By Johnson T.D., Taylor J.M.G.
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Additional resources for [Article] A Bayesian mixture model relating dose to critical organs and functional complication in 3D Conformal Radiation Therapy
23 Take an experiment consisting of tossing a coin infinitely many times. The “natural” sample space S is the space of all infinite sequences 2 = ((1 , E 2 , . ), where Et = 0 or 1 (or any other two distinct symbols representing heads and tails). The “simple” events here are of the form “heads on the nth toss,” that is, sets of all infinite sequences 2 = ([I , (2, . ) with the nth coordinate En satisfying Jn = 0. The events in the field generated by the simple events are of the form “heads on tosses k l , .
Still the disadvantages of both approaches were due to the fact that to determine utilities, one needed to assume knowledge of probabilities by the subject, while conversely, to determine subjective probabilities, one needed to assume knowledge of utilities. The discovery that one can determine both utilities and subjective probabilities of the same person is due to Savage (1954). We present here the basic idea of the experiment rather than formal axioms (to avoid obscuring the issue by technicalities).
It is easy to determine the field generated by all Rectangles: These are events that result from finite operations on Rectangles. 7), whereas the intersection of Rectangles is again a Rectangle (or is empty). 7 Complement of a Rectangle Since unions are reduced to intersections of complements by De Morgan’s laws, every element of the smallest field containing all Rectangles is the union of a finite number of disjoint Rectangles. On the other hand, there exist events that do not belong to this field of events.