open Formula open Eliom_content.Html5.F let title = "Exponential family / Centered Laplace distribution" let path = ["expfam"; "centered_laplace"] let description = [ p [pcdata "Centered Laplace distribution as an exponential family."]; ] let f = Func.def "F" ~doc:"Centered Laplace log-normalizer" ~args:["\\theta"] ~return:Mathset.Real Syntax.( let theta = real ~doc:"natural parameter" "\\theta" in log (- !2 / theta) ) let grad_f = Func.def "\\nabla F" ~doc:"Centered Laplace gradient log-normalizer" ~args:["\\theta"] ~return:Mathset.Real Syntax.( let theta = real ~doc:"natural parameter" "\\theta" in - !1 / theta ) let g = Func.def "F^\\star" ~doc:"Centered Laplace dual log-normalizer" ~args:["\\eta"] ~return:Mathset.Real Syntax.( let eta = real ~doc:"expectation parameter" "\\eta" in - log eta ) let grad_g = Func.def "\\nabla F^\\star" ~doc:"Centered Laplace gradient dual log-normalizer" ~args:["\\eta"] ~return:Mathset.Real Syntax.( let eta = real ~doc:"expectation parameter" "\\eta" in - !1 / eta ) let t = Func.def "t" ~doc:"Sufficient statistic" ~args:["x"] ~return:Mathset.Real Syntax.( let x = real "x" in abs x ) let content = [f; grad_f; g; grad_g; t] let bibliography = [ Card.article ~url:"http://arxiv.org/abs/0911.4863v1" ~where:"arXiv:0911.4863 [cs.LG]" "Statistical exponential families: A digest with flash cards" ["Frank Nielsen"; "Vincent Garcia"]; ]