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\begin{document}

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\section*{information geometry}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Information geometry aims to apply the techniques of [[differential geometry]] to [[statistics]]. Often it is useful to think of a family of probability distributions as a \emph{statistical} [[manifold]]. For example, normal [[Gaussian distribution]]s form a 2-[[dimension]]al manifold, parameterised by $(\mu, \sigma)$, mean and standard deviation. On such manifolds there are notions of [[Riemannian metric]], [[connection]], [[curvature]], and so on, of statistical relevance.

More precisely,

\begin{quote}%
[[Kullback-Leibler information]], or [[relative entropy]], features as a measure of [[divergence]] (not quite a [[metric]], because it's asymmetric), and [[Fisher information]] takes the role of [[curvature]]. One useful aspect of information geometry is that it gives a means to prove results about statistical models, simply by considering them as well-behaved geometrical objects. For instance, it's basically a tautology to say that a [[manifold]] is not changing much in the vicinity of points of low curvature, and changing greatly near points of high curvature. Stated more precisely, and then translated back into probabilistic language, this becomes the [[Cramer-Rao inequality]], that the variance of a parameter estimator is at least the reciprocal of the [[Fisher information]]. (\hyperlink{Shalizi}{Shalizi})


\end{quote}
Founders of the systematical theory are N. N. Chentsov and Shun-ichi \href{http://www.brain.riken.jp/labs/mns/amari/home-E.html}{Amari}.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

For $X$ a [[measurable space]] let $S$ be (a subspace of) the space of probability [[measure]]s on $X$, equipped with the structure of a [[smooth manifold]].

The \textbf{Fisher metric} on $S$ is the [[Riemannian metric]] given on two [[vector field]]s $v,w \in T S$ by

\begin{displaymath}
g(v,w)_s := E_s( v(log s) w(log s))
  \,,
\end{displaymath}
where $E_s(\cdots)$ denotes the [[expectation value]] under the measure $s \in S$ of the function $x \mapsto  v(log s)_x w(log s)_x$ on $X$.

For instance (\hyperlink{AmariTextbook}{Amari, Section 2.1}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

See also [[Fisher metric]], where Fisher metric in other contexts and quantum generalizations are treated. See also [[quantum information]].

Textbooks providing the big picture

\begin{itemize}%
\item Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer, \emph{Information geometry}, Ergeb. der Mathematik and ihrer Grenzgebiete 3. Folge, 64, Springer 2017
\item Shun-ichi Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen, and C. R. Rao, \emph{Differential geometry in statistical inference} (\href{http://projecteuclid.org/euclid.lnms/1215467056}{project Euclid})

\end{itemize}
\begin{itemize}%
\item Shun-ichi Amari, \emph{Information geometry and its applications}, Applied Mathematical Sciences 194, Springer 2016
\item Shun-ichi Amari, Hiroshi Nagaoka, \emph{Methods of information geometry}, Transactions of mathematical monographs; v. 191, American Mathematical Society, 2000.

\end{itemize}
For a series of articles, see

\begin{itemize}%
\item [[John Baez]], \href{http://math.ucr.edu/home/baez/information/}{Information Geometry}

\end{itemize}
Lecture notes include

\begin{itemize}%
\item Shun-ichi Amari, MaxEnt 2007 \href{http://video.google.com/videoplay?docid=2175767568645553282#}{lecture}
\item Shun-ichi Amari, ETVC08 \href{http://videolectures.net/etvc08_amari_igaia/}{lecture}.
\item Sanjoy Dasgupta, \href{http://videolectures.net/mlss05us_dasgupta_ig/}{lecture}.

\end{itemize}
See also

\begin{itemize}%
\item H\^o{}ng V\^a{}n L\^e{}, Statistical manifolds are statistical models, Journal of Geometry 84(1-2), March 2006, pp. 83-93.


\item Blog \href{http://golem.ph.utexas.edu/category/2006/11/infinitedimensional_exponentia.html}{post}.



\end{itemize}
A brief introduction with more references is

\begin{itemize}%
\item Cosma Shalizi, \emph{Information geometry} (\href{http://www.cscs.umich.edu/~crshalizi/notabene/info-geo.html}{webpage})

\end{itemize}
Several people have noted an equivalence between statistic inference as parametric model selection and statistical mechanics on a statistical manifold, e.g.,

\begin{itemize}%
\item [[Vijay Balasubramanian]], \emph{Statistical Inference, Occam’s Razor and Statistical Mechanics on the Space of Probability Distributions}, Neural Comp.9(2)(1997) 349–368, (\href{https://arxiv.org/abs/cond-mat/9601030}{arXiv:cond-mat/9601030}).

\end{itemize}
The interpretation of a [[quantum field theory]] as a probability distribution on the space of field configurations so as to allow the conversion of techniques from information geometry to analogous measures of proximity between QFTs is in

\begin{itemize}%
\item [[Vijay Balasubramanian]], [[Jonathan Heckman]], Alexander Maloney, \emph{Relative Entropy and Proximity of Quantum Field Theories}, (\href{https://arxiv.org/abs/1410.6809}{arXiv:1410.6809})

\end{itemize}
A treatment of collective statistical inference as resulting in the partition function of a non-linear [[sigma model]] is in

\begin{itemize}%
\item [[Jonathan Heckman]], \emph{Statistical Inference and String Theory}, (\href{https://arxiv.org/abs/1305.3621}{arXiv:1305.3621})

\end{itemize}


\end{document}