**3.2. Application to WMAP+SDSS data**

I will use the results from comparison of models to WMAP plus SDSS data
given in
Tegmark et al. (2003,
henceforth T03). Much of the analysis in that paper focusses on a simple
parameter set called the `vanilla' model or sometimes the `six
parameter' model. Confusingly, it actually features seven parameters
(they do not count the bias parameter, although it is an independent fit
parameter). They are not quite the set given in
Table 1; the radiation
density parameter is omitted for reasons I explained above, while the
spectral index *n* is included as an independent parameter. However
*n* - 1 is not actually detected to be non-zero; its 1-sigma
confidence range (table 4, column 6 of T03) is
0.952 < *n* < 1.016. In light of the above discussion, we
might expect that the information criteria reject the inclusion of
*n* - 1 as a useful parameter, and indeed that is the case.

The ^{2}
values quoted by T03 are derived using the WMAP likelihood code, and are
defined as -2 ln (see
Verde et al. (2003)
and Spergel et al. (2003)
for details). The total number of
datapoints *N* (*not* corrected for the number of parameters
in the fit) is
*N* = 1367 (899 WMAP temperature spectrum, 449 WMAP polarization
cross-correlation, 19 SDSS).

As seen in the upper two rows of Table 3, both
information criteria prefer the base model, with *n* fixed at one,
as opposed to letting *n* vary. As
has been remarked before, there is presently no evidence that the parameter
*n* - 1 is needed to fit present data. T03 draw the same conclusion
on subjective grounds, and refer to the base model as `vanilla lite'.

A similar argument applies to other cosmological
parameters. Unfortunately the other models analyzed by T03 include
variation of *n* (their table 3) and so
other parameters are not directly compared with the base model, but
anyway the trend seen in Table 3 is clear - the
more parameters included the
higher the AIC and BIC as compared to the base model. The need for these
additional parameters is strongly rejected by the information criteria,
particularly the BIC which strongly penalizes additional parameters for a
dataset of this size. ^{3} For
example, the information criteria reject the need for
_{k} as an
independent parameter, instead identifying spatially-flat
models as the preferred description of the data.

Model | parameters | -2 ln | AIC | BIC |

Base model | 6 | 1447.9 | 1459.9 | 1491.2 |

Base + n |
7 | 1447.2 | 1461.2 | 1497.7 |

Base + n,
_{k} |
8 | 1445.4 | 1461.4 | 1503.2 |

Base + n,r |
8 | 1446.9 | 1462.9 | 1504.7 |

Base + n,r,
d n / d ln k,
_{k} |
10 | 1444.4 | 1464.4 | 1516.6 |

^{3} It is interesting to note that recent
applications of the Bayesian evidence to cosmological model selection
have also found no significant evidence against the simplest model
considered
(Slosar et al. 2003;
Saini et al. 2003;
Niarchou et al. 2003).
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