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"];
]