{"id":14799,"date":"2021-02-22T14:03:38","date_gmt":"2021-02-22T13:03:38","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=14799"},"modified":"2022-04-24T19:58:21","modified_gmt":"2022-04-24T17:58:21","slug":"fisher-information-between-two-gaussians","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2021\/02\/22\/fisher-information-between-two-gaussians\/","title":{"rendered":"Fisher information between two Gaussians"},"content":{"rendered":"
\"Photo<\/a>
Ronald A. Fisher (1890 - 1962) in 1951.<\/figcaption><\/figure>\n

Fisher information.<\/b> The Fisher information or divergence of a positive Borel measure measure \\( {\\nu} \\) with respect to another one \\( {\\mu} \\) on the same space is<\/p>\n

\\[ \\mathrm{Fisher}(\\nu\\mid\\mu) =\\int\\left|\\nabla\\log\\textstyle\\frac{\\mathrm{d}\\nu}{\\mathrm{d}\\mu}\\right|^2\\mathrm{d}\\nu =\\int\\frac{|\\nabla\\frac{\\mathrm{d}\\nu}{\\mathrm{d}\\mu}|^2}{\\frac{\\mathrm{d}\\nu}{\\mathrm{d}\\mu}}\\mathrm{d}\\mu =4\\int\\left|\\nabla\\sqrt{\\textstyle\\frac{\\mathrm{d}\\nu}{\\mathrm{d}\\mu}}\\right|^2\\mathrm{d}\\mu %=4\\int\\left|\\nabla\\sqrt{\\varphi}\\right|^2\\mathrm{d}\\mu %=\\int_{\\{\\varphi>0\\}}\\frac{\\left|\\nabla \\varphi\\right|^2}{\\varphi}\\mathrm{d}\\mu \\in[0,+\\infty] \\]<\/p>\n

if \\( {\\nu} \\) is absolutey continuous with respect to \\( {\\mu} \\), and \\( {\\mathrm{Fisher}(\\nu\\mid\\mu)=+\\infty} \\) otherwise.<\/p>\n

It plays a role in the analysis and geometry of statistics, information, partial differential equations, and Markov diffusion stochastic processes. It is named after Ronald Aylmer Fisher (1890 \u2013 1962)<\/a>, a British scientist who is also the Fisher of many other objects and concepts including for instance:<\/p>\n