{"product_id":"lower-previsions-9780470723777","title":"Lower Previsions","description":"\u003cp\u003e • Author(s): Matthias C. M. Troffaes\u003cbr\u003e • Publisher: Wiley\u003cbr\u003e • Publisher Imprint: Wiley\u003cbr\u003e • BISAC: Probability \u0026amp; Statistics - General\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eThis book has two main purposes. On the one hand, it provides a\u003cbr\u003econcise and systematic development of the theory of lower previsions,\u003cbr\u003ebased on the concept of acceptability, in spirit of the work of\u003cbr\u003eWilliams and Walley. On the other hand, it also extends this theory to\u003cbr\u003edeal with unbounded quantities, which abound in practical\u003cbr\u003eapplications.\u003c\/p\u003e \u003cp\u003eFollowing Williams, we start out with sets of acceptable gambles. From\u003cbr\u003ethose, we derive rationality criteria---avoiding sure loss and\u003cbr\u003ecoherence---and inference methods---natural extension---for\u003cbr\u003e(unconditional) lower previsions. We then proceed to study various\u003cbr\u003easpects of the resulting theory, including the concept of expectation\u003cbr\u003e(linear previsions), limits, vacuous models, classical propositional\u003cbr\u003elogic, lower oscillations, and monotone convergence. We discuss\u003cbr\u003en-monotonicity for lower previsions, and relate lower previsions with\u003cbr\u003eChoquet integration, belief functions, random sets, possibility\u003cbr\u003emeasures, various integrals, symmetry, and representation theorems\u003cbr\u003ebased on the Bishop-De Leeuw theorem.\u003c\/p\u003e \u003cp\u003eNext, we extend the framework of sets of acceptable gambles to consider\u003cbr\u003ealso unbounded quantities. As before, we again derive rationality\u003cbr\u003ecriteria and inference methods for lower previsions, this time also\u003cbr\u003eallowing for conditioning. We apply this theory to construct\u003cbr\u003eextensions of lower previsions from bounded random quantities to a\u003cbr\u003elarger set of random quantities, based on ideas borrowed from the\u003cbr\u003etheory of Dunford integration.\u003c\/p\u003e \u003cp\u003eA first step is to extend a lower prevision to random quantities that\u003cbr\u003eare bounded on the complement of a null set (essentially bounded\u003cbr\u003erandom quantities). This extension is achieved by a natural extension\u003cbr\u003eprocedure that can be motivated by a rationality axiom stating that\u003cbr\u003eadding null random quantities does not affect acceptability.\u003c\/p\u003e \u003cp\u003eIn a further step, we approximate unbounded random quantities by a\u003cbr\u003esequences of bounded ones, and, in essence, we identify those for\u003cbr\u003ewhich the induced lower prevision limit does not depend on the details\u003cbr\u003eof the approximation. We call those random quantities 'previsible'. We\u003cbr\u003estudy previsibility by cut sequences, and arrive at a simple\u003cbr\u003esufficient condition. For the 2-monotone case, we establish a Choquet\u003cbr\u003eintegral representation for the extension. For the general case, we\u003cbr\u003eprove that the extension can always be written as an envelope of\u003cbr\u003eDunford integrals. We end with some examples of the theory.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Hardcover","offer_id":45201823269015,"sku":"9780470723777","price":6672.0,"currency_code":"INR","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9780470723777.webp?v=1769208070","url":"https:\/\/atlanticbooks.com\/products\/lower-previsions-9780470723777","provider":"Atlantic Books","version":"1.0","type":"link"}