Probable and Improbable Data
In Sect. 3.1 , the Bayesian interval is defined. It contains the probable values of a parameter \(\xi \) and serves as the “error interval” of \(\xi \) . It is the basis of decisions because it allows distinguishing between probable and improbable data. I
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Probable and Improbable Data
In Sect. 3.1, the Bayesian interval is defined. It contains the probable values of a parameter ξ and serves as the “error interval” of ξ. It is the basis of decisions because it allows distinguishing between probable and improbable data. It requires a measure μ(ξ) to be defined in the space of ξ. Examples are discussed in Sect. 3.2. The construction of the Bayesian interval is described in Sect. 3.3. In Sect. 3.4 we formulate a condition for the existence of the Bayesian interval. Its existence is necessary in order to infer ξ. The solutions of the problems suggested to the reader are given in Sect. A.3.
3.1 The Bayesian Interval The conclusions that we draw from perceptions are based on the assignment of probabilities. A text that one reads may be spoiled; one still grasps its message with a reasonably high probability. If the text is ruined, the message becomes ambiguous. Waiting for a friend who is late, one may initially assume that he or she has the usual problems with traffic. When too much time elapses, one assumes that the friend will not come at all. A radioactive substance that emits on average one particle per second should not allow a break of one hour during which no radiation is registered. If that happens, one seeks a flaw in the apparatus. Although the time between two events is random, a break of one hour seems extremely improbable in this case. What is an improbable event? The examples show that its definition is the basis of decisions. If there is a theory predicting ξ pre , and ξ pre turns out to be improbable, one rejects the theory. Let Q(ξ) be a proper distribution of the real parameter ξ. The value ξ pre is improbable if it is outside an interval B(K ) containing the probable events. One is free to choose the probability K with which the probable events fall into B. It has the property B
dξ Q(ξ) = K .
© Springer International Publishing Switzerland 2016 H.L. Harney, Bayesian Inference, DOI 10.1007/978-3-319-41644-1_3
(3.1) 27
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3 Probable and Improbable Data
There is a manifold of areas that fulfil this equation. We require that the smallest one exists and is unique. This is B(K ). We call it the Bayesian interval or more generally the Bayesian area [1]. The construction of the smallest area requires the definition of the length or more generally the volume V =
I
dξ μ(ξ)
(3.2)
of an area I. A measure μ is needed in order to define the volume such that it is independent of reparameterisations. The interested reader should show that V is not changed by the transformation (2.7). Only in combination with a measure is the “smallest interval” a meaningful notion. Decisions and error intervals require a measure. The measure is identified with the prior distribution μ(ξ) defined systematically in Chaps. 6, 9, and 11. Why do we consider the smallest interval B(K ) to be the error interval? There are innumerable areas in which ξ is found with probability K . For the Gaussian distribution (2.16), (2.17) they may extend to infinity. An error interval that e
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