1. Lollipop :

	Mode count	Xqml	Pure	Xpol
LiB
BICEP			Attach:Lol_distrib_BICEPivw_pure.png Δ

Offset of Xpol wrongly evaluated : Must include E-leakage. We thus simulate CMB maps with noise and no B-modes. Then evaluate the variance of the B-modes :

	Xpol corr
LiB
BICEP

Better :)

2. Surveys results :

Error in title : it is not power spectrum, but pulls of 'r' parameter from likelihoods (1000 simulation).

LiteBird	BICEP

2.1 Pulls and stats :

Pull a = pull asymetric likelihood curve (old version used in the Template Cleaning Likelihood https://cmb.lal.in2p3.fr/lalwiki/Cosmo/LikelihoodComponentSeparation#toc8)
Pull b = usual pull, based on the variance of 'r' obtained using the min chi2 of mc simulations : pull = (r - r_{true})/std(r)
Profile mc = mean of the variance obtained from profile on 1000 mc. I do a mean of the 1000 variances obtained from the 1000 profile, should be corrected !
Profile th = variance computed from the \Delta \chi^2 = 1 using the spectrum model C_\ell^{th} and the analitical spectrum covariance matrix (~ inverse fisher).
std = standard deviation computed from the 'r' 's obtained using the min chi2 of mc simulations (= mc std)
I neglect the first and last bin in the likelihood for LiteBird and Bicep. I also neglect the second bin of Bicep. See estimator results in Surveys

Litebird

neglecting first and last bin in the likelihood

0.1 muK	Pull a = -0.049 +/- 1.016	Pull b = -0.048 +/- 1.000	mean r = 0.0010	-0.0004 +0.0004 (profiles mc)	-0.0004 +0.0004 (profile th)	or +/- 0.0004 (std)
1.0 muK	Pull a = -0.047 +/- 0.997	Pull b = -0.047 +/- 1.000	mean r = 0.0010	-0.0004 +0.0004 (profiles mc)	-0.0004 +0.0004 (profile th)	or +/- 0.0004 (std)
5.0 muK	Pull a = -0.019 +/- 0.983	Pull b = -0.019 +/- 1.000	mean r = 0.0010	-0.0007 +0.0007 (profiles mc)	-0.0008 +0.0008 (profile th)	or +/- 0.0007 (std)
10.0 muK	Pull a = 0.031 +/- 1.021	Pull b = 0.031 +/- 1.000	mean r = 0.0010	-0.0016 +0.0016 (profiles mc)	-0.0016 +0.0016 (profile th)	or +/- 0.0016 (std)
20.0 muK	Pull a = 0.061 +/- 1.048	Pull b = 0.058 +/- 1.000	mean r = 0.0013	-0.0047 +0.0047 (profiles mc)	-0.0046 +0.0046 (profile th)	or +/- 0.0049 (std)
50.0 muK	Pull a = 0.073 +/- 1.049	Pull b = 0.070 +/- 1.000	mean r = 0.0028	-0.0247 +0.0247 (profiles mc)	-0.0249 +0.0249 (profile th)	or +/- 0.0259 (std)

pulls 'b' have exactly the expected std (=1). The means are around zeros, as expected as well.
pulls 'a' have a std near but not exactly unity. This is probably due to the too short sample step, and the method used to evalue the variance on the \Delta \chi^2 =1 ? (to be verified).
profile std and 'mc std' agree on the error on 'r'.
a priori no source of bias on r. only for 20muK and 50muK, but could it be statistical since the std is also huge for those two cases ? test with more simulation
all error on 'r' are consistent with each others
update with 100000 simulations, but 100 intervals for r (instead of 1000) ;

20.0 muK

Pull a = 0.004 +/- 1.073

Pull b = 0.004 +/- 1.000

mean r = 0.0010

-0.0044 +0.0044 (profiles mc)

-0.0045 +0.0045 (profile th)

or +/- 0.0047 (std)

conclusion : adding more simulations corrects the bias.

BICEP :

neglecting two first and two last bins in the likelihood

0.1 muK	Pull a = -0.037 +/- 1.037	Pull b = -0.037 +/- 1.000	mean r = 0.0009	-0.0029 +0.0029 (profiles mc)	-0.0029 +0.0029 (profile th)	or +/- 0.0030 (std)
1.0 muK	Pull a = -0.053 +/- 1.038	Pull b = -0.051 +/- 1.000	mean r = 0.0008	-0.0032 +0.0032 (profiles mc)	-0.0032 +0.0032 (profile th)	or +/- 0.0033 (std)
5.0 muK	Pull a = -0.072 +/- 0.999	Pull b = -0.073 +/- 1.000	mean r = 0.0005	-0.0071 +0.0071 (profiles mc)	-0.0071 +0.0071 (profile th)	or +/- 0.0071 (std)
10.0 muK	Pull a = -0.073 +/- 1.004	Pull b = -0.074 +/- 1.000	mean r = -0.0004	-0.0195 +0.0195 (profiles mc)	-0.0196 +0.0196 (profile th)	or +/- 0.0196 (std)
20.0 muK	Pull a = -0.070 +/- 1.001	Pull b = -0.069 +/- 1.000	mean r = -0.0038	-0.0693 +0.0692 (profiles mc)	-0.0694 +0.0694 (profile th)	or +/- 0.0693 (std)
50.0 muK	Pull a = -0.063 +/- 1.011	Pull b = -0.063 +/- 1.000	mean r = -0.0244	-0.4014 +0.4012 (profiles mc)	-0.3998 +0.3998 (profile th)	or +/- 0.4056 (std)

pulls 'b' have exactly the expected std (=1). The means are around the same level as for LiB, all negative.
error level on 'r' is around one order higher than for LiB.
small bias on r? probably due to the binning of the spectrum model + the higher std of 'r' ?
update with 100000 simulations (instead of 1000), but 100 intervals for r (instead of 1000) ;

1.0 muK	Pull a = 0.001 +/- 1.062	Pull b = 0.001 +/- 1.000	mean r = 0.0010	-0.0032 +0.0032 (profiles mc)	-0.0034 +0.0034 (profile th)	or +/- 0.0034 (std)
5.0 muK	Pull a = 0.006 +/- 1.072	Pull b = 0.006 +/- 1.000	mean r = 0.0010	-0.0073 +0.0073 (profiles mc)	-0.0076 +0.0076 (profile th)	or +/- 0.0078 (std)
20.0 muK	Pull a = 0.009 +/- 1.071	Pull b = 0.008 +/- 1.000	mean r = 0.0016	-0.0711 +0.0711 (profiles mc)	-0.0758 +0.0758 (profile th)	or +/- 0.0762 (std)

conclusion : adding simulations correct the bias. The binning approximation seems ok when neglecting the first and last bins.

3. Preliminary tests

basic likelihood computed as (\hat C_\ell - C_\ell^{th})^T \cdot V^{-1} \cdot (\hat C_\ell - C_\ell^{th})

where \hat C_\ell are the MC spectra estimated with the xqml estimator (500 simulations)
and V^{-1} is the inverse of the covariance matrix of the spectrum, computed analytically : V = F^{-1} G F^{-1} + F^{-1} where F is the fisher matrix.

MC distribution and variance computed from the covariance matrix are consistant.
No bias found
the fsky likelihood gets more variance than the full fsky case, as expected.

Pulls and profiles (for muk = 1 ) :

We now compare the mean and standard deviation of the MC data with the profile likelihood based on the same data. Results are consistant. The profile std is computed on the left and on the right of the likelihood, and give also consistant results :

rought mean Full r=', 0.0997	std r = 0.006576
profile mean Full r=' 0.0997	std r = [ 0.006, 0.0060]
rought mean fsky r=' 0.100	std r = 0.0099
profile mean fsky r=' 0.100	std r = [ 0.009 , 0.0096]

All pulls give a mean near zero and a vairance equal to unity.