This paper attempts to reconstruct the large scale primordial power spectrum from WMAP using a smoothing prior.
The results look a bit odd to me. Are they doing the right simulations? i.e. from Section 3 it sounds as though they are simulating posterior C_l from Eq. 18, whereas in the frequentist analysis I'd have though they should be simulating skies, computing psuedoC_l estimators, then using these for the mock Monte Carlo reconstructions.
This might explain why they seem to get biased results when using the correct theorymodel scaling in the posterior, otherwise this is also rather odd.
Since the reconstruction is likely very sensitive to the quadrupole likelihood function, would be nice if they tried using the more conservative code of Ref. astroph/0404567 (as used in e.g. the SDSS and B03 analyses).
[astroph/0509478] Nonparametric reconstruction of the primordial power spectrum at horizon scales from WMAP data
Authors:  Domenico TocchiniValentini (Oxford), Yehuda Hoffman (HU, Jesuralem), Joseph Silk (Oxford) 
Abstract:  We extend to large scales a method proposed in previous work that reconstructs nonparametrically the primordial power spectrum from cosmic microwave background data at high resolution. The improvement is necessary to account for the nongaussianity of the Wilkinson Microwave Anisotropy Probe (WMAP) likelihood due primarily to cosmic variance. We assume the concordance LambdaCDM cosmology, utilise a smoothing prior and perform Monte Carlo simulations around an initial power spectrum that is scalefree and with spectral index n_s=0.97, very close to the concordance spectrum. The horizon scale for the model we are considering corresponds to the wavenumber k_h=4.52 * 10^4 Mpc^1. We find some evidence for the presence of features and we quantify the probabilities of exceeding the observed deviations in WMAP data with respect to the fiducial models. We detect the following potential departures from a scalefree (spectral index n_s=0.97) initial spectrum: a cutoff at 0.0001<k<0.001 Mpc^1 at 0.43% (0.20%), a dip at $0.001<k<0.003 Mpc^1 at 4.71% (0.40%) and, to a lesser extent, a bump at 0.003<k<0.004 Mpc^1 at 16.3% (39.9%) confidence level. These frequentist confidence levels are calculated by integrating over the distribution of the Monte Carlo reconstructions built around the fiducial models. The frequentist analysis finds the low k cutoff of the estimated power spectrum to be about 5 sigma away from the n_s=0.97 model, while in the Bayesian analysis the model is about 3 sigma away from the estimated spectrum. (The sigma's are different for the two different methods.) 
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