Numerical Version of the Non-Uniform Method for Finding Point Estimates of Uncertain Scaling Constants

2013 ◽  
pp. 259-292
Author(s):  
Natalia D. Nikolova ◽  
Kiril I. Tenekedjiev
Keyword(s):  
Analytical Solution ◽  
Probability Distributions ◽  
Numerical Procedure ◽  
Statistical Test ◽  
Numerical Realization ◽  
Uniform Method ◽  
Uncertainty Interval ◽  
Additional Information ◽  
Point Estimates ◽  
Finding Point

The chapter focuses on the analysis of scaling constants when constructing a utility function over multi-dimensional prizes. Due to fuzzy rationality, those constants are elicited in an interval form. It is assumed that the decision maker has provided additional information describing the uncertainty of the scaling constants’ values within their uncertainty interval. The non-uniform method is presented to find point estimates of the interval scaling constants and to test their unit sum. An analytical solution of the procedure to construct the distribution of the interval scaling constants is provided, along with its numerical realization. A numerical procedure to estimate pvalue of the statistical test is also presented. The method allows making an uncertainty description of constants through different types of probability distributions and fuzzy sets.

Field-theoretic density estimation for biological sequence space with applications to 5′ splice site diversity and aneuploidy in cancer

2021 ◽  
Vol 118 (40) ◽  
pp. e2025782118
Author(s):  
Wei-Chia Chen ◽  
Juannan Zhou ◽  
Jason M. Sheltzer ◽  
Justin B. Kinney ◽  
David M. McCandlish
Keyword(s):  
Probability Distribution ◽  
Maximum Entropy ◽  
Density Estimation ◽  
Sequence Space ◽  
Chromosomal Abnormalities ◽  
Fundamental Problem ◽  
Probability Distributions ◽  
Biological Sequence ◽  
Point Estimates ◽  
Site Diversity

Density estimation in sequence space is a fundamental problem in machine learning that is also of great importance in computational biology. Due to the discrete nature and large dimensionality of sequence space, how best to estimate such probability distributions from a sample of observed sequences remains unclear. One common strategy for addressing this problem is to estimate the probability distribution using maximum entropy (i.e., calculating point estimates for some set of correlations based on the observed sequences and predicting the probability distribution that is as uniform as possible while still matching these point estimates). Building on recent advances in Bayesian field-theoretic density estimation, we present a generalization of this maximum entropy approach that provides greater expressivity in regions of sequence space where data are plentiful while still maintaining a conservative maximum entropy character in regions of sequence space where data are sparse or absent. In particular, we define a family of priors for probability distributions over sequence space with a single hyperparameter that controls the expected magnitude of higher-order correlations. This family of priors then results in a corresponding one-dimensional family of maximum a posteriori estimates that interpolate smoothly between the maximum entropy estimate and the observed sample frequencies. To demonstrate the power of this method, we use it to explore the high-dimensional geometry of the distribution of 5′ splice sites found in the human genome and to understand patterns of chromosomal abnormalities across human cancers.

scholarly journals Bayesian estimation of the seroprevalence of antibodies to SARS-CoV-2

JAMIA Open ◽  
2020 ◽  
Author(s):  
Qunfeng Dong ◽  
Xiang Gao
Keyword(s):  
Probability Distribution ◽  
Large Scale ◽  
Prior Probability ◽  
Probability Distributions ◽  
Single Point ◽  
Point Estimates ◽  
Specificity And Sensitivity ◽  
Large Scale Dataset ◽  
The Us ◽  
Antibody Tests

Abstract Accurate estimations of the seroprevalence of antibodies to severe acute respiratory syndrome coronavirus 2 need to properly consider the specificity and sensitivity of the antibody tests. In addition, prior knowledge of the extent of viral infection in a population may also be important for adjusting the estimation of seroprevalence. For this purpose, we have developed a Bayesian approach that can incorporate the variabilities of specificity and sensitivity of the antibody tests, as well as the prior probability distribution of seroprevalence. We have demonstrated the utility of our approach by applying it to a recently published large-scale dataset from the US CDC, with our results providing entire probability distributions of seroprevalence instead of single-point estimates. Our Bayesian code is freely available at https://github.com/qunfengdong/AntibodyTest.

Non-parametric Bayesian density estimation for biological sequence space with applications to pre-mRNA splicing and the karyotypic diversity of human cancer

2020 ◽  
Author(s):  
Wei-Chia Chen ◽  
Juannan Zhou ◽  
Jason M Sheltzer ◽  
Justin B Kinney ◽  
David M McCandlish
Keyword(s):  
Probability Distribution ◽  
Maximum Entropy ◽  
Density Estimation ◽  
Cancer Progression ◽  
Sequence Space ◽  
Chromosomal Abnormalities ◽  
Fundamental Problem ◽  
Human Cancer ◽  
Probability Distributions ◽  
Point Estimates

AbstractDensity estimation in sequence space is a fundamental problem in machine learning that is of great importance in computational biology. Due to the discrete nature and large dimensionality of sequence space, how best to estimate such probability distributions from a sample of observed sequences remains unclear. One common strategy for addressing this problem is to estimate the probability distribution using maximum entropy, i.e. calculating point estimates for some set of correlations based on the observed sequences and predicting the probability distribution that is as uniform as possible while still matching these point estimates. Building on recent advances in Bayesian field-theoretic density estimation, we present a generalization of this maximum entropy approach that provides greater expressivity in regions of sequence space where data is plentiful while still maintaining a conservative maximum entropy char-acter in regions of sequence space where data is sparse or absent. In particular, we define a family of priors for probability distributions over sequence space with a single hyper-parameter that controls the expected magnitude of higher-order correlations. This family of priors then results in a corresponding one-dimensional family of maximum a posteriori estimates that interpolate smoothly between the maximum entropy estimate and the observed sample frequencies. To demonstrate the power of this method, we use it to explore the high-dimensional geometry of the distribution of 5′ splice sites found in the human genome and to understand the accumulation of chromosomal abnormalities during cancer progression.

Three and Four Centre Oscillators and Their Application in Nuclear Physics

1974 ◽  
Vol 29 (7) ◽  
pp. 1003-1010 ◽  
Author(s):  
Peter Bergmann ◽  
Hans-Joachim Scheefer
Keyword(s):  
Analytical Solution ◽  
Schrödinger Equation ◽  
Nuclear Physics ◽  
Schrodinger Equation ◽  
Numerical Procedure ◽  
Orthogonal Basis ◽  
Two Dimensional ◽  
Schmidt's Method ◽  
Symmetry Properties

The extension of the nuclear two-centre-oscillator to three and four centres is investigated. Some special symmetry-properties are required. In two cases an analytical solution of the Schrödinger equation is possible. A numerical procedure is developed which enables the diagonalization of the Hamiltonian in a non-orthogonal basis without applying Schmidt's method of orthonormalization. This is important for calculations of arbitrary two-dimensional arrangements of the centres.

Planar distributions of dislocations

1968 ◽  
Vol 305 (1482) ◽  
pp. 387-404 ◽  
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Crack Nucleation ◽  
Physical Phenomenon ◽  
Continuous Distribution ◽  
Numerical Procedure ◽  
Physical Reality ◽  
Cleavage Crack ◽  
Planar Arrays ◽  
High Degree

Many problems in metal physics can be adequately described by planar arrays of dislocations, and in analysing them there are two extreme types of approach: either all the dislocations can be regarded as discrete entities, when a numerical procedure must in general be adopted, or they may be smeared into a continuous distribution, when a simple analytical solution is usually available. The present paper introduces a procedure whereby the majority of the dislocations in an array are smeared into a continuous distribution, while those that are intimately concerned with the particular physical phenomenon under consideration are allowed to remain discrete. The main advantage of this approach is that it allows for ease of solution, while preserving a high degree of physical reality. Particular types of dislocation array are considered, the accuracy of the method being demonstrated by comparing, where possible, the results with those obtained assuming that all the dislocations are discrete. Both singular integral equation and potential theory techniques are employed, and the results are applied in a brief discussion of the problem of cleavage crack nucleation in crystalline solids.

scholarly journals Statistical Inference by Confidence Intervals: Issues of Interpretation and Utilization

1999 ◽  
Vol 79 (2) ◽  
pp. 186-195 ◽  
Author(s):  
Julius Sim ◽  
Norma Reid
Keyword(s):  
Hypothesis Testing ◽  
Statistical Inference ◽  
Hypothesis Test ◽  
Statistical Significance ◽  
Aggregated Data ◽  
Additional Information ◽  
Point Estimates ◽  
Sample Statistic ◽  
Meta Analyses

Abstract This article examines the role of the confidence interval (CI) in statistical inference and its advantages over conventional hypothesis testing, particularly when data are applied in the context of clinical practice. A CI provides a range of population values with which a sample statistic is consistent at a given level of confidence (usually 95%). Conventional hypothesis testing serves to either reject or retain a null hypothesis. A CI, while also functioning as a hypothesis test, provides additional information on the variability of an observed sample statistic (ie, its precision) and on its probable relationship to the value of this statistic in the population from which the sample was drawn (ie, its accuracy). Thus, the CI focuses attention on the magnitude and the probability of a treatment or other effect. It thereby assists in determining the clinical usefulness and importance of, as well as the statistical significance of, findings. The CI is appropriate for both parametric and nonparametric analyses and for both individual studies and aggregated data in meta-analyses. It is recommended that, when inferential statistical analysis is performed, CIs should accompany point estimates and conventional hypothesis tests wherever possible.

scholarly journals M-dwarf exoplanet surface density distribution

2018 ◽  
Vol 612 ◽  
pp. L3 ◽  
Author(s):  
Michael R. Meyer ◽  
Adam Amara ◽  
Maddalena Reggiani ◽  
Sascha P. Quanz
Keyword(s):  
Density Distribution ◽  
Surface Density ◽  
Probability Distributions ◽  
Mass Range ◽  
Normal Function ◽  
Point Estimates ◽  
Points Of View ◽  
Log Normal ◽  
Parameter Values ◽  
M Dwarf

Aims. We fit a log-normal function to the M-dwarf orbital surface density distribution of gas giant planets, over the mass range 1–10 times that of Jupiter, from 0.07 to 400 AU. Methods. We used a Markov chain Monte Carlo approach to explore the likelihoods of various parameter values consistent with point estimates of the data given our assumed functional form. Results. This fit is consistent with radial velocity, microlensing, and direct-imaging observations, is well-motivated from theoretical and phenomenological points of view, and predicts results of future surveys. We present probability distributions for each parameter and a maximum likelihood estimate solution. Conclusions. We suggest that this function makes more physical sense than other widely used functions, and we explore the implications of our results on the design of future exoplanet surveys.

Estimation of the Time-Dependent Body Force Needed to Exert on a Membrane to Reach a Desired State at the Final Time

2019 ◽  
Vol 19 (2) ◽  
pp. 323-339
Author(s):  
Lyubomir Boyadjiev ◽  
Kamal Rashedi ◽  
Mourad Sini
Keyword(s):  
Body Force ◽  
Single Point ◽  
Finite Size ◽  
Numerical Procedure ◽  
Time Dependent ◽  
Final Time ◽  
Initial Shape ◽  
Additional Information ◽  
Boundary Force ◽  
Hölder Stability

AbstractWe are concerned with the wave propagation in a homogeneous 2D or 3D membrane Ω of finite size. We assume that either the membrane is initially at rest or we know its initial shape (but not necessarily both) and its boundary is subject to a known boundary force. We address the question of estimating the needed time-dependent body force to exert on the membrane to reach a desired state at a given final time T. As an additional information, we ask for the displacement on the boundary. We consider the displacement either at a single point of the boundary or on the whole boundary. First, we show the uniqueness of solution of these inverse problems under natural conditions on the final time T. If, in addition, the displacement on the whole boundary is only time dependent (which means that the boundary moves with a constant speed), this condition on T is removed if Ω satisfies Schiffer’s property. Second, we derive a conditional Hölder stability inequality for estimating such a time-dependent force. Third, we propose a numerical procedure based on the application of the satisfier function along with the standard Fourier expansion of the solution to the problems. Numerical tests are given to illustrate the applicability of the proposed procedure.

scholarly journals On the Approximation of Saddlepoint Expansions in Statistics

1994 ◽  
Vol 10 (5) ◽  
pp. 900-916 ◽  
Author(s):  
Offer Lieberman
Keyword(s):  
Analytical Solution ◽  
Explicit Knowledge ◽  
Probability Distributions ◽  
Saddlepoint Approximation ◽  
Accurate Method ◽  
Test Statistic ◽  
Watson Test

The saddlepoint approximation as developed by Daniels [3] is an extremely accurate method for approximating probability distributions. Econometric and statistical applications of the technique to densities of statistics of interest are often hindered by the requirements of explicit knowledge of the c.g.f. and the need to obtain an analytical solution to the saddlepoint defining equation. In this paper, we show the conditions under which any approximation to the saddlepoint is justified and suggest a convenient solution. We illustrate with an approximate saddlepoint expansion of the Durbin-Watson test statistic.

Sign in / Sign up

Export Citation Format

Share Document