Hessian Structures and Non-invariant (F, G)-Geometry on a Deformed Exponential Family
A deformed exponential family has two kinds of dual Hessian structures, the U-geometry and the \(\chi \)-geometry. In this paper, we discuss the relation between the non-invariant (F, G)-geometry and the two Hessian structures on a deformed exponential family. A generalized likelihood function called F-likelihood function is defined and proved that the Maximum F-likelihood estimator is a Maximum a posteriori estimator.
KeywordsGeneralize Product Maximum Likelihood Estimator Likelihood Estimator Suitable Choice Asymptotic Theory
- 3.Naudts, J.: Estimators, escort probabilities, and phi-exponential families in statistical physics. J. Ineq. Pure Appl. Math., 5(4) (2004). Article no 102Google Scholar
- 8.Matsuzoe, H. and Ohara, A. : Geometry for \(q-\)exponential families. In: Proceedings of the 2nd International Colloquium on Differential Geometry and its Related Fields, Veliko Tarnovo, 6–10 September (2010)Google Scholar
- 10.Harsha, K.V., Subrahamanian Moosath, K.S.: Geometry of \(F\)-likelihood Estimators and \(F\)-Max-Ent Theorem. In: AIP Conference Proceedings Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2014), Amboise, France, 21–26 September 2014, vol. 1641, pp. 263–270 (2015)Google Scholar