{
  "id": "1105.1118",
  "version": "v3",
  "published": "2011-05-05T16:33:13.000Z",
  "updated": "2013-09-11T16:55:50.000Z",
  "title": "On $\\varphi$-families of probability distributions",
  "authors": [
    "Rui F. Vigelis",
    "Charles C. Cavalcante"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a $\\varphi$-function, resulting in a $\\varphi$-family of probability distributions. We show how $\\varphi$-families are constructed. In a $\\varphi$-family, the analogue of the cumulant-generating function is a normalizing function. We define the $\\varphi$-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback-Leibler divergence. A formula for the $\\varphi$-divergence where the $\\varphi$-function is the Kaniadakis' $\\kappa$-exponential function is derived.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-09-11T16:55:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "probability distributions",
      "exponential function",
      "normalizing function",
      "kullback-leibler divergence",
      "bregman divergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.1118V"
    }
  }
}