Annu. Rev. Astron. Astrophys. 1992. 30:
653-703
Copyright © 1993 by . All rights reserved |

**2.2 Analysis Procedures**

The question of the best strategy in analyzing microwave background radiation observations is not fundamental but it is important because the data are exceedingly difficult to obtain and, as of this writing, no clear signal due to the microwave background radiation fluctuations has been reported. It is therefore reasonable to push statistical analysis to its limits in extracting information from these data, and parametric methods, which can be pushed much further than nonparemetric methods, are almost always used. Once fluctuations have been measured with good signal-to-noise ratio, questions of statistical procedures will largely disappear and work will focus on the actual distribution of fluctuations.

Considerable disagreement exists about the best techniques to use in
statistical analyses of these types of data. There are two main
points of view, ``frequentist'', and ``Bayesian'', described briefly
below. We refer the reader to
Readhead et al (1989)
and references
therein for more detailed discussions. Despite apparent similarities,
and many common terms and definitions, there is a fundamental
difference between these two points of view. Frequentist methods are
absolute, in the sense that the probability of obtaining a certain set
of data is evaluated by considering all datasets that *might have
been* obtained, given our assumptions about the sky and knowledge of
instrumental errors. Bayesian methods, on the other hand, are
relative, in that the absolute probability of obtaining a certain set
of data is of little interest. What matters is only the
*relative* probability as the assumptions about the sky are varied.
Frequentist methods tend to go awry with unlikely datasets, because
the frequentist credo is good long-run performance, and what happens
to unlikely datasets doesn't make much difference in the evaluation of
long-run performance. Bayesian methods avoid this difficulty, because
they do not consider long-run performance.

2.2.1
*T* ± by *p*(*T*, ), and the variance of the
distribution of microwave background radiation fluctuations on the
sky by *T*^{2}_{mbr}. For Gaussian fluctuations the
probability density is entirely determined by *W(k)* and we have

In observations of *n* fields we have the set
of observations {*T _{i}*}, and the

When the fields are correlated the joint density function must include the
correlation matrix
(Kendall & Stuart 1977
vol 1).
From Bayes's
formula, the probability density of *T*_{mbr} is

where *p*(*T*_{mbr}), called the *prior* probability
density, represents our knowledge of *T*_{mbr} before the
observations. *p*(*T*_{mbr}|{*T _{i}*}) is called the

2.2.2
*T*_{mbr}, we must
find *T*_{mbr}. The test is
simply a criterion by which we decide between an hypothesis *H* (e.g.,
that *T*_{mbr}
15 *µ*K), and an
alternative hypothesis
*K* (e.g., that *T*_{mbr} > 15 *µ*K). The *level of
significance* or *size* of the test is the
probability of rejecting *H* if *H* is true (a so-called Type I
error). The *power* of the test, , is the probability of
rejecting *H* if *K* is true. Thus 1- is the probability of
accepting *H* when *K* is true (a so-called Type II error). Clearly
we would like a test for which
is low, and is
high. It can be shown that a test based on the statistic S,
defined as the ratio of the likelihoods of the observations *T _{i}* under

Such tests are called *powerful* for obvious reasons, but
it is important to remember that maximizing the power does not
guarantee that the power is high. Likelihood ratio tests have low
power, in fact, on data where ^{2} per degree of freedom,
^{2}_{}, is much less than unity.

2.2.3
^{2}_{}
(reduced ^{2}) much less
than unity - which can happen if
measurement errors are overestimated or if the data are simply
atypical
(Readhead et al 1989).
In such cases the frequentist ``likelihood- ratio'' method can give
misleading results if the *power* of the test is low
(Lasenby 1981,
Cottingham 1987,
Lawrence et al 1988,
Readhead et al 1989).
For well-behaved datasets, with
^{2}_{} 1, however, both frequentist and Bayesian methods
give comparable results. In our opinion, *how* a method is used
matters more than *which one* is used.

Given the importance that we attach to observations of the microwave
background radiation, it seems wise to make use of all the available
statistical tools, and to base the interpretation of the observations
upon those aspects upon which different methods agree. Indeed,
disagreement amongst statistical methods is a sure sign of statistical
peculiarities in the data. Nevertheless statistical analyses can
easily obscure, rather than illuminate, the observations themselves.
For this reason we reproduce below, wherever possible, the actual
*T* measurements with
their errors, since these convey, more
clearly than tables of 95% confidence limits, the real progress in
this field over the last decade.