{
  "id": "1407.0836",
  "version": "v2",
  "published": "2014-07-03T09:41:39.000Z",
  "updated": "2014-10-20T12:47:00.000Z",
  "title": "A Lower Bound on the Relative Entropy with Respect to a Symmetric Probability",
  "authors": [
    "Raphaël Cerf",
    "Matthias Gorny"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let $\\rho$ and $\\mu$ be two probability measures on $\\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(\\mu|\\rho)$ the relative entropy of $\\mu$ with respect to $\\rho$. We prove that, if $\\rho$ is symmetric and $\\mu$ has a finite first moment, then \\[ H(\\mu|\\rho)\\geq \\frac{\\displaystyle{(\\int_{\\mathbb{R}}z\\,d\\mu(z))^2}}{\\displaystyle{2\\int_{\\mathbb{R}}z^2\\,d\\mu(z)}}\\,,\\] with equality if and only if $\\mu=\\rho$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-03T09:41:39.000Z",
      "abstract": "Let $\\rho$ and $\\mu$ be two probability measures on $\\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(\\mu|\\rho)$ the relative entropy of $\\mu$ with respect to $\\rho$. We prove that, if $\\rho$ is symmetric and $\\mu$ has a finite first moment, then \\[ H(\\mu|\\rho)\\geq \\frac{\\displaystyle{\\left(\\int_{\\mathbb{R}}z\\,d\\mu(z)\\right)^2}}{\\displaystyle{2\\int_{\\mathbb{R}}z^2\\,d\\mu(z)}}\\,,\\] with equality if and only if $\\mu=\\rho$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-20T12:47:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "60F10",
      "94A17"
    ],
    "keywords": [
      "relative entropy",
      "symmetric probability",
      "lower bound",
      "finite first moment",
      "probability measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.0836C"
    }
  }
}