By von der Linden W., Dose V., von Toussaint U.
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Additional info for Bayesian Probability Theory: Applications in the Physical Sciences
With this definition the outcome of the urn example provides a sample of size 40. 19 (Events) The outcome of an experiment is an event. Events happen or they don’t. A partial event is undefined. A possible event of the urn example is green ball in the ith draw. However, the choice of the urn α is also an event. In the urn example we will relate the unknown events to the known events. Besides events, propositions are of crucial importance in Bayesian probability theory. Possible propositions for the urn experiment are: • • • • The selected urn has label α.
9781107035904ar” — 2014/1/6 — 20:35 — page 38 — #52 ✐ 38 ✐ Bayesian inference How large is the probability that a patient is indeed infected if the test has been positive? 01. 08. 0009 Therefore, out of 100 persons with a positive outcome of the test only eight are infected. 08. In both of the preceding examples the reason for the small effect for a positive response is due to the fact that there are many more possibilities for unjustified positive responses P (T |I). P (D|T , I) P (T |I) than for a true detection P (T |I) One might be bothered that such a ‘poor’ test may under-diagnose the disease.
Henceforward, we will not discuss these aspects further. We refer readers particularly interested in this topic to [118, 120]. The size of the population can be finite or infinite. 16 (Simple events) Events which are indecomposable are simple events. All other events are compound events. For example, the outcome ‘red’ in roulette may happen in 18 different ways because 18 numbers (2, 4, . . , 34, 36) are labelled in red. It is therefore a compound event. The result ‘14’ (which is also red) is, in this example, a simple event.