scholarly journals The Optimal Error of Monte Carlo Integration

1995 ◽  
Vol 11 (4) ◽  
pp. 394-415 ◽  
Author(s):  
Peter Mathé
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Integration ◽  
Optimal Error

scholarly journals Control variate selection for Monte Carlo integration

2021 ◽  
Vol 31 (4) ◽  
Author(s):  
Rémi Leluc ◽  
François Portier ◽  
Johan Segers
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Integration ◽  
Control Variate ◽  
Selection For

scholarly journals Estimating effective population size from samples of sequences: a bootstrap Monte Carlo integration method

1992 ◽  
Vol 60 (3) ◽  
pp. 209-220 ◽  
Author(s):  
Joseph Felsenstein
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood ◽  
Population Size ◽  
Effective Population Size ◽  
Random Mating ◽  
Computational Effort ◽  
Monte Carlo Integration ◽  
Bootstrap Sampling ◽  
Effective Population ◽  
Full Data

SummaryWe would like to use maximum likelihood to estimate parameters such as the effective population size Ne, or, if we do not know mutation rates, the product 4Neμof mutation rate per site and effective population size. To compute the likelihood for a sample of unrecombined nucleotide sequences taken from a random-mating population it is necessary to sum over all genealogies that could have led to the sequences, computing for each one the probability that it would have yielded the sequences, and weighting each one by its prior probability. The genealogies vary in tree topology and in branch lengths. Although the likelihood and the prior are straightforward to compute, the summation over all genealogies seems at first sight hopelessly difficult. This paper reports that it is possible to carry out a Monte Carlo integration to evaluate the likelihoods pproximately. The method uses bootstrap sampling of sites to create data sets for each of which a maximum likelihood tree is estimated. The resulting trees are assumed to be sampled from a distribution whose height is proportional to the likelihood surface for the full data. That it will be so is dependent on a theorem which is not proven, but seems likely to be true if the sequences are not short. One can use the resulting estimated likelihood curve to make a maximum likelihood estimate of the parameter of interest, Ne or of 4Neμ. The method requires at least 100 times the computational effort required for estimation of a phylogeny by maximum likelihood, but is practical on today's work stations. The method does not at present have any way of dealing with recombination.

scholarly journals A neural network approach to estimating a posteriori distributions of Bayesian retrieval problems

2018 ◽  
Vol 11 (8) ◽  
pp. 4627-4643 ◽  
Author(s):  
Simon Pfreundschuh ◽  
Patrick Eriksson ◽  
David Duncan ◽  
Bengt Rydberg ◽  
Nina Håkansson ◽  
...  
Keyword(s):  
Neural Network ◽  
Monte Carlo ◽  
Monte Carlo Integration ◽  
A Posteriori ◽  
Retrieval Performance ◽  
Neural Network Approach ◽  
Novel Approach ◽  
Uncertainty Estimates ◽  
Non Gaussian ◽  
Similar Accuracy

Abstract. A neural-network-based method, quantile regression neural networks (QRNNs), is proposed as a novel approach to estimating the a posteriori distribution of Bayesian remote sensing retrievals. The advantage of QRNNs over conventional neural network retrievals is that they learn to predict not only a single retrieval value but also the associated, case-specific uncertainties. In this study, the retrieval performance of QRNNs is characterized and compared to that of other state-of-the-art retrieval methods. A synthetic retrieval scenario is presented and used as a validation case for the application of QRNNs to Bayesian retrieval problems. The QRNN retrieval performance is evaluated against Markov chain Monte Carlo simulation and another Bayesian method based on Monte Carlo integration over a retrieval database. The scenario is also used to investigate how different hyperparameter configurations and training set sizes affect the retrieval performance. In the second part of the study, QRNNs are applied to the retrieval of cloud top pressure from observations by the Moderate Resolution Imaging Spectroradiometer (MODIS). It is shown that QRNNs are not only capable of achieving similar accuracy to standard neural network retrievals but also provide statistically consistent uncertainty estimates for non-Gaussian retrieval errors. The results presented in this work show that QRNNs are able to combine the flexibility and computational efficiency of the machine learning approach with the theoretically sound handling of uncertainties of the Bayesian framework. Together with this article, a Python implementation of QRNNs is released through a public repository to make the method available to the scientific community.

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