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 AL=0.27 or 0.337 and experiment config 04c (Pole sky coverage) with residual AL=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 )
- r is the likelihood parameter,
- rML is the maximum-likelihood r value,
- σ is σ(r) calculated assuming a fiducial model with r = rf,
- 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 rML = rf = 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.