{"title":"攻击不确定性下的高斯估计","authors":"T. Javidi, Y. Kaspi, Himanshu Tyagi","doi":"10.1109/ITW.2015.7133120","DOIUrl":null,"url":null,"abstract":"We consider the estimation of a standard Gaussian random variable under an observation attack where an adversary may add a zero mean Gaussian noise with variance in a bounded, closed interval to an otherwise noiseless observation. A straightforward approach would entail either ignoring the attack and simply using an optimal estimator under normal operation or taking the worst-case attack into account and using a minimax estimator that minimizes the cost under the worst-case attack. In contrast, we seek to characterize the optimal tradeoff between the MSE under normal operation and the MSE under the worst-case attack. Equivalently, we seek a minimax estimator for any fixed prior probability of attack. Our main result shows that a unique minimax estimator exists for every fixed probability of attack and is given by the Bayesian estimator for a least-favorable prior on the set of possible variances. Furthermore, the least-favorable prior is unique and has a finite support. While the minimax estimator is linear when the probability of attack is 0 or 1, our numerical results show that the minimax linear estimator is far from optimal for all other probabilities of attack and a simple nonlinear estimator does much better.","PeriodicalId":174797,"journal":{"name":"2015 IEEE Information Theory Workshop (ITW)","volume":"60 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2015-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Gaussian estimation under attack uncertainty\",\"authors\":\"T. Javidi, Y. Kaspi, Himanshu Tyagi\",\"doi\":\"10.1109/ITW.2015.7133120\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the estimation of a standard Gaussian random variable under an observation attack where an adversary may add a zero mean Gaussian noise with variance in a bounded, closed interval to an otherwise noiseless observation. A straightforward approach would entail either ignoring the attack and simply using an optimal estimator under normal operation or taking the worst-case attack into account and using a minimax estimator that minimizes the cost under the worst-case attack. In contrast, we seek to characterize the optimal tradeoff between the MSE under normal operation and the MSE under the worst-case attack. Equivalently, we seek a minimax estimator for any fixed prior probability of attack. Our main result shows that a unique minimax estimator exists for every fixed probability of attack and is given by the Bayesian estimator for a least-favorable prior on the set of possible variances. Furthermore, the least-favorable prior is unique and has a finite support. While the minimax estimator is linear when the probability of attack is 0 or 1, our numerical results show that the minimax linear estimator is far from optimal for all other probabilities of attack and a simple nonlinear estimator does much better.\",\"PeriodicalId\":174797,\"journal\":{\"name\":\"2015 IEEE Information Theory Workshop (ITW)\",\"volume\":\"60 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2015-04-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2015 IEEE Information Theory Workshop (ITW)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/ITW.2015.7133120\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2015 IEEE Information Theory Workshop (ITW)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ITW.2015.7133120","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We consider the estimation of a standard Gaussian random variable under an observation attack where an adversary may add a zero mean Gaussian noise with variance in a bounded, closed interval to an otherwise noiseless observation. A straightforward approach would entail either ignoring the attack and simply using an optimal estimator under normal operation or taking the worst-case attack into account and using a minimax estimator that minimizes the cost under the worst-case attack. In contrast, we seek to characterize the optimal tradeoff between the MSE under normal operation and the MSE under the worst-case attack. Equivalently, we seek a minimax estimator for any fixed prior probability of attack. Our main result shows that a unique minimax estimator exists for every fixed probability of attack and is given by the Bayesian estimator for a least-favorable prior on the set of possible variances. Furthermore, the least-favorable prior is unique and has a finite support. While the minimax estimator is linear when the probability of attack is 0 or 1, our numerical results show that the minimax linear estimator is far from optimal for all other probabilities of attack and a simple nonlinear estimator does much better.