Pitfalls of Estimating the Marginal Likelihood Using the Modified Harmonic Mean

2015 ◽  
Author(s):  
Joshua C. C. Chan ◽  
Angelia Grant
Keyword(s):  
Marginal Likelihood ◽  
Harmonic Mean

scholarly journals Integration with an adaptive harmonic mean algorithm

2020 ◽  
Vol 35 (24) ◽  
pp. 1950142
Author(s):  
Allen Caldwell ◽  
Philipp Eller ◽  
Vasyl Hafych ◽  
Rafael Schick ◽  
Oliver Schulz ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Distribution ◽  
Marginal Likelihood ◽  
Harmonic Mean ◽  
High Dimensional ◽  
Test Cases ◽  
Multiple Test ◽  
Mathematical Properties

Numerically estimating the integral of functions in high dimensional spaces is a nontrivial task. A oft-encountered example is the calculation of the marginal likelihood in Bayesian inference, in a context where a sampling algorithm such as a Markov Chain Monte Carlo provides samples of the function. We present an Adaptive Harmonic Mean Integration (AHMI) algorithm. Given samples drawn according to a probability distribution proportional to the function, the algorithm will estimate the integral of the function and the uncertainty of the estimate by applying a harmonic mean estimator to adaptively chosen regions of the parameter space. We describe the algorithm and its mathematical properties, and report the results using it on multiple test cases.

Pitfalls of estimating the marginal likelihood using the modified harmonic mean

2015 ◽  
Vol 131 ◽  
pp. 29-33 ◽  
Author(s):  
Joshua C.C. Chan ◽  
Angelia L. Grant
Keyword(s):  
Marginal Likelihood ◽  
Harmonic Mean

scholarly journals On Certain Differential Subordination of Harmonic Mean Related to a Linear Function

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 966
Author(s):  
Anna Dobosz ◽  
Piotr Jastrzębski ◽  
Adam Lecko
Keyword(s):  
Convex Function ◽  
Linear Function ◽  
Differential Subordination ◽  
Harmonic Mean ◽  
Best Dominant ◽  
Symmetry Properties ◽  
Differential Subordinations ◽  
Difficult Issue

In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot–Bouquet differential subordination.

scholarly journals Linkage disequilibrium and genetic variances under mutation-selection balance.

Genetics ◽  
1989 ◽  
Vol 121 (4) ◽  
pp. 857-860 ◽  
Author(s):  
A Hastings
Keyword(s):  
Linkage Disequilibrium ◽  
Mutation Rate ◽  
Genetic Variance ◽  
Harmonic Mean ◽  
Recombination Rates ◽  
Genetic Variances ◽  
Locus Selection

Abstract I determine the contribution of linkage disequilibrium to genetic variances using results for two loci and for induced or marginal systems. The analysis allows epistasis and dominance, but assumes that mutation is weak relative to selection. The linkage disequilibrium component of genetic variance is shown to be unimportant for unlinked loci if the gametic mutation rate divided by the harmonic mean of the pairwise recombination rates is much less than one. For tightly linked loci, linkage disequilibrium is unimportant if the gametic mutation rate divided by the (induced) per locus selection is much less than one.

scholarly journals Markov-Modulated Continuous-Time Markov Chains to Identify Site- and Branch-Specific Evolutionary Variation in BEAST

2020 ◽  
Vol 70 (1) ◽  
pp. 181-189
Author(s):  
Guy Baele ◽  
Mandev S Gill ◽  
Paul Bastide ◽  
Philippe Lemey ◽  
Marc A Suchard
Keyword(s):  
High Performance ◽  
Markov Models ◽  
Marginal Likelihood ◽  
Likelihood Estimation ◽  
Phylogenetic Inference ◽  
Time Variability ◽  
Substitution Process ◽  
Wide Range ◽  
Markov Modulated ◽  
Over Time

Abstract Markov models of character substitution on phylogenies form the foundation of phylogenetic inference frameworks. Early models made the simplifying assumption that the substitution process is homogeneous over time and across sites in the molecular sequence alignment. While standard practice adopts extensions that accommodate heterogeneity of substitution rates across sites, heterogeneity in the process over time in a site-specific manner remains frequently overlooked. This is problematic, as evolutionary processes that act at the molecular level are highly variable, subjecting different sites to different selective constraints over time, impacting their substitution behavior. We propose incorporating time variability through Markov-modulated models (MMMs), which extend covarion-like models and allow the substitution process (including relative character exchange rates as well as the overall substitution rate) at individual sites to vary across lineages. We implement a general MMM framework in BEAST, a popular Bayesian phylogenetic inference software package, allowing researchers to compose a wide range of MMMs through flexible XML specification. Using examples from bacterial, viral, and plastid genome evolution, we show that MMMs impact phylogenetic tree estimation and can substantially improve model fit compared to standard substitution models. Through simulations, we show that marginal likelihood estimation accurately identifies the generative model and does not systematically prefer the more parameter-rich MMMs. To mitigate the increased computational demands associated with MMMs, our implementation exploits recent developments in BEAGLE, a high-performance computational library for phylogenetic inference. [Bayesian inference; BEAGLE; BEAST; covarion, heterotachy; Markov-modulated models; phylogenetics.]

On Representing an Integer as the Harmonic Mean of Integers

1973 ◽  
Vol 46 (5) ◽  
pp. 241-244
Author(s):  
Solomon W. Golomb
Keyword(s):  
Harmonic Mean

Hyperparameter Estimation Based on Gaussian Process and its Application in Injection Molding

2011 ◽  
Vol 328-330 ◽  
pp. 524-529
Author(s):  
Jun Yan Ma ◽  
Xiao Ping Liao ◽  
Wei Xia ◽  
Xue Lian Yan
Keyword(s):  
Injection Molding ◽  
Gaussian Process ◽  
Process Model ◽  
Marginal Likelihood ◽  
Best Value ◽  
Highly Nonlinear ◽  
General Scientific ◽  
Wolfe Line Search ◽  
Search Approach ◽  
Hyperparameter Estimation

As a powerful modeling tool, Gaussian process (GP) employs a Bayesian statistics approach and adopts a highly nonlinear regression technique for general scientific and engineering tasks. In the first step of constructing Gaussian process model is to estimate the best value of the hyperparameter which turned to be used in the second step where a satisfactory nonlinear model was fitted. In this paper, a modified Wolfe line search approach for hyperparameters estimation by maximizing the marginal likelihood based on conjugate gradient method is proposed. And then we analyze parameter correlation according to the value of hyperparameters to control the warpage which is a main defect for a thin shell structure part in injection molding.

A Neutral Model With Fluctuating Population Size and Its Effective Size

Genetics ◽  
2002 ◽  
Vol 161 (1) ◽  
pp. 381-388
Author(s):  
Masaru Iizuka ◽  
Hidenori Tachida ◽  
Hirotsugu Matsuda
Keyword(s):  
Asymptotic Behavior ◽  
Population Size ◽  
Diffusion Model ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Effective Size ◽  
Neutral Model ◽  
Average Heterozygosity ◽  
Special Case ◽  
Concrete Model

Abstract We consider a diffusion model with neutral alleles whose population size is fluctuating randomly. For this model, the effects of fluctuation of population size on the effective size are investigated. The effective size defined by the equilibrium average heterozygosity is larger than the harmonic mean of population size but smaller than the arithmetic mean of population size. To see explicitly the effects of fluctuation of population size on the effective size, we investigate a special case where population size fluctuates between two distinct states. In some cases, the effective size is very different from the harmonic mean. For this concrete model, we also obtain the stationary distribution of the average heterozygosity. Asymptotic behavior of the effective size is obtained when the population size is large and/or autocorrelation of the fluctuation is weak or strong.

Marginal likelihood, conditional likelihood and empirical likelihood: Connections and applications

Biometrika ◽  
2005 ◽  
Vol 92 (2) ◽  
pp. 251-270 ◽  
Author(s):  
Jing Qin ◽  
Biao Zhang
Keyword(s):  
Empirical Likelihood ◽  
Marginal Likelihood ◽  
Conditional Likelihood
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