Analytic approximation for r likelihood
Colin Bischoff, 2018-11-16 // updated 2018-11-20
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 ) * ( r_{f} + N )^{2} / σ^{2} x = ( r_{ML} + 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 * ( r_{ML} + N )^{2} / σ^{2}
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 | A_{L} | r_{ML} | 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.