ABSTRACT
Bayes’ equation, Equation 11.4a, can be applied to any statement A whether it represents an event, as in the medical test, a property of a system, or an estimate of a parameter, for example, does θ fall in the range θ1 ≤ θ ≤ θ2. In general, we assume that A has a “true value” but that data, D, will have measurement errors, that is
D = M(A|θ , E) +
but p(D|A, E) = p(M(A|θ , E) + ) = p()
since D is presumed to be deterministic. Consequently, in order to evaluate the likelihood, p(D|A, E), we will need to spec-
ify a statistical model for . Furthermore, it is rare that the prior, π(A|E), can be a precise number, as we used in the examples in Section 11.2. Instead our information will either come from previous tests or subjective beliefs in which the prior will be defined in terms of statistics, as in the coin tossing experiment, Section 11.3. If the prior is based upon previous samples, as in the example of estimating a mean, Section 11.2.5, the prior estimate of the expected value of A and its standard deviation will often be obtained through the inverse probability method, see Sections 12.4.1.3 and 12.4.2.
