Maximum a posteriori estimators as a limit of Bayes estimators

Bassett, Robert; Deride, Julio

doi:10.1007/s10107-018-1241-0

Cited by 98 publications

(48 citation statements)

References 15 publications

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“…This optimization called the maximum a posteriori (MAP) estimation. [68][69][70] (2) by-instance-based, which is the standard kNN approach, in which a new case is classified by a majority vote of its neighbors, with the case being assigned to a class that is most common among its k nearest neighbors measured by a distance function. If k is 1, then the case is simply assigned to the class of its nearest neighbor.…”

Section: Discussionmentioning

confidence: 99%

Development of use-specific high-performance cyber-nanomaterial optical detectors by effective choice of machine learning algorithms

Hejazi

Liu

Farnoosh

et al. 2020

Mach. Learn.: Sci. Technol.

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P( |T) (nm) Unknown Narrow-Band Light arXiv:1912.11751v3 [physics.app-ph] 3 Jan 2020 term reliability can be equal or more important considerations. Although various machine learning (ML) tools are frequently used on sensor and detector networks to address these considerations and dramatically enhance their functionalities, nonetheless, their effectiveness on nanomaterials-based sensors has not been explored. Here, we show that the best choice of ML algorithm in a cyber-nanomaterial detector is largely determined by the specific useconsiderations, including accuracy, computational cost, speed, and resilience against drifts and long-term ageing effects. When sufficient data and computing resources are provided, the highest sensing accuracy can be achieved by the k-nearest neighbors (kNN) and Bayesian inference algorithms, however, these algorithms can be computationally expensive for real-time applications. In contrast, artificial neural networks (ANN) are computationally expensive to train (off-line), but they provide the fastest result under testing conditions (on-line) while remaining reasonably accurate. When access to data is limited, support vector machines (SVMs) can perform well even with small training sample sizes, while other algorithms show considerable reduction in accuracy if data is scarce, hence, setting a lower limit on the size of required training data. We also show by tracking and modeling the long-term drifts of the detector performance over large (i.e. one year) time-frame, it is possible to dramatically improve the predictive accuracy without the need for any recalibration. Our research shows for the first time that if the ML algorithm is chosen specific to the use-case, lowcost solution-processed cyber-nanomaterial detectors can be practically implemented under diverse operational requirements, despite their inherent variabilities.

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Section: Discussionmentioning

confidence: 99%

Development of use-specific high-performance cyber-nanomaterial optical detectors by effective choice of machine learning algorithms

Hejazi

Liu

Farnoosh

et al. 2020

Mach. Learn.: Sci. Technol.

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“…The analysis is further complicated by the fact that p(x|y) may have an infinite number of maximisers in disconnected areas of the parameter space. As mentioned previously, the derivation of MAP estimation as an approximation arising from the degenerate loss L (u, x) = 1 x−u < with → 0 also fails in this case [8]. Interestingly, MMSE estimation may also struggle here given that models in this class may not have a posterior mean [38].…”

Section: Models With Heavy-tailsmentioning

confidence: 92%

“…Currently the predominant view is that MAP estimation is not formal Bayesian estimation in the decision-theoretic sense postulated by Definition 1.1 because it does not generally minimise a known expected loss. The prevailing interpretation is that MAP estimation is in fact an approximation arising from the degenerate loss L (u, x) = 1 x−u < with → 0 [38] (this derivation holds for all log-concave models, but is not generally true [8]). However, this asymptotic derivation does not lead to a proper Bayesian estimator.…”

Section: Introductionmentioning

confidence: 99%

Revisiting Maximum-A-Posteriori Estimation in Log-Concave Models

Pereyra

2019

SIAM J. Imaging Sci.

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Maximum-a-posteriori (MAP) estimation is the main Bayesian estimation methodology in imaging sciences, where high dimensionality is often addressed by using Bayesian models that are log-concave and whose posterior mode can be computed efficiently by convex optimisation. However, despite its success and wide adoption, MAP estimation is not theoretically well understood yet. In particular, the prevalent view in the community is that MAP estimation is not proper Bayesian estimation in the sense of Bayesian decision theory because it does not minimise a meaningful expected loss function (unlike the minimum mean squared error (MMSE) estimator that minimises the mean squared loss). This paper addresses this theoretical gap by presenting a general decision-theoretic derivation of MAP estimation in Bayesian models that are log-concave. A main novelty is that our analysis is based on differential geometry, and proceeds as follows. First, we use the underlying convex geometry of the Bayesian model to induce a Riemannian geometry on the parameter space. We then use differential geometry to identify the so-called natural or canonical loss function to perform Bayesian point estimation in that Riemannian manifold. For log-concave models, this canonical loss coincides with the Bregman divergence associated with the negative log posterior density. Following on from this, we show that the MAP estimator is the only Bayesian estimator that minimises the expected canonical loss, and that the posterior mean or MMSE estimator minimises the dual canonical loss. We then study the question of MAP and MSSE estimation performance in high dimensions. Precisely, we establish a universal bound on the expected canonical error as a function of image dimension, providing new insights the good empirical performance observed in convex problems. Together, these results provide a new understanding of MAP and MMSE estimation in log-concave settings, and of the multiple beneficial roles that convex geometry plays in imaging problems. Finally, we illustrate this new theory by analysing the regularisation-by-denoising Bayesian models, a class of state-of-the-art imaging models where priors are defined implicitly through image denoising algorithms, and an image denoising model with a wavelet shrinkage prior.

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“…The methods stated here like the least squares estimators (LSE) is commonly used in estimating parameters in linear regression models assuming a normal distribution of errors of observation and the condition of linear combination of the values with error variance being minimum. The maximum likelihood estimates (MLE) which yields an unbiased estimator, the method of moments, Bayesian estimators, minimum variance estimators [6,7] and Cramer Rao lower bound [8] that is used to obtain the Uniformly Minimum variance unbiased estimator (UMVUE) are some of the techniques employed when estimating the unknown parameters.…”

Section: Methodsmentioning

confidence: 99%

Parametric Point Estimation of the Geeta Distribution

Korir¹

2018

IJSDA

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Geeta distribution is a new discrete random variable distribution defined over all the positive integers with two parameters. This distribution belongs to the family of Location-parameter (LDPD) system and is of the form L -shaped model. Pareto and Yule distributions belong to the same family but these distributions have a disadvantage of having a single parameter which makes them not versatile to meet the needs of modern complex data sets. Geeta distribution is found to be very versatile and flexible to fit observed count data sets and can be used efficiently to model different types of sets. This paper investigates the characteristics of Geeta distribution, such as the existence of the mean, variance, moment generating function, probability generating function and that the sum of probabilities for all values of X for Geeta Distribution model is unity. It is well known that the sample mean is the estimator of a population mean from a given population of interest as a point estimator which assume a single number that is obtained by taking a random sample of a specified size from the entire population, depending on whether the population mean and variance is known or unknown These point estimators were obtained by employing the method of Moments, Maximum Likelihood (MLE) and Bayesian estimator. Further the estimators were subjected to the conditions like unbiasedness, efficiency, sufficiency and completeness which are properties of a good estimator. For the first aspect, the results of the mean, variance, moments and generating functions were achieved that proves the distribution is a probability density function (pdf). The methods of moments and the maximum likelihood and their properties were applied and yielded the desired and expected results for any given probability distribution. The best estimator obtained is best linear unbiased estimator (BLUE).

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Maximum a posteriori estimators as a limit of Bayes estimators

Cited by 98 publications

References 15 publications

Development of use-specific high-performance cyber-nanomaterial optical detectors by effective choice of machine learning algorithms

Development of use-specific high-performance cyber-nanomaterial optical detectors by effective choice of machine learning algorithms

Revisiting Maximum-A-Posteriori Estimation in Log-Concave Models

Parametric Point Estimation of the Geeta Distribution

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