QALY gained. Conclusion At a conservative willingness to pay of $50 000/QALY, there is room to expand services to improve patient adherence.
