LEADER 05041nam 22008655 450 001 9910986137903321 005 20251107122524.0 010 $a9789819788804 024 7 $a10.1007/978-981-97-8880-4 035 $a(CKB)37836602700041 035 $a(MiAaPQ)EBC31957540 035 $a(Au-PeEL)EBL31957540 035 $a(DE-He213)978-981-97-8880-4 035 $a(EXLCZ)9937836602700041 100 $a20250312d2025 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aMinimum Gamma-Divergence for Regression and Classification Problems /$fby Shinto Eguchi 205 $a1st ed. 2025. 210 1$aSingapore :$cSpringer Nature Singapore :$cImprint: Springer,$d2025. 215 $a1 online resource (212 pages) 225 1 $aJSS Research Series in Statistics,$x2364-0065 311 08$a9789819788798 327 $a1. Introduction -- 2. Framework of gamma-divergence -- 2.1. Scale invariance -- 2.2 GM divergence and HM divergence -- 3. Minimum divergence methods for generalized linear models -- 3.1. Bernoulli logistic model -- 3.2. Poisson log-linear model -- 3.3. Poisson point process model -- 4. Minimum divergence methods in machine leaning -- 4.1. Multi-class AdaBoost -- 4.2. Boltzmann machine -- 5. gamma-divergence for real valued functions -- 6. Discussion. 330 $aThis book introduces the gamma-divergence, a measure of distance between probability distributions that was proposed by Fujisawa and Eguchi in 2008. The gamma-divergence has been extensively explored to provide robust estimation when the power index ? is positive. The gamma-divergence can be defined even when the power index ? is negative, as long as the condition of integrability is satisfied. Thus, the authors consider the gamma-divergence defined on a set of discrete distributions. The arithmetic, geometric, and harmonic means for the distribution ratios are closely connected with the gamma-divergence with a negative ?. In particular, the authors call the geometric-mean (GM) divergence the gamma-divergence when ? is equal to -1. The book begins by providing an overview of the gamma-divergence and its properties. It then goes on to discuss the applications of the gamma-divergence in various areas, including machine learning, statistics, and ecology. Bernoulli, categorical, Poisson, negative binomial, and Boltzmann distributions are discussed as typical examples. Furthermore, regression analysis models that explicitly or implicitly assume these distributions as the dependent variable in generalized linear models are discussed to apply the minimum gamma-divergence method. In ensemble learning, AdaBoost is derived by the exponential loss function in the weighted majority vote manner. It is pointed out that the exponential loss function is deeply connected to the GM divergence. In the Boltzmann machine, the maximum likelihood has to use approximation methods such as mean field approximation because of the intractable computation of the partition function. However, by considering the GM divergence and the exponential loss, it is shown that the calculation of the partition function is not necessary, and it can be executed without variational inference. . 410 0$aJSS Research Series in Statistics,$x2364-0065 606 $aStatistics 606 $aStochastic models 606 $aMathematical statistics 606 $aMachine learning 606 $aRegression analysis 606 $aBiometry 606 $aStatistical Theory and Methods 606 $aStochastic Modelling in Statistics 606 $aParametric Inference 606 $aMachine Learning 606 $aLinear Models and Regression 606 $aBiostatistics 606 $aEstadística$2thub 606 $aEstadística matemàtica$2thub 606 $aAprenentatge automàtic$2thub 606 $aAnàlisi de regressió$2thub 606 $aBiometria$2thub 606 $aModels lineals (Estadística)$2thub 608 $aLlibres electrònics$2thub 615 0$aStatistics. 615 0$aStochastic models. 615 0$aMathematical statistics. 615 0$aMachine learning. 615 0$aRegression analysis. 615 0$aBiometry. 615 14$aStatistical Theory and Methods. 615 24$aStochastic Modelling in Statistics. 615 24$aParametric Inference. 615 24$aMachine Learning. 615 24$aLinear Models and Regression. 615 24$aBiostatistics. 615 7$aEstadística 615 7$aEstadística matemàtica 615 7$aAprenentatge automàtic. 615 7$aAnàlisi de regressió 615 7$aBiometria 615 7$aModels lineals (Estadística) 676 $a519.5 700 $aEguchi$b Shinto$0781822 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910986137903321 996 $aMinimum Gamma-Divergence for Regression and Classification Problems$94327847 997 $aUNINA