ABSTRACT
The full information gained from a measurement of some physical quantity x
is not limited to just a single value. Rather, a measurement yields information
about a set of discrete probabilities Pj , when the possible values of x form a
discrete set. Similarly, when the possible values of x form a continuum with
probability density function p (x), a measurement yields information about a
set of infinitesimal probabilities p (x) dx of the true value lying between x and
x+dx. Note that x need not be a bona-fide random quantity for the probabil-
ity distribution to exist, but could be (and in science and technology it usually
is) an imperfectly known constant. In practice, users of measured data seldom
require knowledge of the complete posterior distribution, but usually request
Evaluation and
a “recommended value” for the respective quantity, accompanied by “error
bars” or some suitably equivalent summary of the posterior distribution. De-
cision theory can provide such a summary, since it describes the penalty for
bad estimates by a loss function. Since the true value is never known in prac-
tice, it is not possible to avoid a loss completely, but it is possible to minimize
the expected loss, which is what an optimal estimate must accomplish. As will
be shown in this Chapter, in the practically most important case of “quadratic
loss” involving a multivariate posterior distribution, the “recommended value”
turns out to be the vector of mean values, while the “error bars” are provided
by the corresponding covariance matrix. Conversely, an experimental result
reported in the form 〈x〉 ± ∆x, where ∆x represents the standard deviation (i.e., the root-mean-square error), is customarily interpreted as a short-hand
notation representing a distribution of possible values x that cannot be re-
covered in detailed form, but is characterized by the mean 〈x〉 and standard deviation ∆x.
