Analytic approximation for r likelihood

Colin Bischoff, 2018-11-16 // updated 2018-11-20

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In a previous posting, I ran BICEP/Keck-style CosmoMC likelihood analysis for bandpowers corresponding to the model expectation value for r=0.003 + foregrounds + lensing residual. This allowed me to calculate the detection significance for r in those particular scenarios: experiment config 04b (Chile sky coverage) with residual A_L=0.27 or 0.337 and experiment config 04c (Pole sky coverage) with residual A_L=0.081 or 0.106. Victor's Fisher analysis allows us to calculate σ(r) for other scenarios, but we would like to estimate statistics like detection significance which rely on the non-Gaussian shape of the likelihood.

My ansatz is that the shape of the r likelihood can be well described with the H-L likelihood (Hamimeche & Lewis; PRD 77, 10, 103013; 2008). That likelihood is meant to describe CMB power spectra, but the BB spectrum is linear in r (more or less) so we might expect this choice to work well. For a one-dimensional likelihood with r as the only parameter, the form of the H-L likelihood simplifies (scalar multiplication commutes) and we can write it as

 -log(L) = ( x - log x - 1 ) * ( rf + N )2 / σ2
 x = ( rML + N ) / ( r + N )

where

• r is the likelihood parameter,
• r_ML is the maximum-likelihood r value,
• σ is σ(r) calculated assuming a fiducial model with r = r_f,
• N is a "noise bias" that contains contributions from instrumental noise, residual foregrounds, and residual lensing.

In practice, if we want a representative likelihood curve for a particular value of r, we can use a Fisher code to calculate σ(r) then set r_ML = r_f = r and σ = σ(r). However, we still need to get an estimate of parameter N from the CosmoMC-derived likelihoods.

For the four scenarios shown in my previous posting, I reran the CosmoMC likelihood with somewhat tighter convergence criteria and at a higher temperature to get a better measurement in the tails of the distribution. Then I fit each curve to the model by minimizing a K-S statistic. The results are shown in the following table, along with detection significance calculated both from the CosmoMC likelihood and from the analytic fit.

The table also includes a degrees of freedom statistic calculated as

 k = 2 * ( rML + N )2 / σ2

We might expect that this parameter should come out with a common value for the two Chile scenarios and a common value for the two Pole scenarios, with a ratio that corresponds to the relative sky area. I do find that both Pole scenarios correspond to k ~ 525 but I find very high and inconsistent values of k for the two Chile scenarios. I think the reason is that parameter N mostly affects the skewness of the distribution. The Pole likelihoods have significant skewness and I get a reliable fit. The Chile likelihoods have less skewness, so there is equal preference for any large value of N. Perhaps the best estimate of k for the Chile mask would be to start from the Pole mask value of k, then multiply by some map-derived estimate of the relative sky areas.

UPDATE 2018-11-20: I added new columns to the table and new lines to the figures showing how the likelihood shape and detection significance change if we fix k to 525 for Pole and 10000 for Chile.

Using the degrees of freedom parameter, we can write down an analytic model for the r likelihood by picking r, doing a Fisher calculation for σ(r), and then calculating N from the above equation.

Site	AL		rML	σ	N	k	significance (original)	significance (fit)	k*	significance (fix k=k*)
Chile	0.270		0.00256	0.00102	0.195	28446	2.5 σ	2.5 σ	10000	2.6
Chile	0.337		0.00255	0.00112	0.076	9803	2.3 σ	2.3 σ	10000	2.3
Pole	0.081		0.00268	0.00080	0.010	492	3.8 σ	4.0 σ	525	3.9
Pole	0.106		0.00269	0.00088	0.012	569	3.5 σ	3.5 σ	525	3.5

The figures below show the CosmoMC likelihood (blue) and the analytic model (orange) for the four scenarios that I used. The lower panels of each figure shows the fractional difference between the original likelihood and the model, which does increase out in the tails.

Figure 1: Marginalized r likelihood for experiment config 04b and A_L=0.270

Figure 2: Marginalized r likelihood for experiment config 04b and A_L=0.337

Figure 3: Marginalized r likelihood for experiment config 04c and A_L=0.081

Figure 4: Marginalized r likelihood for experiment config 04c and A_L=0.106