{
  "id": "1907.03753",
  "version": "v1",
  "published": "2019-07-08T10:00:02.000Z",
  "updated": "2019-07-08T10:00:02.000Z",
  "title": "Foundations for conditional probability",
  "authors": [
    "Ladislav Mečíř"
  ],
  "categories": [
    "math.LO",
    "math.PR"
  ],
  "abstract": "We analyze several formalizations of conditional probability and find a new one that encompasses all. Our main result is that a preference relation on random quantities called a plausible preorder induces a coherent conditional expectation; and vice versa, that every coherent function can be extended to a conditional expectation induced by a plausible preorder. The advantages of our approach include a convenient justification of probability laws by the properties of plausible preorders, independence on probability interpretations, or the ability to extend conditional probability to any nonzero condition. In particular, if C is a nonzero condition and \\Prob is coherent, then it can be extended so that \\Prob(0|C)=0, \\Prob(C|C)=1 and \\Prob(1|C)=1, no matter whether \\Prob(C) is zero or whether it is defined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-08T10:00:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "foundations",
      "nonzero condition",
      "extend conditional probability",
      "coherent conditional expectation",
      "preference relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}