scholarly journals Unbiased estimation of the gradient of the log-likelihood in inverse problems

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Ajay Jasra ◽  
Kody J. H. Law ◽  
Deng Lu
Keyword(s):  
Numerical Approximation ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Multilevel Monte Carlo ◽  
Gradient Algorithms ◽  
Log Likelihood ◽  
Bayesian Inverse Problem ◽  
Unbiased Estimates ◽  
The Cost ◽  
Single Processor

AbstractWe consider the problem of estimating a parameter $$\theta \in \Theta \subseteq {\mathbb {R}}^{d_{\theta }}$$ θ ∈ Θ ⊆ R d θ associated with a Bayesian inverse problem. Typically one must resort to a numerical approximation of gradient of the log-likelihood and also adopt a discretization of the problem in space and/or time. We develop a new methodology to unbiasedly estimate the gradient of the log-likelihood with respect to the unknown parameter, i.e. the expectation of the estimate has no discretization bias. Such a property is not only useful for estimation in terms of the original stochastic model of interest, but can be used in stochastic gradient algorithms which benefit from unbiased estimates. Under appropriate assumptions, we prove that our estimator is not only unbiased but of finite variance. In addition, when implemented on a single processor, we show that the cost to achieve a given level of error is comparable to multilevel Monte Carlo methods, both practically and theoretically. However, the new algorithm is highly amenable to parallel computation.

scholarly journals Unbiased estimation of the solution to Zakai’s equation

2020 ◽  
Vol 26 (2) ◽  
pp. 113-129
Author(s):  
Hamza M. Ruzayqat ◽  
Ajay Jasra
Keyword(s):  
Continuous Time ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Linear Filtering ◽  
Multilevel Monte Carlo ◽  
Zakai Equation ◽  
Unbiased Estimators ◽  
First Order ◽  
Multilevel Monte Carlo Method ◽  
Filtering Problem

AbstractIn the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.

Unbiased Estimation of Total Rural Loans Advanced and Incurred in an Indian State Along with Unbiased Estimation of Variance in Estimation

2017 ◽  
Vol 69 (1) ◽  
pp. 71-75
Author(s):  
Arijit Chaudhuri
Keyword(s):  
Sampling Technique ◽  
Unbiased Estimation ◽  
Approximately Unbiased ◽  
Rural Farmers ◽  
Statistical Institute ◽  
Indian State ◽  
Unbiased Estimates ◽  
Indian Statistical ◽  
Year 2000 ◽  
Estimation Of Variance

Around the year 2000, the problem of reconciling the estimate of loans advanced by the banks and the estimate of loans incurred by the rural farmers was studied in the Indian Statistical Institute. Some approximately unbiased estimates were examined along with approximately unbiased estimates of their approximate variances. Utilizing “Constrained Network” sampling technique exactly unbiased counterparts are presented as alternatives.

scholarly journals HEDGE-FUND MANAGEMENT WITH LIQUIDITY CONSTRAINT

2019 ◽  
Vol 22 (06) ◽  
pp. 1950026
Author(s):  
HUGO E. RAMIREZ ◽  
PETER DUCK ◽  
PAUL V. JOHNSON ◽  
SYDNEY HOWELL
Keyword(s):  
Numerical Approximation ◽  
Optimal Allocation ◽  
Hedge Fund ◽  
Rate Of Return ◽  
Liquidity Constraint ◽  
Control Techniques ◽  
Trading Decisions ◽  
The Difference ◽  
Hamilton Jacobi Bellman ◽  
The Cost

We propose a model for a manager of a hedge fund with a liquidity constraint, where he is seeking to optimize his utility of wealth, with one and multiple period horizons. By using stochastic control techniques, we state the corresponding multi-dimensional Hamilton–Jacobi–Bellman partial differential equation and we use a robust numerical approximation to obtain its unique viscosity solution. We examine the effects of the liquidity constraint on managerial trading decisions and optimal allocation, finding that the manager behaves in a less risky manner. We also calculate the cost of being at sub-optimal positions as the difference in the certainty equivalent payoff for the manager. Moreover, we compare the values of a benchmark hedge fund with another one having a risky asset with a higher rate of return but less liquidity, finding that higher rate of return with a liquidity constraint does not always lead to greater return.

scholarly journals Modeling Parent-Specific Genetic Nurture in Families with Missing Parental Genotypes: Application to Birthweight and BMI

2020 ◽  
Author(s):  
Justin D. Tubbs ◽  
Liang-Dar Hwang ◽  
Justin Luong ◽  
David M. Evans ◽  
Pak C. Sham
Keyword(s):  
Genetic Effects ◽  
Unbiased Estimation ◽  
Paternal Effects ◽  
Genetic Transmission ◽  
Nuclear Families ◽  
Genome Wide ◽  
Unbiased Estimates ◽  
Related Individuals ◽  
Missing Genotypes ◽  
The Uk

AbstractDisaggregation and estimation of genetic effects from offspring and parents has long been of interest to statistical geneticists. Recently, technical and methodological advances have made the genome-wide and loci-specific estimation of direct offspring and parental genetic nurture effects more possible. However, unbiased estimation using these methods requires datasets where both parents and at least one child have been genotyped, which are relatively scarce. Our group has recently developed a method and accompanying software (IMPISH; Hwang et al., 2020) which is able to impute missing parental genotypes from observed data on sibships and estimate their effects on an offspring phenotype conditional on the effects of genetic transmission. However, this method is unable to disentangle maternal and paternal effects, which may differ in magnitude and direction. Here, we introduce an extension to the original IMPISH routine which takes advantage of all available nuclear families to impute parent-specific missing genotypes and obtain asymptotically unbiased estimates of genetic effects on offspring phenotypes. We apply this this method to data from related individuals in the UK Biobank, showing concordance with previous estimates of maternal genetic effects on offspring birthweight. We also conduct the first GWAS jointly estimating offspring-, maternal-, and paternal-specific genetic effects on body mass index.

scholarly journals New high order FDTD method to solve EMC problems

2015 ◽  
Vol 4 (2) ◽  
pp. 17 ◽  
Author(s):  
N. Deymier ◽  
T. Volpert ◽  
X. Ferrieres ◽  
V. Mouysset ◽  
B. Pecqueux
Keyword(s):  
Electromagnetic Compatibility ◽  
Numerical Approximation ◽  
Fdtd Method ◽  
Higher Order ◽  
Small Cell ◽  
Small Cell Size ◽  
Dg Method ◽  
The Time Domain ◽  
Spatial Approximation ◽  
The Cost

In electromagnetic compatibility (EMC) context, we are interested in developing new ac- curate methods to solve efficiently and accurately Maxwell’s equations in the time domain. Indeed, usual methods such as FDTD or FVTD present im- portant dissipative and/or dispersive errors which prevent to obtain a good numerical approximation of the physical solution for a given industrial scene unless we use a mesh with a very small cell size. To avoid this problem, schemes like the Discontinuous Galerkin (DG) method, based on higher order spa- tial approximations, have been introduced and stud- ied on unstructured meshes. However the cost of this kind of method can become prohibitive accord- ing to the mesh used. In this paper, we first present a higher order spatial approximation method on carte- sian meshes. It is based on a finite element ap- proach and recovers at the order 1 the well-known Yee’s schema. Next, to deal with EMC problem, a non-oriented thin wire formalism is proposed for this method. Finally, several examples are given to present the benefits of this new method by compar- ison with both Yee’s schema and DG approaches.

scholarly journals Multivariate General Spatial Three-Stage Least Squares Fixed Effect Panel Simultaneous Models and Estimation of Their Parameters

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Least Squares ◽  
Numerical Approximation ◽  
Generalized Least Squares ◽  
Simultaneous Equation ◽  
Fixed Effect ◽  
Log Likelihood ◽  
New Models ◽  
Spatial Autoregressive ◽  
Three Stage Least Squares ◽  
Simultaneous Equation Models

Simultaneous equation models describe a two-way flow of influence among variables. Simultaneous equation models using panel data, especially for fixed effect where there are spatial autoregressive and spatial errors with exact solutions, still require to be developed. In this paper, we develop the new models that it consist of spatial autoregressive and spatial errors. We call it as general spatial. This paper proposes feasible generalized least squares-three-stage least squares (FGLS-3SLS) to find all the estimators with exact solution and the numerical approximation estimators by concentrated log-likelihood formulation with method of forming sequence. All proposed estimators especially for closed-form estimators are proved to be consistent.

Edge-correction needs in estimating indices of spatial forest structure

2006 ◽  
Vol 36 (7) ◽  
pp. 1723-1739 ◽  
Author(s):  
Arne Pommerening ◽  
Dietrich Stoyan
Keyword(s):  
Forest Structure ◽  
Estimation Error ◽  
Buffer Zone ◽  
Nearest Neighbour ◽  
Reflection Method ◽  
Limited Size ◽  
Neighbour Effects ◽  
Unbiased Estimates ◽  
The Cost ◽  
Algorithmic Structure

Indices quantifying spatial forest structure are frequently used to monitor spatial aspects of tree attributes including biodiversity in research plots of limited size. The treatment of edge trees, which are close to the plot boundaries, can affect the estimation of such indices that include neighbour effects, since some of their neighbours are likely to fall outside the plot. This paper investigates whether and under what circumstances edge-correction methods are necessary and evaluates the performance of six different approaches: no edge correction, translation, reflection, buffer zone, and two new nearest-neighbour methods. The performance of edge-correction methods depends strongly on the algorithmic structure of the indices and the spatial pattern of tree positions involved. Some edge-correction methods introduce more error than ignoring edge bias altogether. For indices accounting for the diversity of tree positions and especially for those computing angles, translation or buffer zone methods reduce the estimation error regardless of the sample size. The use of the reflection method is associated with large bias values. One of the new nearest-neighbour edge-correction methods proves to be capable of reducing the bias considerably. The results confirm the need for sufficiently large monitoring plots to avoid bias from edge effects. Where this is impossible, neighbours beyond the plot boundary need to be included in the survey, thus providing unbiased estimates but at the cost of extra measurements. Sensitivity analysis is required for newly introduced indices prior to their first application.

Simultaneous unbiased estimates of multiple downed wood attributes in perpendicular distance sampling

2008 ◽  
Vol 38 (7) ◽  
pp. 2044-2051 ◽  
Author(s):  
Mark J. Ducey ◽  
Michael S. Williams ◽  
Jeffrey H. Gove ◽  
Harry T. Valentine
Keyword(s):  
Surface Area ◽  
Unbiased Estimator ◽  
Distance Sampling ◽  
Area Coverage ◽  
Unbiased Estimation ◽  
General Design ◽  
Perpendicular Distance ◽  
Simple Measurement ◽  
Unbiased Estimates ◽  
Downed Wood

Perpendicular distance sampling (PDS) is a fast probability-proportional-to-size method for inventory of downed wood. However, previous development of PDS had limited the method to estimating only one variable (such as volume per hectare, or surface area per hectare) at a time. Here, we develop a general design-unbiased estimator for PDS. We then show how that estimator can be used to develop simple measurement protocols that allow simultaneous, unbiased estimation of multiple downed wood variables, including logs per hectare, length of logs per hectare, surface area or area coverage per hectare, and volume per hectare.

scholarly journals Multilevel Monte Carlo methods

Acta Numerica ◽  
2015 ◽  
Vol 24 ◽  
pp. 259-328 ◽  
Author(s):  
Michael B. Giles
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Low Cost ◽  
Computational Cost ◽  
High Accuracy ◽  
Stochastic Pdes ◽  
Multilevel Monte Carlo ◽  
The Cost ◽  
Multilevel Monte Carlo Method ◽  
Very High

Monte Carlo methods are a very general and useful approach for the estimation of expectations arising from stochastic simulation. However, they can be computationally expensive, particularly when the cost of generating individual stochastic samples is very high, as in the case of stochastic PDEs. Multilevel Monte Carlo is a recently developed approach which greatly reduces the computational cost by performing most simulations with low accuracy at a correspondingly low cost, with relatively few simulations being performed at high accuracy and a high cost.In this article, we review the ideas behind the multilevel Monte Carlo method, and various recent generalizations and extensions, and discuss a number of applications which illustrate the flexibility and generality of the approach and the challenges in developing more efficient implementations with a faster rate of convergence of the multilevel correction variance.

scholarly journals pixy: Unbiased estimation of nucleotide diversity and divergence in the presence of missing data

2020 ◽  
Author(s):  
Katharine L Korunes ◽  
Kieran Samuk
Keyword(s):  
Population Genetics ◽  
Missing Data ◽  
Population Genetic ◽  
Unbiased Estimation ◽  
Systematic Bias ◽  
Summary Statistics ◽  
Reference Allele ◽  
Genetic Analyses ◽  
Unbiased Estimates ◽  
Population Genetic Analyses

AbstractPopulation genetic analyses often use summary statistics to describe patterns of genetic variation and provide insight into evolutionary processes. Among the most fundamental of these summary statistics are π and dXY, which are used to describe genetic diversity within and between populations, respectively. Here, we address a widespread issue in π and dXY calculation: systematic bias generated by missing data of various types. Many popular methods for calculating π and dXY operate on data encoded in the Variant Call Format (VCF), which condenses genetic data by omitting invariant sites. When calculating π and dXY using a VCF, it is often implicitly assumed that missing genotypes (including those at sites not represented in the VCF) are homozygous for the reference allele. Here, we show how this assumption can result in substantial downward bias in estimates of π and dXY that is directly proportional to the amount of missing data. We discuss the pervasive nature and importance of this problem in population genetics, and introduce a user-friendly UNIX command line utility, pixy, that solves this problem via an algorithm that generates unbiased estimates of π and dXY in the face of missing data. We compare pixy to existing methods using both simulated and empirical data, and show that pixy alone produces unbiased estimates of π and dXY regardless of the form or amount of missing data. In sum, our software solves a long-standing problem in applied population genetics and highlights the importance of properly accounting for missing data in population genetic analyses.

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