3 Testing the approximation

In order to quantitatively test the proposed ansatz, we have calculated the complete likelihood function for several experiments. Our aim will be to compare the true likelihoods to the approximation. Figures 2-5 summarize our comparisons with the Saskatoon and MAX data sets. In all cases, the complete likelihood functions have been calculated as outlined in Sect. 2 above.

Figure 4: Comparison to the MAX ID likelihood function. This is the combined likelihood for the 3 frequency channels, 3.5, 6 and 9  ${\rm cm}^{-1}$. Linestyles are the same as in the previous two figures, and here $N_{\rm pix}=21$ applies to the dot-dashed (blue) line

The first comparison will be made to the Saskatoon Q-band 1995 4-point and 10-point differences (experimental information can be found in Netterfield et al. 1997; all relevant information concerning the experiment can be found on the group's web page; for useful and detailed information on a number of experiments, see Caltech's web page). This particular choice of window functions was arbitrary. The approximation, applied using the constraints (19) and (20), is shown in Figs. 2 and 3 as the dashed (red) curve. We see that it provides a good representation of the complete likelihood functions, traced by the solid (black) curves in each figure; in fact, the fit is truly spectacular for the 10-point difference. Taking as a benchmark the rule-of-thumb that 1, 2 and 3 $\sigma$ confidence intervals may be estimated by $2\Delta\ln{\cal L} = 1$, 4 and 9, respectively, we see that the approximation reproduces almost perfectly all of these, and more.

Consider now setting $\nu =N_{\rm pix}=24$ and 48, for the 4-point and 10-point differences, respectively, and then adjusting $\beta$ to the 68% confidence interval. In so doing, we obtain the dot-dashed (blue) curves, which in fact are not too bad in both cases. These values of $N_{\rm pix}$ should be compared to the values of $\nu=16$ and 41 found previously by adjusting to two confidence intervals. Thus, we see that the effective number of degrees-of-freedom describing these Saskatoon likelihood functions is indeed $\nu\le N_{\rm pix}$, as we expected from the above discussion.

Finally, the 3-dot-dashed (green) curves show "2-winged'' Gaussians with separate positive- and negative- going variances, sometimes employed in traditional $\chi^2$-analyses. This is also a fare representation of the two likelihood functions, although the proposed ansatz does perform slightly better. We will return to this point, but we should not be too surprised that the Gaussian works reasonably well when, as here, $\nu$ becomes large (all the same, notice that the curves are not symmetric and that a single Gaussian, with a single $\sigma$, would not fare particularly well).

Comparison to the MAX experiment is shown in Fig. 4 for the region ID (experimental details can be found in Tanaka et al. 1996); we have combined all three frequency channels to construct the complete likelihood function. The scan strategy consisted in taking $N_{\rm pix}=21$ single differences aligned along a unidimensional scan. Once again, the approximation, applied using Eqs. (19, 20), supplies an excellent representation of the likelihood function, down to values well below "$3\sigma$'' (0.01 of the peak). The effective number of degrees-of-freedom is $\nu=8.5$, demonstrating again that $\nu\le N_{\rm pix}$. Here, the difference is rather large, due to the significant overlap between adjacent pixels along the scan, and we see that the ansatz with $\nu=N_{\rm pix}$ does not produce a good approximation.

Figure 5: Comparison to a Saskatoon bin consisting of the 10-, 11- and 12-point differences (bands) from the K-band 1994, Q-band 1994 and Q-band 1995 CAP data. As previously, the solid black line shows the true likelihood function, while the dashed (red) curve displays the approximation based on two confidence intervals. This figure demonstrates that the approximation works well even when several individual bands are combined to form a band-power likelihood

Could there a way to proceed if only one confidence interval is given? This would require a choice for one of the parameters, say $\nu$, based on some knowledge of the scan strategy. We have just seen that for MAX $\nu=N_{\rm pix}$leads to a bad representation of the likelihood function. One might be tempted to try instead $\nu=$(scan length)/(beam FWHM) =8.8, which is in fact very close to the best value of $\nu$ found from adjusting to two confidence intervals. Although this is successful in this case, it is nevertheless guess-work, the problem being that it is really not clear if there is a unique rule for judiciously choosing $\nu$. For Saskatoon, $\nu=N_{\rm pix}$ worked reasonably well, while here it does not, something much less being required because of the significant redundancy in the scan. We have found that it is difficult to justify a priori a general rule for choosing $\nu$ when lacking two confidence intervals. The most sure way of finding the effective number of degrees-of-freedom to be used in the ansatz remains the use of two confidence intervals, via Eqs. (19, 20).

A noteworthy aspect of this MAX likelihood function is its asymmetry, i.e., it is manifestly non-Gaussian. Even a "2-winged'' Gaussian is clearly a very bad representation. As the number of statistically independent elements entering the power estimation increases, we should expect the likelihood function to approach a Gaussian distribution. The question is, what is meant by statistically independent elements? It is obviously not something like 2l+1, for MAX covers multipoles near 100; rather, as we have argued above, it is really the parameter $\nu$which measures this, what we have been calling the effective number of degrees-of-freedom. The fact that $\nu\le N_{\rm pix}$ tells us that the number of generalized pixels is an upper limit to this number degrees-of-freedom determining the non-Gaussian nature of the likelihood function. We make the connection to the familiar 2l+1-rule only when we have full-sky coverage and bands consisting of single multipoles; then, the number of generalized pixels defining each (single multipole) band corresponds to 2l+1. In the general case, it is more useful and correct to reason with the number of pixels (really, $\nu$). We may also conclude from this that although experiments with relatively large sky coverage should provide Gaussian likelihood functions on scales much smaller than the survey area, band-power estimates on scales approaching the survey area will always be non-Gaussian. The proposed ansatz represents a substantial improvement over either a single or "2-winged'' Gaussian in such cases.

These comparisons focus on simple cases where the power over a single band defined by the observing strategy is to be estimated, although in the MAX case the analysis did include three frequency channels simultaneously. A more subtle test of the approximation is its extension to a power estimate over several bands defined by different window functions. Such is the situation presented by the five standard Saskatoon power bins. Each bin comprises several bands, of the type considered above, and the bin power is estimated using the joint likelihood of the contributing bands, including all band-band correlations. One could worry that the information carried by several bands might not be adequately incorporated by the two parameters of the ansatz.

In Fig. 5 we compare the approximation to the likelihood function of a combination of 10-, 11- and 12-point differences. Included are the K-band 1994 and Q-band 1994 and 1995 CAP data. The true likelihood function for this bin is calculated from the complete covariance matrix accounting for all correlations, and the approximation was fit using two confidence intervals. Even in this more complicated situation we see that the ansatz continues to work quite well, once the appropriate best power estimate and errors for the complete bin are used to find $\nu$ and $\beta$.

It is on the basis of such comparisons that we believe the proposed ansatz and method of application produces acceptable likelihood functions. Besides the comparisons shown here, we have also tested the approximation against 11 other complete likelihood functions, all kindly provided by K. Ganga; these comparisons may be viewed on our web page. The approximation works well in all cases. We emphasize again that the particular value of the proposed ansatz resides in its simplicity - we obtain very good approximations with little effort.