{"title":"CALCULATING PROBABILITY DENSITIES WITH HOMOTOPY, AND APPLICATIONS TO PARTICLE FILTERS","authors":"Juan Restrepo","doi":"10.1615/int.j.uncertaintyquantification.2022038553","DOIUrl":null,"url":null,"abstract":"We explore a homotopy sampling procedure and its generalization, loosely based on importance sampling, known as annealed importance sampling. The procedure makes use of a known probability distribution to\nfind, via homotopy, the unknown normalization of a target distribution, as well as samples of the target distribution.\nIn the context of stationary distributions that are associated with physical systems the method is an alternative way to estimate an unknown microcanonical ensemble.\nWe make the connection between the homotopy and a dynamics problem explicit. Further, we propose a reformulation of the method that leads to a rejection sampling alternative. We derive the error incurred in computing the target distribution normalization, when sample inversion is not possible.\nThe error in the procedure depends on the errors incurred in sample averaging and the number of stages used in the\ncomputational implementation of the process. However, we show that it is possible to exchange the number of homotopy stages and the total number of samples needed at each stage in order to enhance the computational efficiency of the implemented algorithm. Estimates of the error as a function of stages and sample averages are derived. These could guide computational efficiency decisions on how the calculation would be mapped to a given computer architecture.\nConsideration is given to how the procedure can be adapted to Bayesian estimation problems, both stationary and non-stationary. The connection between homotopy sampling and thermodynamic integration is made. Emphasis is placed on the non-stationary problems, and in particular, on a sequential estimation technique know","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":"94 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal for Uncertainty Quantification","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1615/int.j.uncertaintyquantification.2022038553","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We explore a homotopy sampling procedure and its generalization, loosely based on importance sampling, known as annealed importance sampling. The procedure makes use of a known probability distribution to
find, via homotopy, the unknown normalization of a target distribution, as well as samples of the target distribution.
In the context of stationary distributions that are associated with physical systems the method is an alternative way to estimate an unknown microcanonical ensemble.
We make the connection between the homotopy and a dynamics problem explicit. Further, we propose a reformulation of the method that leads to a rejection sampling alternative. We derive the error incurred in computing the target distribution normalization, when sample inversion is not possible.
The error in the procedure depends on the errors incurred in sample averaging and the number of stages used in the
computational implementation of the process. However, we show that it is possible to exchange the number of homotopy stages and the total number of samples needed at each stage in order to enhance the computational efficiency of the implemented algorithm. Estimates of the error as a function of stages and sample averages are derived. These could guide computational efficiency decisions on how the calculation would be mapped to a given computer architecture.
Consideration is given to how the procedure can be adapted to Bayesian estimation problems, both stationary and non-stationary. The connection between homotopy sampling and thermodynamic integration is made. Emphasis is placed on the non-stationary problems, and in particular, on a sequential estimation technique know
期刊介绍:
The International Journal for Uncertainty Quantification disseminates information of permanent interest in the areas of analysis, modeling, design and control of complex systems in the presence of uncertainty. The journal seeks to emphasize methods that cross stochastic analysis, statistical modeling and scientific computing. Systems of interest are governed by differential equations possibly with multiscale features. Topics of particular interest include representation of uncertainty, propagation of uncertainty across scales, resolving the curse of dimensionality, long-time integration for stochastic PDEs, data-driven approaches for constructing stochastic models, validation, verification and uncertainty quantification for predictive computational science, and visualization of uncertainty in high-dimensional spaces. Bayesian computation and machine learning techniques are also of interest for example in the context of stochastic multiscale systems, for model selection/classification, and decision making. Reports addressing the dynamic coupling of modern experiments and modeling approaches towards predictive science are particularly encouraged. Applications of uncertainty quantification in all areas of physical and biological sciences are appropriate.